Gliding, Climbing, and Turning Flight Performance! Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2014!

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Gliding, Climbing, and Turning Flight Performance!

Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2014

!

http://www.princeton.edu/~stengel/MAE331.html!

http://www.princeton.edu/~stengel/FlightDynamics.html!

•! Conditions for gliding ﬂight"

•! Parameters for maximizing climb angle and rate"

•! Review the V-n diagram"

•! Energy height and speciﬁc excess power"

•! Alternative expressions for steady turning ﬂight"

•! The Herbst maneuver!

Learning Objectives! Reading:! Flight Dynamics ! Aerodynamic Coefﬁcients, 130-141! 1!

Gliding Flight!

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D = C_D 1 2!V2S = "W sin# CL 1₂ !V2S =W cos# ! h =V sin# ! r =V cos#

Equilibrium Gliding Flight"

3!

Gliding Flight"

•! Thrust = 0"

•! Flight path angle < 0 in gliding ﬂight"

•! Altitude is decreasing"

•! Airspeed ~ constant"

•! Air density ~ constant "

tan! = "D L = "CCDL = h! ! r = dhdr; ! = "tan "1 D L # $ % & ' ( = "cot"1 L D # $ % & ' (

Gliding ﬂight path angle "

Corresponding airspeed "

Vglide = 2W

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Maximum Steady Gliding Range"

•! Glide range is maximum when !!_{is least}

negative, i.e., most positive"

•! This occurs at (L/D)max " 5!

Maximum Steady Gliding Range"

•! Glide range is maximum when !!_{is least}

negative, i.e., most positive"

•! This occurs at (L/D)max "

tan! = h! ! r =negative constant = h "h_o ( ) r"r_o ( ) #r= #h tan! = "#h

"tan! =maximum when LD =maximum

!_max ="tan"1 D L # $ % & ' ( min ="cot"1 L D # $ % & ' ( max

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Sink Rate, m/s "

•! Lift and drag deﬁne !!_{and V in gliding equilibrium}_"

D =C_D 1 2!V2S= "W sin# sin# =" D W L =C_L 1 2!V2S =W cos" V = 2W cos" CL!S ! h=Vsin! =" 2W cos! CL#S D W $ % & ' ( )=" 2W cos! CL#S L W $ % & ' ( ) D L $ % & ' ( ) =" 2W cos! CL#S cos! 1 L D $ % & ' ( )

•! Sink rate = altitude rate, dh/dt (negative)"

7!

•! Minimum sink rate provides maximum endurance"

•! Minimize sink rate by setting "(dh/dt)/"CL = 0 (cos !! ~1)"

Conditions for Minimum Steady Sink Rate"

! h=! 2W cos" CL#S cos" CD CL $ % & ' ( ) =! 2W cos3" #S C_D CL3/2 $ % & ' ( ) * ! 2 # W S $ % & ' ( ) CD CL3/2 $ % & ' ( ) CLME = 3CDo ! and CDME = 4CDo

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L/D and V_ME for Minimum Sink Rate" VME = 2W !S C_D2_ME+C_L2_ME" 2 W S( ) ! # 3CDo "0.76V_{L D}_max L _D

( )

ME = 1 4 !C3_D_o = 3 2

( )

L D max "0.86 L D

( )

max 9!

L/D for Minimum Sink Rate"

•! For L/D < L/Dmax, there are two

solutions"

•! Which one produces minimum

sink rate?"

L D

( )

ME ! 0.86

( )

LD max

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Climbing Flight!

11!

•! Rate of climb, dh/dt = Speciﬁc Excess Power "

Climbing Flight" ! V=0= (T !D!W sin") m sin" =(T !D) W ; " =sin!1 T !D ( ) W ! !=0=(L"W cos!) mV L=W cos! ! h=Vsin! =V T( "D) W = Pthrust "Pdrag ( ) W

Specific Excess Power (SEP)= Excess Power

Unit Weight #

Pthrust "Pdrag

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W •! Flight path angle " •! Required lift"

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•! Note signiﬁcance of thrust-to-weight ratio and wing loading" Steady Rate of Climb"

! h=V sin! =V T W " # $ % & '( CDo +)CL 2

(

)

q W S ( ) * + , , -. / / L=C_Lq S=W cos" CL= W_S # $ % & ' ( cos" q V= 2 W S # $ % & ' ( cos" CL) ! h=V T W ! " # $ % &' CDoq W S ( ) ' ((W S)cos2) q * + , -. / =V T( )h W ! " # $ % &'CDo0( )h V 3 2 W S( ) ' 2((W S)cos2) 0( )h V •! Climb rate" 13!

•! Necessary condition for a maximum with

respect to airspeed"

Condition for Maximum Steady Rate of Climb"

! h=V T W ! " # $ % &' CDo(V 3 2 W S( )' 2)₍W S₎cos2* (V !h! !V =0 = T W " # $ % & '+V !T /!V W " # $ % & ' ( ) * +_,-. 3CDo/V 2 2(W S) + 20(W S)cos21 /V2

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Maximum Steady " Rate of Climb: " Propeller-Driven Aircraft" !P_thrust !V =0= T W " # $ % & '+V !T /!V W " # $ % & ' ( ) * +_, -•! At constant power" !h! !V =0=" 3CDo#V 2 2 W S( ) + 2$(W S) #V2

•! With cos2_!!_{~ 1,}_{optimality condition reduces to"}

•! Airspeed for maximum rate of climb at maximum power,Pmax"

V4 ₌ 4 3 ! " # $ % &'(W S) 2 CDo( 2 ; V = 2 W S( ₍ ) ' 3CDo =V_ME 15! Maximum Steady Rate of Climb: " Jet-Driven Aircraft"

•! Condition for a maximum at constant thrust and cos2_!!_{~ 1}_"

•! Airspeed for maximum rate of climb at maximum thrust, T_max"

!h! !V =0 0 = ax2 +bx+c and V = + x =! 3CDo" 2 W S( )V4 + WT # $ % & ' (V2+2)(W S) " =! 3CDo" 2 W S( ) V 2 ( )2+ T W # $ % & ' (( )V2 +2)(W S) "

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Optimal Climbing Flight!

17!

What is the Fastest Way to Climb from One Flight Condition to

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•! Speciﬁc Energy "

•! = (Potential + Kinetic Energy) per Unit Weight"

•! = Energy Height"

Energy Height"

•! Could trade altitude with airspeed with no change in energy

height if thrust and drag were zero"

Total Energy

Unit Weight !Specific Energy= mgh

+mV2 2 mg =h+V 2 2g ! Energy Height,E_h, ft or m 19!

Speciﬁc Excess Power" dEh dt = dtd h+V 2 2g ! " # $ % &= dh_dt +!V_g " # $ % &dV_dt

Rate of change of Speciﬁc Energy "

=V sin! + V g " #$ % &' T (_D(_mgsin! m " #$ %&' =V T( _W(D)=V CT (CD ( )1₂)(h)V2_S W

=Specific Excess Power(SEP)

= Excess Power

Unit Weight !

Pthrust"Pdrag

( )

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Contours of Constant Speciﬁc Excess Power"

•! Speciﬁc Excess Power is a function of altitude and airspeed"

•! SEPis maximized at each altitude, h, when" _{d SEP(h)}_[ _]

dV =0

21!

Subsonic Energy Climb"

•! Objective:Minimize time or fuel to climb to desired altitude and airspeed"

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Supersonic Energy Climb"

•! Objective:Minimize time or fuel to climb to desired altitude and airspeed"

23!

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•! Maneuvering envelope: limits on normal load factor and allowable equivalent airspeed"

–! Structural factors"

–! Maximum and minimum achievable lift coefﬁcients"

–! Maximum and minimum airspeeds"

–! Protection against

overstressing due to gusts"

–! Corner Velocity: Intersection of maximum lift coefﬁcient and maximum load factor"

Typical Maneuvering Envelope:

V-n Diagram"

•! Typical positive load factor limits"

–! Transport: > 2.5"

–! Utility: > 4.4"

–! Aerobatic: > 6.3"

–! Fighter: > 9"

•! Typical negative load factor limits"

–! Transport: < –1"

–! Others: < –1 to –3"

C-130 exceeds maneuvering envelope!

http://www.youtube.com/watch?v=4bDNCac2N1o&feature=related! 25!

Maneuvering Envelopes (V-n Diagrams)

for Three Fighters of the Korean War Era" Republic F-84!

North American F-86!

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Turning Flight!

27!

•! Vertical force equilibrium"

Level Turning Flight"

L cosµ =W

•! Load factor"

n= L_W = L_mg =secµ,"g"s

•! Thrust required to maintain level ﬂight"

Treq = CDo +!CL 2 ( )1₂"V2S= D_o + 2! "V2S W cosµ # $% & '( 2 = D_o+ 2! "V2S(nW) 2 µ : Bank Angle

•! Level ﬂight = constant altitude"

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•! Bank angle"

Maximum Bank Angle in Level Flight" cosµ= W CLqS =1 n=W 2 ! Treq"Do ( )#V2S µ=cos"1 W CLqS $ % & ' ( ) =cos"1 1 n $ % & ' ( ) =cos"1 W 2! Treq"Do ( )#V2S * + , , -. / /

•! Bank angle is limited by "

µ: Bank Angle

C_L_max or T_max or n_max

29!

•! Turning rate"

Turning Rate and Radius in Level Flight" ! ! = CLqSsinµ mV = W tan µ mV = gtan µ V = L 2_"_W2 mV = W n 2_"₁ mV = Treq " Do ( )#V2S 2$ "W2 mV

•! Turning rate is limited by "

CLmax or Tmax or nmax

•! Turning radius "

Rturn =V_!

! = V2

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Maximum Turn Rates"

“Wind-up turns”!

31!

•! Corner velocity"

Corner Velocity Turn"

•! Turning radius " Rturn = V

2 _cos2_!

g n_max2 _" _cos2_!

V_corner = 2nmaxW

C_L_mas!S

•! For steady climbing

or diving ﬂight" sin! = Tmax

" D

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Corner Velocity Turn"

•! Time to complete a

full circle " t2! =

V cos"

g nmax2 # cos2"

•! Altitude gain/loss " !h₂_" = t₂_"V sin#

•! Turning rate " !! = g nmax

2 _" _cos2_#

(

)

V cos#

33!

Herbst Maneuver"

•! Minimum-time reversal of direction"

•! Kinetic-/potential-energy exchange"

•! Yaw maneuver at low airspeed"

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Next Time:!

Aircraft Equations of Motion!

Reading:! Flight Dynamics, ! Section 3.1, 3.2, pp. 155-161! Airplane Stability and Control !

Chapter 5!

35!

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Gliding Flight of the P-51 Mustang" Loaded Weight=9,200 lb (3,465 kg) L / D ( )_max= 1 2 !CDo =16.31 "_MR= #cot#1 L D $ % & ' ( ) max = #cot#1_(16.31)_{= #}_3.51_° CD ( )L/Dmax=2CDo=0.0326 CL ( )L/Dmax= CDo ! =0.531 VL/Dmax= 76.49 * m / s ! hL/Dmax=V sin"= # 4.68 * m / s Rho=10km=(16.31)( )10 =163.1 km

Maximum Range Glide!

Loaded Weight=9,200 lb (3,465 kg) S=21.83 m2 CDME=4CDo =4 0.0163( )=0.0652 CLME = 3CDo ! = 3 0.0163( ) 0.0576 =0.921 L D ( )_ME=14.13 ! hME=" 2_# W_S $ % & ' ( ) CDME CLME 3/2 $ % && ' ( ))="4.11 # m / s *_ME ="4.05° VME =58.12_# m / s

Maximum Endurance Glide!