Gliding, Climbing, and Turning Flight Performance! Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2016 Learning Objectives 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 Reading:! Flight Dynamics! Aerodynamic Coefficients, 130-141! Copyright 2016 by Robert Stengel. All rights reserved. For educational use only. http://www.princeton.edu/~stengel/mae331.html http://www.princeton.edu/~stengel/flightdynamics.html 1 Review Questions!!! How does air density decrease with altitude?!!! What are the different definitions of airspeed?!!! What is a lift-drag polar?!!! Power and thrust: How do they vary with altitude?!!! What factors define the flight envelope?!!! What were some features of the first commercial transport aircraft?!!! What are the important parameters of the Breguet Range Equation?!!! What is a step climb, and why is it important?! 2
Gliding Flight! 3 Equilibrium Gliding Flight D = C D 1 2!V 2 S = "W sin# C L 1 2!V 2 S = W cos#!h = V sin#!r = V cos# 4
Gliding Flight Thrust = 0 Flight path angle < 0 in gliding flight Altitude is decreasing Airspeed ~ constant Air density ~ constant Gliding flight path angle tan! = " D L = " C D h =! C L!r = dh dr ;! = # D "tan"1 % $ L Corresponding airspeed & # L & ( = "cot "1 % ( ' $ D ' V glide = 2W!S C D 2 + C L 2 5 Maximum Steady Gliding Range Glide range is maximum when! is least negative, i.e., most positive This occurs at (L/D) max 6
Maximum Steady Gliding Range Glide range is maximum when! is least negative, i.e., most positive This occurs at (L/D) max # D &! max = "tan "1 % ( $ L ' min # L & = "cot "1 % ( $ D ' h tan! =!!r = negative constant = h " h o r " r o #r = #h tan! = "#h " tan! = maximum when L D = maximum max 7 Sink Rate, m/s Lift and drag define! and V in gliding equilibrium D = C D 1 2!V 2 S = "W sin# sin# = " D W!h = V sin! L = C L 1 2!V 2 S = W cos" V = 2W cos" C L!S Sink rate = altitude rate, dh/dt (negative) = " 2W cos! C L #S $ D ' & ) = " % W ( 2W cos! C L #S $ L & % W ' ( ) $ D ' & % L ) ( = " 2W cos! C L #S cos! $ 1 ' & ) % L D ( 8
Conditions for Minimum Steady Sink Rate Minimum sink rate provides maximum endurance Minimize sink rate by setting!(dh/dt)/!c L = 0 (cos! ~1)!h =! 2W cos" C L #S =! 2W cos3 " $ & #S % cos" $ C ' D & ) % ( C D 3/2 C L C L ' ) *! 2 $ & ( # % W S ' $ )& (% C D 3/2 C L ' ) ( C LME = 3C D o! and C DME = 4C Do 9 L/D and V ME for Minimum Sink Rate ( L D ) = 1 ME 4 3 = 3!C Do 2 ( L D ) " 0.86 L max ( D) max V ME = 2W!S C 2 DME + C " 2 W S 2 LME! # 3C Do " 0.76V L Dmax 10
L/D for Minimum Sink Rate For L/D < L/D max, there are two solutions Which one produces smaller sink rate? ( L D )! 0.86 L ME ( D) max V ME! 0.76V L Dmax 11 Checklist! "!Steady flight path angle?! "!Corresponding airspeed?! "!Sink rate?! "!Maximum-range glide?! "!Maximum-endurance glide?! 12
!is"rical Fac"ids #if$ng-body Reen%y Vehicles M2-F1 HL-10 M2-F2 X-24A M2-F3 X-24B 13 Climbing Flight! 14
!V = 0 = ( T! D! W sin" ) m sin" = ( T! D) W ; " = T! D sin!1 W Climbing Flight Flight path angle Required lift! = 0 = L " W cos! mv L = W cos! Rate of climb, dh/dt = Specific Excess Power!h = V sin! = V ( T " D) W Specific Excess Power (SEP) = ( = P " P ) thrust drag W Excess Power Unit Weight ( # P " P thrust drag ) W 15 Steady Rate of Climb Climb rate *"!h = V sin! = V T % $ '( C 2 ( D o +)C L )q -, / +,# W & ( W S)./ L = C L qs = W cos! " C L = W % # $ S & ' cos! q V = " 2 W % # $ S & ' cos! C L ( Note significance of thrust-to-weight ratio and wing loading *! T $!h = V " # W % & ' C D o q ( W S) ' ( W S )cos 2 ) -, / + q.! = V T ( h) $ " # W % & ' C D o 0 ( h)v 3 ' 2( ( W S )cos 2 ) 2 W S 0 ( h)v 16
Condition for Maximum Steady Rate of Climb!!h = V T $ # &' C D o (V 3 " W % 2 W S ' 2) W S cos 2 * (V Necessary condition for a maximum with respect to airspeed! h!!v = 0 = (" T % "!T /!V * $ '+V $ )# W & # W % + '-. 3C D o /V 2 &, 2 W S + 20 W S cos 2 1 /V 2 17 Maximum Steady Rate of Climb: Propeller-Driven Aircraft At constant power! P thrust!v = 0 = (" T % "!T /!V * $ '+V $ )# W & # W With cos 2! ~ 1, optimality condition reduces to! h!!v = 0 = " 3C D o #V 2 2 W S + 2$ ( W S ) #V 2 Airspeed for maximum rate of climb at maximum power, P max % + '- &, 2! V 4 = 4 $ # & ' W S ; V = 2 W S " 3% C Do ( 2 ( ' 3C Do = V ME 18
Maximum Steady Rate of Climb: Jet-Driven Aircraft Condition for a maximum at constant thrust and cos 2! ~ 1!!h!V = 0! 3C D o " 2 W S V 4 + T $ W! 3C D o " 2( W S) V 2 Airspeed for maximum rate of climb at maximum thrust, T max # % 2 + # T $ % W Quadratic in V 2 & ' ( V 2 + 2) W S & ' ( V 2 " = 0 + 2) W S " = 0 0 = ax 2 + bx + c and V = + x 19 Checklist! "!Specific excess power?! "!Maximum steady rate of climb?! "!Velocity for maximum climb rate?! 20
Optimal Climbing Flight! 21 What is the Fastest Way to Climb from One Flight Condition to Another? 22
Energy Height Specific Energy = (Potential + Kinetic Energy) per Unit Weight = Energy Height Total Energy Unit Weight = mgh + mv 2 2 mg! Specific Energy = h + V 2 2g! Energy Height, E h, ft or m Can trade altitude for airspeed with no change in energy height if thrust and drag are zero 23 Specific Excess Power Rate of change of Specific Energy de h dt " = V sin! + V % # $ g & ' = d! h + V 2 $ # & = dh dt " 2g % dt +! V $ # & dv " g % dt " # $ T ( D ( mgsin! m % & ' = V T ( D W = Specific Excess Power (SEP) = Excess Power Unit Weight! P thrust " P drag W = V ( C T! C D ) 1 2 "(h)v 2 S W 24
Contours of Constant Specific Excess Power Specific Excess Power is a function of altitude and airspeed SEP is maximized at each altitude, h, when max wrt V [ SEP(h) ] d[ SEP(h) ] = 0 dv 25 Subsonic Minimum-Time Energy Climb Objective: Minimize time to climb to desired altitude and airspeed Minimum-Time Strategy: Zoom climb/dive to intercept SEP max (h) contour Climb at SEP max (h) Zoom climb/dive to intercept target SEP max (h) contour Bryson, Desai, Hoffman, 1969 26
Subsonic Minimum-Fuel Energy Climb Objective: Minimize fuel to climb to desired altitude and airspeed Minimum-Fuel Strategy: Zoom climb/dive to intercept [SEP (h)/(dm/dt)] max contour Climb at [SEP (h)/(dm/dt)] max Zoom climb/dive to intercept target[sep (h)/(dm/dt)] max contour Bryson, Desai, Hoffman, 1969 27 Supersonic Minimum-Time Energy Climb Objective: Minimize time to climb to desired altitude and airspeed Minimum-Time Strategy: Intercept subsonic SEP max (h) contour Climb at SEP max (h) to intercept matching zoom climb/dive contour Zoom climb/dive to intercept supersonic SEP max (h) contour Climb at SEP max (h) to intercept target SEP max (h) contour Zoom climb/dive to intercept target SEP max (h) contour Bryson, Desai, Hoffman, 1969 28
Checklist! "!Energy height?! "!Contours?! "!Subsonic minimum-time climb?! "!Supersonic minimum-time climb?! "!Minimum-fuel climb?! de h = de h dm fuel dt dt dm fuel = 1 ' dh!m fuel dt +! V $ " # g % & dv * ), ( dt + 29 SpaceShipOne" Ansari X Prize, December 17, 2003 Brian Binnie, *78 Pilot, Astronaut 30
SpaceShipOne Altitude vs. Range! MAE 331 Assignment #4, 2010 31 SpaceShipOne State Histories 32
SpaceShipOne Dynamic Pressure and Mach Number Histories 33 The Maneuvering Envelope! 34
Typical Maneuvering Envelope: V-n Diagram 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 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 35 Maneuvering Envelopes (V-n Diagrams) for Three Fighters of the Korean War Era Republic F-84 Lockheed F-94 North American F-86 36
Turning Flight! 37 Level Turning Flight Level flight = constant altitude Sideslip angle = 0 Vertical force equilibrium L cos µ = W Load factor µ : Bank Angle n = L W = L mg = sec µ,"g"s Thrust required to maintain level flight 2 T req = ( C Do +!C L ) 1 2 "V 2 S = D o + 2! = D o + 2! ( "V 2 S nw )2 # W & "V 2 S $ % cos µ ' ( 2 38
µ : Bank Angle Maximum Bank Angle in Steady Level Flight = W cos µ = W C L qs = 1 n 2! ( T req " D o )#V 2 S Bank angle * = cos!1, W +, " W % µ = cos!1 # $ C L qs & ' 1 = cos!1 " % # $ n& ' 2( - / ( T req! D o ))V 2 S. / Bank angle is limited by C Lmax or T max or n max 39 Turning Rate and Radius in Level Flight Turning rate!! = C L qssin µ mv = W tan µ mv = g tan µ V = L2 " W 2 mv Turning rate is limited by or T max or n max C Lmax = = W n2 "1 mv ( T req " D o )#V 2 S 2$ " W 2 mv Turning radius V 2 R turn = V! = g n 2 " 1 40
Maximum Turn Rates Wind-up turns 41 Corner Velocity Turn Corner velocity V corner = 2n maxw C Lmas!S For steady climbing or diving flight sin! = T max " D W Turning radius R turn = V 2 cos 2! g n 2 max " cos 2! 42
Time to complete a full circle Corner Velocity Turn Turning rate! g n 2 max " cos 2 # = V cos# t 2! = V cos" g n 2 max # cos 2 " Altitude gain/loss!h 2" = t 2" V sin# 43 Checklist! "! V-n diagram?! "! Maneuvering envelope?! "! Level turning flight?! "! Limiting factors?! "! Wind-up turn?! "! Corner velocity?! 44
Herbst Maneuver Minimum-time reversal of direction Kinetic-/potential-energy exchange Yaw maneuver at low airspeed X-31 performing the maneuver 45 Next Time:! Aircraft Equations of Motion! Reading:! Flight Dynamics,! Section 3.1, 3.2, pp. 155-161! Learning Objectives What use are the equations of motion? How is the angular orientation of the airplane described? What is a cross-product-equivalent matrix? What is angular momentum? How are the inertial properties of the airplane described? How is the rate of change of angular momentum calculated? 46
&upplemental Ma'rial 47 Gliding Flight of the P-51 Mustang Maximum Range Glide Loaded Weight = 9, 200 lb (3,465 kg) ( L / D) max = L " MR = # cot #1 $ ' % & D( ) C D max 1 2!C Do = 16.31 = # cot #1 (16.3) = #3.5 L/Dmax = 2C Do = 0.0326 ( C L ) L/Dmax = C D o! = 0.531 V L/Dmax = 76.49 * m/s!h L/Dmax = V sin" = # 4.68 * m/s R ho =10km = ( 16.31) ( 10) = 163.1 km Maximum Endurance Glide Loaded Weight = 9, 200 lb (3,465 kg) S = 21.83 m 2 C DME = 4C Do = 4 0.0163 C LME = 3C D o! ( L D) ME = 14.13!h ME = " 2 $ # % & * ME = "4.05 = 0.0652 = 3( 0.0163) 0.0576 = 0.921 W S ' ( ) V ME = 58.12 # m/s $ & % C DME 3/2 C LME ' ) ( = " 4.11 # m/s 48