Gliding, Climbing, and Turning Flight Performance Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2018

Transcription

1 Gliding, Climbing, and Turning Flight Performance Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2018 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, Copyright 2018 by Robert Stengel. All rights reserved. For educational use only Review Questions How does air density decrease with altitude? hat are the different definitions of airspeed? hat is a lift-drag polar? Power and thrust: How do they vary with altitude? hat factors define the flight envelope? hat were some features of the first commercial transport aircraft? hat are the important parameters of the Breguet Range Equation? hat is a step climb, and why is it important? 2 1

2 Gliding Flight 3 Equilibrium Gliding Flight C D 1 2 ρv 2 S = sinγ 1 2 ρv 2 S = cosγ!h = V sinγ!r = V cosγ 4 2

3 Gliding Flight Thrust = 0 Flight path angle < 0 in gliding flight Altitude is decreasing Airspeed ~ constant Air density ~ constant Gliding flight path angle tanγ = D L = C D h = r = dh dr ; γ = # D & # L & tan 1 % ( = cot 1 % ( $ L ' $ D ' Corresponding airspeed V glide = 2 ρs C D Maximum Steady Gliding Range Glide range is maximum when γ is least negative, i.e., most positive This occurs at (L/D) max 6 3

4 Maximum Steady Gliding Range Glide range is maximum when γ is least negative, i.e., most positive This occurs at (L/D) max # D & γ max = tan 1 % ( $ L ' min # L & = cot 1 % ( $ D ' h tanγ =!!r = negative constant = h h o r r o Δr = Δh tanγ = Δh tanγ = maximum when L D = maximum max 7 Sink Rate, m/s Lift and drag define γ and V in gliding equilibrium D = C D 1 2 ρv 2 S = sinγ sinγ = D h = V sinγ L = 1 2 ρv 2 S = cosγ V = 2 cosγ ρs Sink rate = altitude rate, dh/dt (negative) = = 2 cosγ ρs $ D ' & ) = % ( 2 cosγ ρs cosγ $ 1 ' & ) % L D ( 2 cosγ ρs $ L ' & % ( ) $ D ' & % L ) ( 8 4

5 Conditions for Minimum Steady Sink Rate Minimum sink rate provides maximum endurance Minimize sink rate by setting (dh/dt)/ = 0 (cos γ ~1) h = 2 cosγ ρs cosγ $ C D & % = 2 cos3 γ ρs $ & % C D 3/2 ' ) ( ' ) 2 $ & ( ρ % S ' $ )& (% C D 3/2 ' ) ( ME = 3C D o ε and C DME = 4C Do 9 L/D and V ME for Minimum Sink Rate ( L D ) = 1 ME 4 3 = 3 εc Do 2 ( L D ) 0.86 L max D max V ME = 2 ρs C 2 DME + C 2 S 2 LME ρ ε 3C Do 0.76V L Dmax 10 5

6 L/D for Minimum Sink Rate For L/D < L/D max, there are two solutions hich one produces smaller sink rate? ( L D ) 0.86 L ME ( D) max V ME 0.76V L Dmax 11 Historical Factoids Lifting-Body Reentry Vehicles M2-F1 HL- 10 M2-F2 X-24A M2-F3 X-24B 12 6

7 Climbing Flight 13!V = 0 = ( T D sinγ ) m sinγ = ( T D) ; γ = T D sin 1 Climbing Flight Flight path angle Required lift γ! = 0 = L cosγ mv L = cosγ Rate of climb, dh/dt = Specific Excess Power!h = V sinγ = V ( T D) Specific Excess Power (SEP) = ( = P P ) thrust drag Excess Power Unit eight ( P thrust P drag ) 14 7

8 Steady Rate of Climb Climb rate *" h = V sinγ = V T % $ ' C 2 ( D o +ε )q -, / +,# & ( S)./ L = qs = cosγ = S cosγ q Note significance of thrust-to-weight ratio and wing loading V = 2 S cosγ ρ T!h = V C D o q ( S) ε ( S)cos2 γ q = V T ( h) C D o ρ ( h)v 3 2ε ( S)cos2 γ 2 S ρ ( h)v 15 Condition for Maximum Steady Rate of Climb! h = V T $ # & C D o ρv 3 " % 2 S 2ε S cos 2 γ ρv Necessary condition for a maximum with respect to airspeed h V = 0 = (" * $ T )# % " '+V $ & # T / V % + '- 3C D o ρv 2 &, 2 S + 2ε S cos 2 γ ρv

9 At constant power Maximum Steady Rate of Climb: Propeller-Driven Aircraft P thrust V = 0 = (" T % " T / V % + * $ '+V $ '- )# & # &, ith cos 2 γ ~ 1, optimality condition reduces to h V = 0 = 3C D o ρv 2 2 S + 2ε ( S ) ρv 2 Airspeed for maximum rate of climb at maximum power, P max 2! V 4 = 4 $ # & ε S ; V = 2 S " 3% C Do ρ 2 ρ ε 3C Do = V ME 17 Maximum Steady Rate of Climb: Jet-Driven Aircraft Condition for a maximum at constant thrust and cos 2 γ ~ 1 h V = 0 3C D o ρ 2 S V 4 + T 3C D o ρ 2( S) V 2 Quadratic in V 2 Airspeed for maximum rate of climb at maximum thrust, T max 2 + T V 2 + 2ε S V 2 ρ = 0 + 2ε S ρ = 0 0 = ax 2 + bx + c and V = + x 18 9

10 Optimal Climbing Flight 19 hat is the Fastest ay to Climb from One Flight Condition to Another? 20 10

11 Energy Height Specific Energy = (Potential + Kinetic Energy) per Unit eight = Energy Height Specific Energy Total Energy Unit eight = mgh + mv 2 2 mg = 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 21 Specific Excess Power Rate of change of Specific Energy de h dt = d! h + V 2 $ # & = dh dt " 2g % dt +! V $ # & dv " g % dt = V sinγ + V g T D mgsinγ m = V T D = Specific Excess Power (SEP) = Excess Power Unit eight P thrust P drag = V 1 2 ρ(h)v 2 S C T C D 22 11

12 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 23 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,

13 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, 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,

14 Checklist q Energy height? q Contours? q Subsonic minimum-time climb? q Supersonic minimum-time climb? q Minimum-fuel climb? de h = de h dt = 1 dh dm fuel dt dm fuel!m fuel dt + V g dv dt 27 SpaceShipOne Ansari X Prize, December 17, 2003 Brian Binnie, *78 Pilot, Astronaut 28 14

15 SpaceShipOne Altitude vs. Range MAE 331 Assignment #4, SpaceShipOne State Histories 30 15

16 SpaceShipOne Dynamic Pressure and Mach Number Histories 31 The Maneuvering Envelope 32 16

17 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 33 Maneuvering Envelopes (V-n Diagrams) for Three Fighters of the Korean ar Era Republic F-84 Lockheed F-94 North American F

18 Turning Flight 35 Level Turning Flight Level flight = constant altitude Sideslip angle = 0 Vertical force equilibrium Load factor L cos µ = µ : Bank Angle n = L = L mg = sec µ,"g"s Thrust required to maintain level flight 2 T req = ( C Do + ε ) 1 2 ρv 2 S = D o + 2ε = D o + 2ε ( ρv 2 S n )2 ρv 2 S cosµ

19 Maximum Bank Angle in Steady Level Flight Bank angle µ : Bank Angle cosµ = qs = 1 n = 2ε ( T req D o )ρv 2 S µ = cos 1 qs 1 = cos 1 n = cos 1 Bank angle is limited by 2ε ( T req D o )ρv 2 S max or T max or n max 37 Turning Rate and Radius in Level Flight Turning rate!ξ = qssin µ mv = tan µ mv = g tan µ V = L2 2 mv = n2 1 mv ( T req D o )ρv 2 S 2ε 2 = mv Turning rate is limited by max or T max or n max Turning radius V 2 R turn = V = ξ g n

20 Maximum Turn Rates ind-up turns 39 Corner Velocity Turn Corner velocity V corner = 2n max mas ρs For steady climbing or diving flight sinγ = T max D Turning radius R turn = V 2 cos 2 γ g n 2 max cos 2 γ 40 20

21 Corner Velocity Turn Turning rate g n 2 max cos 2 γ ξ = V cosγ Time to complete a full circle t 2π = V cosγ g n 2 max cos 2 γ Altitude gain/loss Δh 2π = t 2π V sinγ 41 Checklist q V-n diagram? q Maneuvering envelope? q Level turning flight? q Limiting factors? q ind-up turn? q Corner velocity? 42 21

22 Herbst Maneuver Minimum-time reversal of direction Kinetic-/potential-energy exchange Yaw maneuver at low airspeed X-31 performing the maneuver 43 Next Time: Aircraft Equations of Motion Reading: Flight Dynamics, Section 3.1, 3.2, pp Learning Objectives hat use are the equations of motion? How is the angular orientation of the airplane described? hat is a cross-product-equivalent matrix? hat is angular momentum? How are the inertial properties of the airplane described? How is the rate of change of angular momentum calculated? 44 22

23 Supplemental Material 45 Gliding Flight of the P-51 Mustang Maximum Range Glide Loaded eight = 9,200 lb (3,465 kg) 1 ( L / D) max = = εc Do L γ MR = cot 1 D C D max = cot 1 (16.3) = 3.5 L/Dmax = 2C Do = ( ) L/Dmax = C D o = ε V L/Dmax = ρ m/s!h L/Dmax = V sinγ = 4.68 ρ m/s R ho =10km = ( 16.31) ( 10) = km Maximum Endurance Glide Loaded eight = 9,200 lb (3,465 kg) S = m 2 C DME = 4C Do = ME = 3C D o ε ( L D) ME = 14.13!h ME = 2 ρ S γ ME = 4.05 V ME = ρ m/s = = 3( ) = C DME ME 3/2 = 4.11 ρ m/s 46 23