Gliding, Climbing, and Turning Flight Performance! Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2014!
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1 Gliding, Climbing, and Turning Flight Performance! Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2014! 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 2014 by Robert Stengel. All rights reserved. For educational use only.! 1! Gliding Flight! 2!
2 Equilibrium Gliding Flight" D C D 1 2!V 2 S " sin# 1 2!V 2 S cos#!h V sin#!r V cos# 3! 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!!r dh dr ;! # D "tan"1 % $ L Corresponding airspeed " & # L & ( "cot "1 % ( ' $ D ' V glide 2!S C D !
3 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 " # D &! max "tan "1 % ( $ L ' min # L & "cot "1 % ( $ D ' max h tan!!!r negative constant h " h o r " r o #r #h tan! "#h "tan! maximum when L D maximum 6!
4 Sink Rate, m/s " Lift and drag define! and V in gliding equilibrium" D C D 1 2!V 2 S " sin# sin# " D!h V sin! L 1 2!V 2 S cos" V 2 cos"!s Sink rate altitude rate, dh/dt (negative)" " 2 cos! #S $ D ' & ) " % ( 2 cos! #S $ L & % ' ( ) $ D ' & % L ) ( " 2 cos! #S cos! $ 1 ' & ) % L D ( 7! Conditions for Minimum Steady Sink Rate" Minimum sink rate provides maximum endurance" Minimize sink rate by setting "(dh/dt)/" 0 (cos! ~1)"!h! 2 cos" #S! 2 cos3 " $ & #S % cos" $ C ' D & ) % ( C D 3/2 ' ) *! 2 $ & ( # % S ' $ )& (% C D 3/2 ' ) ( ME 3C D o! and C DME 4C Do 8!
5 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 2!S C 2 DME + C " 2 S 2 LME! # 3C Do " 0.76V L Dmax 9! L/D for Minimum Sink Rate" For L/D < L/D max, there are two solutions" hich one produces minimum sink rate?" ( L D )! 0.86 L ME ( D) max V ME! 0.76V L Dmax 10!
6 Climbing Flight! 11! Climbing Flight" Flight path angle " Required lift" ( T! D! sin" )!V 0 m ( sin" T! D ) ; " T! D sin!1!! 0 L cos! ( L " cos! ) mv Rate of climb, dh/dt Specific Excess Power "!h V sin! V ( T " D) ( P " P ) thrust drag Specific Excess Power (SEP) Excess Power Unit eight ( # P thrust " P drag ) 12!
7 Steady Rate of Climb" Climb rate " *"!h V sin! V, $ T +,# % '( C 2 ( D o +) )q - / & ( S)./ L q S cos" # & % ( cos" $ S ' q # V 2 & % ( cos" $ S ' ) Note significance of thrust-to-weight ratio and wing loading" *!!h V T $ # &' C D o q " % ( S) ' ( S )cos 2 ) -, / + q.! V T ( h ) $ # &' C D o 0 ( h)v 3 ' 2( ( S )cos 2 ) " % 2( S) 0 ( h)v 13! Condition for Maximum Steady Rate of Climb"!!h V T $ # &' C D o (V 3 " % 2 S ' 2) S cos 2 * (V Necessary condition for a maximum with respect to airspeed"! h!!v 0 (" T % "!T /!V * $ '+V $ )# & # % + '-. 3C D o /V 2 &, 2 S + 20 S cos 2 1 /V 2 14!
8 Maximum Steady" Rate of Climb: " Propeller-Driven Aircraft" At constant power"! P thrust!v 0 (" T % "!T /!V * $ '+V $ )# & # ith cos 2! ~ 1, optimality condition reduces to"! h!!v 0 " 3C D o #V 2 2 S + 2$ ( S ) #V 2 % + '- &, Airspeed for maximum rate of climb at maximum power, P max " 2! V 4 4 $ # & ' S ; V 2 S " 3% C Do ( 2 ( ' 3C Do V ME 15! 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 S V 4 + T $! 3C D o " 2( S) V 2 Airspeed for maximum rate of climb at maximum thrust, T max " # % 2 + % T $ & (V 2 + 2) S ' " 0 ax 2 + bx + c and V + x # & (( V 2 )+ 2) S ' " 16!
9 Optimal Climbing Flight! 17! hat is the Fastest ay to Climb from One Flight Condition to Another?" 18!
10 Energy Height" Specific Energy " (Potential + Kinetic Energy) per Unit eight" Energy Height" Total Energy Unit eight! Specific Energy mgh + mv 2 2 mg! Energy Height, E h, ft or m h + V 2 2g Could trade altitude with airspeed with no change in energy height if thrust and drag were zero" 19! 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 sin! + V % # $ g & ' " # $ T ( D ( mgsin! m % & ' V T ( D Specific Excess Power (SEP) Excess Power Unit eight! P thrust " P drag ( C V T ( C D ) 1 2 )(h)v 2 S 20!
11 Contours of Constant Specific Excess Power" Specific Excess Power is a function of altitude and airspeed" SEP is maximized at each altitude, h, when" d SEP(h) 0 [ ] dv 21! Subsonic Energy Climb" Objective: Minimize time or fuel to climb to desired altitude and airspeed" 22!
12 Supersonic Energy Climb" Objective: Minimize time or fuel to climb to desired altitude and airspeed" 23! The Maneuvering Envelope! 24!
13 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" C-130 exceeds maneuvering envelope! 25! Maneuvering Envelopes (V-n Diagrams) for Three Fighters of the Korean ar Era" Republic F-84! Lockheed F-94! North American F-86! 26!
14 Turning Flight! 27! Level Turning Flight" Level flight constant altitude" Sideslip angle 0" Vertical force equilibrium" L cos µ Load factor" µ : Bank Angle n L L mg sec µ,"g"s Thrust required to maintain level flight" 2 T req ( C Do +! ) 1 2 "V 2 S D o + 2! "V 2 S D o + 2! ( "V 2 S n )2 # & $ % cos µ ' ( 2 28!
15 Maximum Bank Angle in Level Flight" µ : Bank Angle Bank angle" cosµ qs 1 n 2! ( T req " D o )#V 2 S $ ' $ 1 µ cos "1 & ) cos "1 & ' * % qs ( % n cos"1, +, 2! - / ( T req " D o )#V 2 S./ Bank angle is limited by " max or T max or n max 29! Turning Rate and Radius in Level Flight" Turning rate"!! qssin µ mv tan µ n2 " 1 mv Turning rate is limited by " mv g tan µ L2 " 2 V mv ( T req " D o )#V 2 S 2$ " 2 mv max or T max or n max Turning radius " V 2 R turn V! g n 2 " 1 30!
16 Maximum Turn Rates" ind-up turns! 31! Corner Velocity Turn" Corner velocity" V corner 2n max mas!s For steady climbing or diving flight" sin! T max " D Turning radius " R turn V 2 cos 2! g n 2 max " cos 2! 32!
17 Corner Velocity Turn" Turning rate "! g n 2 max " cos 2 # V cos# Time to complete a full circle " t 2! V cos" g n 2 max # cos 2 " Altitude gain/loss "!h 2" t 2" V sin# 33! Herbst Maneuver" Minimum-time reversal of direction" Kinetic-/potential-energy exchange" Yaw maneuver at low airspeed" X-31 performing the maneuver" 34!
18 Next Time:! Aircraft Equations of Motion! Reading:! Flight Dynamics,! Section 3.1, 3.2, pp ! Airplane Stability and Control! Chapter 5! 35!!upplemental Ma"rial # 36!
19 Gliding Flight of the P-51 Mustang" Loaded eight 9,200 lb (3,465 kg) ( L / D) max $ L ' " MR #cot #1 & ) % D ( C D Maximum Range Glide! 1 2!C Do max L/Dmax 2C Do ( ) L/Dmax C D o 0.531! V L/Dmax * m / s!h L/Dmax V sin" # 4.68 * m / s #cot #1 (16.31) #3.51 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! L D ME ( ) !h ME " 2 $ ' $ & ) C ' D ME 3/2 # % S (& % C ) " 4.11 LME ( # m / s * ME "4.05 V ME # m / s 37!
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