Upload
beatrice-firan
View
215
Download
2
Embed Size (px)
Citation preview
Gliding, Climbing, and Turning Flight Performance
Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2012!
Copyright 2012 by Robert Stengel. All rights reserved. For educational use only.!http://www.princeton.edu/~stengel/MAE331.html!
http://www.princeton.edu/~stengel/FlightDynamics.html!
• Flight envelope"• Minimum glide angle/rate"• Maximum climb angle/rate"• V-n diagram"• Energy climb"• Corner velocity turn"• Herbst maneuver"
The Flight Envelope�
Flight Envelope Determined by Available Thrust"
• Flight ceiling defined by available climb rate"– Absolute: 0 ft/min"– Service: 100 ft/min"– Performance: 200 ft/min" • Excess thrust provides the
ability to accelerate or climb"
• Flight Envelope: Encompasses all altitudes and airspeeds at which an aircraft can fly "– in steady, level flight "– at fixed weight"
Additional Factors Define the Flight Envelope"
• Maximum Mach number"• Maximum allowable
aerodynamic heating"• Maximum thrust"• Maximum dynamic
pressure"• Performance ceiling"• Wing stall"• Flow-separation buffet"
– Angle of attack"– Local shock waves"
Piper Dakota Stall Buffet"http://www.youtube.com/watch?v=mCCjGAtbZ4g!
Boeing 787 Flight Envelope (HW #5, 2008)"
Best Cruise Region"
Gliding Flight�
D = CD12ρV 2S = −W sinγ
CL12ρV 2S =W cosγ
h = V sinγr = V cosγ
Equilibrium Gliding Flight" Gliding Flight"• Thrust = 0"• Flight path angle < 0 in gliding flight"• Altitude is decreasing"• Airspeed ~ constant"• Air density ~ constant "
tanγ = −D
L= −
CD
CL
=hr=dhdr; γ = − tan−1 D
L#
$%
&
'(= −cot−1
LD#
$%
&
'(
• Gliding flight path angle "
• Corresponding airspeed "
Vglide =2W
ρS CD2 + CL
2
Maximum Steady Gliding Range"
• Glide range is maximum when γ is least negative, i.e., most positive"
• This occurs at (L/D)max "
Maximum Steady Gliding Range"
• Glide range is maximum when γ is least negative, i.e., most positive"
• This occurs at (L/D)max "
tanγ =hr= negative constant =
h−ho( )r − ro( )
Δr = Δhtanγ
=−Δh− tanγ
= maximum when LD= maximum
γmax = − tan−1 D
L#
$%
&
'(min
= −cot−1 LD#
$%
&
'(max
Sink Rate "• Lift and drag define γ and V in gliding equilibrium"
D = CD12ρV 2S = −W sinγ
sinγ = − DW
L = CL12ρV 2S =W cosγ
V =2W cosγCLρS
h =V sinγ
= −2W cosγCLρS
DW$
%&
'
()= −
2W cosγCLρS
LW$
%&
'
()DL
$
%&
'
()
= −2W cosγCLρS
cosγ 1L D$
%&
'
()
• Sink rate = altitude rate, dh/dt (negative)"
• Minimum sink rate provides maximum endurance"• Minimize sink rate by setting ∂(dh/dt)/dCL = 0 (cos γ ~1)"
Conditions for Minimum Steady Sink Rate"
h = − 2W cosγCLρS
cosγ CD
CL
$
%&
'
()
= −2W cos3γ
ρSCD
CL3/2
$
%&
'
() ≈ −
2ρWS
$
%&
'
()CD
CL3/2
$
%&
'
()
CLME=
3CDo
εand CDME
= 4CDo
L/D and VME for Minimum Sink Rate"
VME =2W
ρS CDME
2 + CLME2≈
2 W S( )ρ
ε3CDo
≈ 0.76VL Dmax
LD( )
ME=14
3εCDo
=32
LD( )
max≈ 0.86 L
D( )max
L/D for Minimum Sink Rate"• For L/D < L/Dmax, there are two solutions"• Which one produces minimum sink rate?"
LD( )
ME≈ 0.86 L
D( )max
VME ≈ 0.76VL Dmax
Gliding Flight of the P-51 Mustang"
Loaded Weight = 9,200 lb (3,465 kg)
L /D( )max =1
2 εCDo
=16.31
γMR = −cot−1 L
D$
%&
'
()max
= −cot−1(16.31)= −3.51°
CD( )L/Dmax = 2CDo= 0.0326
CL( )L/Dmax =CDo
ε= 0.531
VL/Dmax =76.49ρ
m / s
hL/Dmax =V sinγ = −4.68ρm / s
Rho=10km = 16.31( ) 10( ) =163.1 km
Maximum Range Glide"
Loaded Weight = 9,200 lb (3,465 kg)S = 21.83m2
CDME= 4CDo
= 4 0.0163( ) = 0.0652
CLME=
3CDo
ε=
3 0.0163( )0.0576
= 0.921
L D( )ME =14.13
hME = −2ρWS
$
%&
'
()CDME
CLME3/2
$
%&&
'
())= −
4.11ρm / s
γME = −4.05°
VME =58.12ρ
m / s
Maximum Endurance Glide"
Climbing Flight�
• Rate of climb, dh/dt = Specific Excess Power "
Climbing Flight"
V = 0 =T −D−W sinγ( )
m
sinγ =T −D( )W
; γ = sin−1T −D( )W
γ = 0 =L −W cosγ( )
mVL =W cosγ
h = V sinγ = VT − D( )W
=Pthrust − Pdrag( )
W
Specific Excess Power (SEP) = Excess PowerUnit Weight
≡Pthrust − Pdrag( )
W
• Flight path angle " • Required lift"
• Note significance of thrust-to-weight ratio and wing loading"
Steady Rate of Climb"
h =V sinγ =V TW"
#$
%
&'−
CDo+εCL
2( )qW S( )
*
+,,
-
.//
€
L = CLq S = W cosγ
CL =WS
#
$ %
&
' ( cosγ
q
V = 2 WS
#
$ %
&
' ( cosγCLρ
h =V TW!
"#
$
%&−
CDoq
W S( )−ε W S( )cos2 γ
q
*
+,
-
./
=VT h( )W
!
"#
$
%&−
CDoρ h( )V 3
2 W S( )−2ε W S( )cos2 γ
ρ h( )V
• Climb rate "
• Necessary condition for a maximum with respect to airspeed"
Condition for Maximum Steady Rate of Climb"
h =V TW!
"#
$
%&−
CDoρV 3
2 W S( )−2ε W S( )cos2 γ
ρV
∂ h∂V
= 0 = TW"
#$
%
&'+V
∂T /∂VW
"
#$
%
&'
(
)*
+
,-−3CDo
ρV 2
2 W S( )+2ε W S( )cos2 γ
ρV 2
Maximum Steady "Rate of Climb: "
Propeller-Driven Aircraft"
∂Pthrust∂V
= 0 = TW"
#$
%
&'+V
∂T /∂VW
"
#$
%
&'
(
)*
+
,-• At constant power"
∂ h∂V
= 0 = −3CDo
ρV 2
2 W S( )+2ε W S( )ρV 2
• With cos2γ ~ 1, optimality condition reduces to"
• Airspeed for maximum rate of climb at maximum power, Pmax"
V 4 =43!
"#$
%&ε W S( )2
CDoρ2
; V = 2W S( )ρ
ε3CDo
=VME
Maximum Steady Rate of Climb: "
Jet-Driven Aircraft"• Condition for a maximum at constant thrust and cos2γ ~ 1"
• Airspeed for maximum rate of climb at maximum thrust, Tmax"
∂ h∂V
= 0
0 = ax2 + bx + c and V = + x
= −3CDo
ρ
2 W S( )V 4 +
TW#
$%
&
'(V 2 +
2ε W S( )ρ
= −3CDo
ρ
2 W S( )V 2( )2 + T
W#
$%
&
'( V 2( )+ 2ε W S( )
ρ
Optimal Climbing Flight�
What is the Fastest Way to Climb from One Flight Condition to Another?" • 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 EnergyUnit Weight
≡ Specific Energy =mgh + mV 2 2
mg= h +
V 2
2g≡ Energy Height, Eh , ft or m
Specific Excess Power"
dEh
dt=ddt
h+V2
2g!
"#
$
%&=
dhdt+Vg!
"#
$
%&dVdt
• Rate of change of Specific Energy "
=V sinγ + Vg"
#$
%
&'T −D−mgsinγ
m"
#$
%
&'=V
T −D( )W
=VCT −CD( ) 1
2ρ(h)V 2S
W
= Specific Excess Power (SEP)= Excess PowerUnit Weight
≡Pthrust −Pdrag( )
W
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)[ ]
dV= 0
Subsonic Energy Climb"• Objective: Minimize time or fuel to climb to desired altitude
and airspeed"
Supersonic Energy Climb"• Objective: Minimize time or fuel to climb to desired altitude
and airspeed"
The Maneuvering Envelope�
• 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!
Maneuvering Envelopes (V-n Diagrams) for Three Fighters of the Korean War Era"
Republic F-84"
North American F-86"
Lockheed F-94"
Turning Flight�
• Vertical force equilibrium"
Level Turning Flight"
L cosµ =W
• Load factor"
n = LW = L mg = secµ,"g"s
• Thrust required to maintain level flight"
Treq = CDo+ εCL
2( ) 12 ρV2S = Do +
2ερV 2S
Wcosµ
#
$%&
'(
2
= Do +2ερV 2S
nW( )2
µ :Bank Angle
• Level flight = constant altitude"• Sideslip angle = 0"
• Bank angle"
Maximum Bank Angle in Level Flight"
cosµ = WCLqS
=1n=W 2ε
Treq −Do( )ρV 2S
µ = cos−1 WCLqS
$
%&
'
()= cos−1
1n$
%&'
()= cos−1 W
2εTreq −Do( )ρV 2S
*
+,,
-
.//
• Bank angle is limited by "
€
µ :Bank Angle
CLmaxor Tmax or nmax
• Turning rate"
Turning Rate and Radius in Level Flight"
ξ =CLqS sinµ
mV=W tanµmV
=g tanµV
=L2 −W 2
mV
=W n2 −1
mV=
Treq − Do( )ρV 2S 2ε −W 2
mV
• Turning rate is limited by "
CLmaxor Tmax or nmax
• Turning radius "
Rturn =
Vξ=
V 2
g n2 −1
Maximum Turn Rates"
“Wind-up turns”"
• Corner velocity"
Corner Velocity Turn"
• Turning radius "
Rturn =V 2 cos2 γ
g nmax2 − cos2 γ
Vcorner =2nmaxWCLmas
ρS
• For steady climbing or diving flight"sinγ = Tmax − D
W
Corner Velocity Turn"
• Time to complete a full circle "
t2π =V cosγ
g nmax2 − cos2 γ
• Altitude gain/loss "Δh2π = t2πV sinγ
• Turning rate "
ξ =g nmax
2 − cos2 γ( )V cosγ
�Not a turning rate comparison�"http://www.youtube.com/watch?v=z5aUGum2EiM!
Herbst Maneuver"• Minimum-time reversal of direction"• Kinetic-/potential-energy exchange"• Yaw maneuver at low airspeed"• X-31 performing the maneuver"
Next Time:�Aircraft Equations of Motion�
�
Reading�Flight Dynamics, 155-161 �
Virtual Textbook, Parts 8,9 �