7_gliding_climbing_turning.pdf
TRANSCRIPT
Gliding, Climbing, and TurningFlight Performance
Robert Stengel, Aircraft Flight Dynamics
MAE 331, 2010
Copyright 2010 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 byavailable 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 altitudesand airspeeds at which an aircraft can fly
– in steady, level flight
– at fixed weight
Additional Factors Define
the Flight Envelope• Maximum Mach number
• Maximum allowableaerodynamic heating
• Maximum thrust
• Maximum dynamicpressure
• Performance ceiling
• Wing stall
• Flow-separation buffet– Angle of attack
– Local shock waves
Piper Dakota Stall Buffethttp://www.youtube.com/watch?v=mCCjGAtbZ4g
Boeing 787 FlightEnvelope (HW #5, 2008)
Best
Cruise
Region
Gliding Flight
D = CD
1
2!V 2
S = "W sin#
CL
1
2!V 2
S =W cos#
!h = V sin#
!r = V cos#
Equilibrium Gliding Flight Gliding Flight
• Thrust = 0
• Flight path angle < 0 in gliding flight
• Altitude is decreasing
• Airspeed ~ constant
• Air density ~ constant
tan! = "D
L= "
CD
CL
=
!h
!r=dh
dr
! = " tan"1 D
L
#$%
&'(= " cot"1
L
D
#$%
&'(
• Gliding flight path angle
• Corresponding airspeed
Vglide =2W
!S CD
2+ CL
2
Maximum Steady Gliding Range
• Glide range is maximum when ! is least negative, i.e.,most positive
• This occurs at (L/D)max
Maximum SteadyGliding 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 " ho( )r " ro( )
#r =#h
tan!=
"#h
" tan!= maximum when
L
D= maximum
!max
= " tan"1 D
L
#$%
&'(min
= " cot"1L
D
#$%
&'(max
Sink Rate• Lift and drag define ! and V in gliding equilibrium
D = CD
1
2!V 2
S = "W sin#
sin# = "D
W
L = CL
1
2!V 2
S =W cos"
V =2W cos"
CL!S
!h = V sin!
= "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)
• Minimum sink rate provides maximum endurance
• Minimize sink rate by setting ! (dh/dt)/dCL = 0 (cos ! ~1)
• See Mathematica performance calculations in BlackboardCourse Materials
Conditions for MinimumSteady Sink Rate
!h = !2W cos"CL#S
cos"CD
CL
$
%&'
()
= !2W cos
3 "#S
CD
CL
3/2
$
%&'
()* !
2
#W
S
$%&
'()
CD
CL
3/2
$
%&'
()
CLME
=3C
Do
!and C
DME
= 4CDo
L/D and VME for
Minimum Sink Rate
VME
=2W
!S CDME
2+ C
LME
2
"2 W S( )
!
#
3CDo
" 0.76VL Dmax
LD( )
ME
=1
4
3
!CDo
=3
2
LD( )
max
" 0.86 LD( )
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 LD( )
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 !CDo
= 16.31
" MR = # cot#1L
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 = 21.83m2
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
Climbing Flight
• 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 = V sin! = VT " D( )
W=
Pthrust " Pdrag( )W
Specific Excess Power (SEP) =Excess Power
Unit Weight#
Pthrust " Pdrag( )W
• Note significance of thrust-to-weight ratio and wing loading
Steady Rate of Climb
!h = V sin! = VT
W
"#$
%&'(CDo
+ )CL
2( )qW S( )
*
+,,
-
.//
L = CLq S = W cos!
CL =W
S
"
# $
%
& ' cos!
q
V = 2W
S
"
# $
%
& ' cos!CL(
!h = VT
W
!"#
$%&'CDo
q
W S( )'( W S( )cos2 )
q
*
+,
-
./
= VT
W
!"#
$%&'CDo
0V 3
2 W S( )'2( W S( )cos2 )
0V
• Necessary condition for a maximum with respectto airspeed
Condition for Maximum
Steady Rate of Climb
!h = VT
W
!"#
$%&'CDo(V 3
2 W S( )'2) W S( )cos2 *
(V
! !h!V
= 0 =T
W
"#$
%&'+V
!T / !VW
"#$
%&'
(
)*
+
,- .
3CDo/V 2
2 W S( )+20 W S( )cos2 1
/V 2
Maximum SteadyRate of Climb:
Propeller-Driven Aircraft
!Pthrust
!V=
T
W
"#$
%&'+V
!T / !VW
"#$
%&'
(
)*
+
,- = 0
• At constantpower
! !h
!V= 0 = "
3CDo#V 2
2 W S( )+2$ W S( )
#V 2
• With cos2! ~ 1
• Airspeed for maximum rate of climb at maximum power, Pmax
V4=4
3
!"#
$%&' W S( )
2
CDo(2
; V = 2W S( )(
'3C
Do
= VME
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, Tmax
! !h!V
= 0
0 = "3C
Do#
2 W S( )V4+
T
W
$%&
'()V2+2* W S( )
#
= "3C
Do#
2 W S( )V2( )2
+T
W
$%&
'()V2( ) +
2* W S( )#
0 = ax2+ bx + c and V = + x
Optimal Climbing Flight
What is the Fastest Way to Climb fromOne 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 energyheight if thrust and drag were zero
Total Energy
Unit Weight! Specific Energy =
mgh + mV22
mg= h +
V2
2g
! Energy Height, Eh , ft or m
Specific Excess Power
dEh
dt=d
dth +
V2
2g
!"#
$%&=dh
dt+
V
g
!"#
$%&dV
dt
= V sin' +V
g
!"#
$%&T ( D ( mgsin'
m
!"#
$%&= V
T ( D( )
W= V
CT ( CD( )1
2)(h)V 2
S
W
= Specific Excess Power (SEP) =Excess Power
Unit Weight*
Pthrust ( Pdrag( )W
Contours of ConstantSpecific Excess Power
• Specific Excess Power is a function of altitude and airspeed
• SEP is maximized at each altitude, h, whend SEP(h)[ ]
dV= 0
Subsonic Energy Climb
• Objective: Minimize time or fuel to climb to desired altitudeand airspeed
Supersonic Energy Climb
• Objective: Minimize time or fuel to climb to desired altitudeand airspeed
The Maneuvering Envelope
• Maneuvering envelopedescribes limits on normalload factor and allowableequivalent airspeed– Structural factors
– Maximum and minimumachievable lift coefficients
– Maximum and minimumairspeeds
– Protection againstoverstressing due to gusts
– Corner Velocity:Intersection of maximum liftcoefficient and maximumload factor
Typical Maneuvering Envelope:
V-n Diagram
• Typical positive load factor limits– Transport: > 2.5
– Utility: > 4.4
– Aerobatic: > 6.3
– Fighter: > 9
• Typical negative load factor limits– Transport: < –1
– Others: < –1 to –3
C-130 exceeds maneuvering envelopehttp://www.youtube.com/watch?v=4bDNCac2N1o&feature=related
Maneuvering Envelopes (V-n Diagrams)for Three Fighters of the Korean War Era
Republic F-84
North American F-86
Lockheed F-94
Turning Flight
• Vertical force equilibrium
Level Turning Flight
L cosµ =W
• Load factor
n = LW
= Lmg
= secµ,"g"s
• Thrust required to maintain level flight
Treq = CDo+ !CL
2( )1
2"V 2
S = Do +2!
"V 2S
W
cosµ
#$%
&'(
2
= Do +2!
"V 2SnW( )
2
µ :Bank Angle
• Level flight = constant altitude
• Sideslip angle = 0
• Bank angle
Maximum Bank
Angle in Level Flight
cosµ =W
CLqS=1
n=W
2!Treq " Do( )#V 2
S
µ = cos"1W
CLqS
$
%&'
()= cos"1
1
n
$%&
'()= cos"1 W
2!Treq " Do( )#V 2
S
*
+,,
-
.//
• Bank angle is limited by
µ :Bank Angle
CLmax
or Tmax
or nmax
• Turning rate
Turning Rate and Radius in Level Flight
!! =CLqS sinµ
mV=W tanµ
mV=g tanµ
V=
L2 "W 2
mV
=W n
2 "1
mV=
Treq " Do( )#V 2S 2$ "W 2
mV
• Turning rate is limited by
CLmax
or Tmax
or nmax
• Turning radius
Rturn
=V
!!=
V2
g n2 "1
Maximum Turn Rates
• Corner velocity
Corner Velocity Turn
• Turning radius
Rturn
=V2cos
2 !
g nmax
2" cos
2 !
Vcorner
=2n
maxW
CLmas
!S
• For steady climbing or diving flight
sin! =Tmax
" D
W
Corner Velocity Turn
• Time to complete a full circle
t2! =
V cos"
g nmax
2# cos
2 "
• Altitude gain/loss
!h2" = t
2"V sin#
• Turning rate
!! =g n
max
2 " cos2 #( )V cos#
“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
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