7. Gliding Climbing Turning Flight Performance - 2016stengel/MAE331Lecture7.pdfGliding, Climbing,...
Transcript of 7. Gliding Climbing Turning Flight Performance - 2016stengel/MAE331Lecture7.pdfGliding, Climbing,...
Gliding, Climbing, and Turning Flight Performance!
Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2016
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
•! 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
Learning Objectives
Reading:!Flight Dynamics !
Aerodynamic Coefficients, 130-141!
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
D = CD12!V 2S = "W sin#
CL12!V 2S =W cos#
!h =V sin#!r =V cos#
Equilibrium Gliding Flight
4
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=dhdr; ! = " tan"1 D
L#
$%
&
'(= "cot"1
LD#
$%
&
'(
Gliding flight path angle
Corresponding airspeed
Vglide =2W
!S CD2 + CL
25
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
tan! =!h!r= negative constant =
h " ho( )r " ro( )
#r = #htan!
= "#h" tan!
= maximum when LD
= maximum
!max = " tan"1 D
L#
$%
&
'(min
= "cot"1 LD#
$%
&
'(max
7
Sink Rate, m/s •! Lift and drag define !! and V in gliding equilibrium
D = CD12!V 2S = "W sin#
sin# = " DW
L = CL12!V 2S =W cos"
V = 2W cos"CL!S
!h =V sin!
= "2W cos!CL#S
DW$
%&
'
()= "
2W cos!CL#S
LW$
%&
'
()DL
$
%&
'
()
= "2W cos!CL#S
cos! 1L D$
%&
'
()
•! Sink rate = altitude rate, dh/dt (negative)
8
•! Minimum sink rate provides maximum endurance•! Minimize sink rate by setting !(dh/dt)/!CL = 0 (cos !! ~1)
Conditions for Minimum Steady Sink Rate
!h = ! 2W cos"CL#S
cos" CD
CL
$
%&
'
()
= !2W cos3"
#SCD
CL3/2
$
%&
'
() * !
2#WS
$
%&
'
()CD
CL3/2
$
%&
'
()
CLME=
3CDo
!and CDME
= 4CDo9
L/D and VME for Minimum Sink Rate
VME =2W
!S CDME
2 + CLME2"
2 W S( )!
#3CDo
" 0.76VL Dmax
LD( )
ME=14
3!CDo
=32
LD( )
max" 0.86 L
D( )max
10
L/D for Minimum Sink Rate•! For L/D < L/Dmax, there are
two solutions•! Which one produces smaller
sink rate?
LD( )
ME! 0.86 L
D( )max
VME ! 0.76VL Dmax
11
Checklist!"!Steady flight path angle?!"!Corresponding airspeed?!"!Sink rate?!"!Maximum-range glide?!"!Maximum-endurance glide?!
12
HL-10M2-F1
M2-F2
M2-F3
X-24A
X-24B
!!iiss""rriiccaall FFaacc""iiddss##iiff$$nngg--BBooddyy RReeeenn%%yy VVeehhiicclleess
13
Climbing Flight!
14
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!( )
mVL =W cos!
!h =V sin! =VT " D( )W
=Pthrust " Pdrag( )
W
Specific Excess Power (SEP) = Excess PowerUnit Weight
#Pthrust " Pdrag( )
W
•! Flight path angle •! Required lift
15
Note significance of thrust-to-weight ratio and wing loading
Steady Rate of Climb
!h =V sin! =V TW"
#$
%
&'(
CDo+)CL
2( )qW S( )
*
+,,
-
.//
L = CLqS =W cos!
CL =WS
"#$
%&'cos!q
V = 2 WS
"#$
%&'cos!CL(
!h =V TW
!"#
$%& '
CDoq
W S( ) '( W S( )cos2 )
q*
+,
-
./
=VT h( )W
!"#
$%&'CDo
0 h( )V 3
2 W S( ) '2( W S( )cos2 )
0 h( )V
Climb rate
16
Necessary condition for a maximum with respect to airspeed
Condition for Maximum Steady Rate of Climb
!h =V TW!
"#
$
%&'
CDo(V 3
2 W S( )'2) W S( )cos2 *
(V
! !h!V
= 0 = TW"
#$
%
&'+V
!T /!VW
"
#$
%
&'
(
)*
+
,-.3CDo
/V 2
2 W S( )+20 W S( )cos2 1
/V 2
17
Maximum Steady Rate of Climb:
Propeller-Driven Aircraft
!Pthrust!V
= 0 = TW"
#$
%
&'+V
!T /!VW
"
#$
%
&'
(
)*
+
,-•! At constant power
! !h!V
= 0 = "3CDo
#V 2
2 W S( )+2$ W S( )#V 2
•! With cos2!! ~ 1, optimality condition reduces to
•! Airspeed for maximum rate of climb at maximum power, Pmax
V 4 =43!
"#$
%&' W S( )2
CDo(2
; V = 2W S( )(
'3CDo
=VME
18
Maximum Steady Rate of Climb:
Jet-Driven AircraftCondition for a maximum at constant thrust and cos2!! ~ 1
Airspeed for maximum rate of climb at maximum thrust, Tmax
! !h!V
= 0
0 = ax2 + bx + c and V = + x
!3CDo
"2 W S( )V
4 + TW
#$%
&'(V
2 +2) W S( )
"= 0
19
!3CDo
"2 W S( ) V
2( )2 + TW
#$%
&'( V
2( ) + 2) W S( )"
= 0
Quadratic in V2
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
•! Specific Energy •! = (Potential + Kinetic
Energy) per Unit Weight•! = Energy Height
Energy Height
Can trade altitude for airspeed with no change in energy height if thrust and drag are zero
Total EnergyUnit Weight
! Specific Energy
= mgh +mV2 2
mg= h + V
2
2g
23
! 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 sin! + Vg
"#$
%&'T ( D (mgsin!
m"#$
%&' =V
T ( D( )W
= Specific Excess Power (SEP)
= Excess PowerUnit Weight
!Pthrust " Pdrag( )
W
24
=VCT !CD( ) 1
2"(h)V 2S
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
25
maxwrt V
SEP(h)[ ]
Subsonic Minimum-Time Energy ClimbObjective: Minimize time to climb to desired
altitude and airspeed
26
•! Minimum-Time Strategy:•! Zoom climb/dive to intercept SEPmax(h) contour•! Climb at SEPmax(h)•! Zoom climb/dive to intercept target SEPmax(h) contour
Bryson, Desai, Hoffman, 1969
Subsonic Minimum-Fuel Energy ClimbObjective: Minimize fuel to climb to desired
altitude and airspeed
27
•! 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
Supersonic Minimum-Time Energy Climb
Objective: Minimize time to climb to desired altitude and airspeed
28
•! Minimum-Time Strategy:•! Intercept subsonic SEPmax(h) contour•! Climb at SEPmax(h) to intercept matching zoom climb/dive contour•! Zoom climb/dive to intercept supersonic SEPmax(h) contour•! Climb at SEPmax(h) to intercept target SEPmax(h) contour•! Zoom climb/dive to intercept target SEPmax(h) contour
Bryson, Desai, Hoffman, 1969
Checklist!"!Energy height?!"!Contours?!"!Subsonic minimum-time climb?!"!Supersonic minimum-time climb?!"!Minimum-fuel climb?!
29
dEh
dmfuel
= dEh
dtdt
dmfuel
= 1!mfuel
dhdt
+ Vg
!"#
$%&dVdt
'
()
*
+,
SpaceShipOne"Ansari X Prize, December 17, 2003
Brian Binnie, *78Pilot, Astronaut
30
SpaceShipOne Altitude vs. Range!MAE 331 Assignment #4, 2010
31
SpaceShipOne State Histories
32
SpaceShipOne Dynamic Pressure and Mach Number Histories
33
The Maneuvering Envelope!
34
•! Maneuvering envelope: limits on normal load factor and allowable equivalent airspeed–! Structural factors–! Maximum and minimum
achievable lift coefficients–! Maximum and minimum
airspeeds–! Protection against
overstressing due to gusts–! Corner Velocity: Intersection
of maximum lift coefficient and maximum load factor
Typical Maneuvering Envelope: V-n Diagram
•! Typical positive load factor limits–! Transport: > 2.5–! Utility: > 4.4–! Aerobatic: > 6.3–! Fighter: > 9
•! Typical negative load factor limits–! Transport: < –1–! Others: < –1 to –3
35
Maneuvering Envelopes (V-n Diagrams) for Three Fighters of the Korean War Era
Republic F-84
North American F-86
Lockheed F-94
36
Turning Flight!
37
•! Vertical force equilibrium
Level Turning Flight
L cosµ =W•! Load factor
n = LW = L mg = secµ,"g"s
•! Thrust required to maintain level flight
Treq = CDo+ !CL
2( ) 12 "V2S = Do +
2!"V 2S
Wcosµ
#$%
&'(
2
= Do +2!
"V 2SnW( )2
µ :Bank Angle
•! Level flight = constant altitude•! Sideslip angle = 0
38
Bank angle
Maximum Bank Angle in Steady Level Flight
cosµ = WCLqS
= 1n
=W 2!Treq " Do( )#V 2S
Bank angle is limited by
µ :Bank Angle
CLmaxor Tmax or nmax
39
µ = cos!1 WCLqS
"#$
%&'
= cos!1 1n
"#$
%&'
= cos!1 W 2(Treq ! Do( ))V 2S
*
+,,
-
.//
Turning rate
Turning Rate and Radius in Level Flight
!! = CLqS sinµmV
= W tanµmV
= g tanµV
= L2 "W 2
mV
= W n2 "1mV
=Treq " Do( )#V 2S 2$ "W 2
mV
Turning rate is limited by
CLmaxor Tmax or nmax
Turning radius
Rturn =
V!!=
V 2
g n2 "140
Maximum Turn Rates
“Wind-up turns”
41
•! Corner velocity
Corner Velocity Turn
•! Turning radius Rturn =V 2 cos2 !
g nmax2 " cos2 !
Vcorner =2nmaxWCLmas
!S
•! For steady climbing or diving flight sin! = Tmax " D
W
42
Corner Velocity Turn
•! Time to complete a full circle
t2! =V cos"
g nmax2 # cos2 "
•! Altitude gain/loss !h2" = t2"V sin#
•! Turning rate
!! =g nmax
2 " cos2 #( )V cos#
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!
46
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?
Learning Objectives
&&uupppplleemmeennttaall MMaa''rriiaall
47
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#1 LD
$%&
'() max
= #cot#1(16.3) = #3.5°
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#
WS
$%&
'()
CDME
CLME3/2
$
%&'
()= " 4.11
#m/s
* ME = "4.05°
VME =58.12
#m/s
Maximum Endurance Glide
48