Gliding, Climbing, and Turning
Flight Performance!
Robert Stengel, Aircraft Flight Dynamics,
MAE 331, 2016
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, 130-141!
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
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
Equilibrium Gliding Flight
1
D = C D !V 2 S = "W sin #
2
1
C L !V 2 S = W cos #
2
h! = V sin #
r! = V cos #
4
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&
D
C
h! dh
= " D = = ; ! = " tan "1 % ( = "cot "1 % (
$L'
$D'
L
C L r! dr
Corresponding airspeed
Vglide =
2W
!S C D2 + C L2
5
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
#D&
#L&
! max = " tan "1 % ( = "cot "1 % (
$ L 'min
$ D 'max
( h " ho )
h!
tan ! = = negative constant =
r!
( r " ro )
#r =
#h
"#h
L
=
= maximum when = maximum
tan ! " tan !
D
7
Sink Rate, m/s
•! Lift and drag define ! and V in gliding equilibrium
D = CD
1
!V 2 S = "W sin #
2
D
sin # = "
W
1
!V 2 S = W cos "
2
2W cos "
V=
CL ! S
L = CL
•! Sink rate = altitude rate, dh/dt (negative)
h! = V sin !
="
2W cos ! $ D '
2W cos ! $ L '$ D '
& )="
& )& )
CL #S % W (
C L #S % W (% L (
="
$ 1 '
2W cos !
cos ! &
)
CL #S
%L D(
8
Conditions for Minimum
Steady Sink Rate
•! Minimum sink rate provides maximum endurance
•! Minimize sink rate by setting !(dh/dt)/!CL = 0 (cos ! ~1)
$ '
!h = ! 2W cos " cos " & C D )
CL #S
% CL (
2 $ W ' $ CD '
2W cos 3 " $ C D '
=!
& )&
& 3/2 ) * !
)
#S
# % S ( % C L3/2 (
% CL (
C LME =
3C Do
!
and C DME = 4C Do
9
L/D and VME for Minimum
Sink Rate
L
( D)
VME =
ME
=
3
1
3 L
=
D
2
4 ! C Do
( )
2W
"
!S C D2 ME+ C L2ME
max
" 0.86 L D
( )
max
2 (W S )
#
" 0.76VL Dmax
!
3C Do
10
L/D for Minimum Sink Rate
•! For L/D < L/Dmax, there are
two solutions
•! Which one produces smaller
sink rate?
( L D)
ME
VME
( )
! 0.86 L D
! 0.76VL Dmax
max
11
Checklist!
"! Steady flight path angle?!
"! Corresponding airspeed?!
"! Sink rate?!
"! Maximum-range glide?!
"! Maximum-endurance glide?!
12
!is"rical Fac"ids
#if$ng-Body Reen%y Vehicles
M2-F1
HL-10
M2-F2
M2-F3
X-24A
X-24B
13
Climbing Flight!
14
Climbing Flight
•! Flight path angle
•! Required lift
(T ! D ! W sin " )
V! = 0 =
m
(T ! D ) ; " = sin !1 (T ! D )
sin " =
W
W
( L " W cos ! )
!! = 0 =
mV
L = W cos !
Rate of climb, dh/dt = Specific Excess Power
(
(T " D ) = Pthrust " Pdrag
h! = V sin ! = V
W
W
Specific Excess Power (SEP) =
)
(
Pthrust " Pdrag
Excess Power
#
Unit Weight
W
)
15
Steady Rate of Climb
Climb rate
*" T % (C + ) C 2 ) q Do
L
/
h! = V sin ! = V ,$ ' (
#
&
W
W
S
( ) /.
,+
L = C L qS = W cos !
" W % cos !
CL = $ '
# S& q
" W % cos !
V = 2$ '
# S & CL (
Note significance of thrust-to-weight ratio and wing loading
*! T $ C Do q ( (W S ) cos 2 ) '
h! = V ,# & '
/
"
%
W
q
W
S
(
)
+
.
3
2( (W S ) cos 2 )
! T ( h ) $ C Do 0 ( h )V
=V#
'
'
" W &%
2 (W S )
0 ( h )V
16
Condition for Maximum
Steady Rate of Climb
3
2
(
V
C
!
$
2
)
W
S
cos
*
T
(
)
D
o
'
h! = V # & '
" W % 2 (W S )
(V
Necessary condition for a maximum with
respect to airspeed
(" T %
" ! T / ! V %+ 3C Do /V 2 20 (W S ) cos 2 1
! h!
= 0 = *$ ' +V $
+
'- .
# W &, 2 (W S )
/V 2
!V
)# W &
17
Maximum Steady
Rate of Climb:
Propeller-Driven Aircraft
•! At constant power
(" T %
" ! T / ! V %+
! Pthrust
= 0 = *$ ' +V $
'# W &,
!V
)# W &
•! With cos2! ~ 1, optimality condition reduces to
3C #V
2$ (W S )
! h!
= 0 = " Do
+
!V
2 (W S )
#V 2
2
•! Airspeed for maximum rate of climb at maximum power, Pmax
2
! 4 $ ' (W S )
(W S ) ' = V
V =# &
;
V
=
2
ME
" 3 % C Do ( 2
(
3C Do
4
18
Maximum Steady
Rate of Climb:
Jet-Driven Aircraft
Condition for a maximum at constant thrust and cos2! ~ 1
! h!
=0
!V
!
!
2) (W S )
#T&
V4 +% (V2 +
=0
$W '
2 (W S )
"
3C Do "
3C Do "
(V )
2 (W S )
2 2
2) (W S )
#T&
+ % ( (V 2 ) +
=0
$W '
"
Quadratic in V2
Airspeed for maximum rate of climb at maximum thrust, Tmax
0 = ax 2 + bx + c and V = + x
19
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
Energy Height
•! Specific Energy
•! = (Potential + Kinetic
Energy) per Unit Weight
•! = Energy Height
Total Energy
! Specific Energy
Unit Weight
V2
mgh + mV 2 2
= h+
=
mg
2g
! Energy Height, Eh ,
ft or m
Can trade altitude for airspeed with no change
in energy height if thrust and drag are zero
23
Specific Excess Power
Rate of change of Specific Energy
dEh d ! V 2 $ dh ! V $ dV
= #h + & = + # &
dt dt " 2g % dt " g % dt
" V % " T ( D ( mgsin ! %
(T ( D )
= V sin ! + $ ' $
=V
'
&
# g&#
m
W
= Specific Excess Power (SEP)
=
(
Pthrust " Pdrag
Excess Power
!
Unit Weight
W
)
=V
(CT ! CD ) 12 " (h)V 2 S
W
24
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 [ SEP(h)]
wrt V
d [ SEP(h)]
=0
dV
25
Subsonic Minimum-Time Energy Climb
Objective: Minimize time to climb to desired
altitude and airspeed
•! 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
26
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
27
Bryson, Desai, Hoffman, 1969
Supersonic Minimum-Time
Energy Climb
Objective: Minimize time to climb to desired
altitude and airspeed
•! 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
28
Checklist!
"! Energy height?!
"! Contours?!
"! Subsonic minimum-time climb?!
"! Supersonic minimum-time climb?!
"! Minimum-fuel climb?!
dEh
dE dt
1
= h
=
dm fuel
dt dm fuel m! fuel
' dh ! V $ dV *
) dt + #" g &%
,
dt +
(
29
SpaceShipOne"
Ansari X Prize, December 17, 2003
Brian Binnie, *78
Pilot, 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
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
35
Maneuvering Envelopes (V-n Diagrams)
for Three Fighters of the Korean War Era
Republic F-84
Lockheed F-94
North American F-86
36
Turning Flight!
37
Level Turning Flight
•! Level flight = constant altitude
•! Sideslip angle = 0
•! Vertical force equilibrium
µ : Bank Angle
L cos µ = W
•! Load factor
n = L W = L mg = sec µ,"g"s
•! Thrust required to maintain level flight
(
Treq = C Do + ! C L
2
)
1
2! # W &
"V 2 S = Do +
2
"V 2 S %$ cos µ ('
= Do +
2!
2
nW )
2 (
"V S
2
38
µ : Bank Angle
Maximum Bank Angle in
Steady Level Flight
Bank angle
=
=W
(
" W %
µ = cos !1 $
# C L qS '&
W
C L qS
cos µ =
" 1%
= cos !1 $ '
# n&
1
n
2!
Treq " Do #V 2 S
)
*
= cos !1 ,W
,+
(
2(
/
Treq ! Do )V 2 S /
.
)
Bank angle is limited by
C Lmax or Tmax or nmax
39
Turning Rate and Radius in
Level Flight
Turning rate
C qS sin µ
!! = L
mV
W tan µ
=
mV
g tan µ
=
V
=
=
=
(T
L2 " W 2
mV
W n2 " 1
mV
)
2
2
req " Do #V S 2$ " W
mV
Turning rate is
limited by
C Lmax or Tmax or nmax
Turning radius
V
V2
Rturn = ! =
! g n2 " 1
40
Maximum Turn Rates
“Wind-up
turns”
41
Corner Velocity Turn
•! Corner velocity
Vcorner =
2nmaxW
C Lmas !S
•! For steady climbing sin ! = Tmax " D
or diving flight
W
•! Turning radius
Rturn =
V 2 cos 2 !
2
g nmax
" cos 2 !
42
Corner Velocity Turn
!! =
•! Turning rate
•! Time to complete a
full circle
•! Altitude gain/loss
t2! =
2
" cos 2 #
g nmax
(
V cos #
)
V cos "
2
g nmax
# cos 2 "
!h2 " = t 2 " V sin #
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!
Learning Objectives
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?
46
&upplemental Ma'rial
47
Gliding Flight of the
P-51 Mustang
Maximum Range Glide
Loaded Weight = 9,200 lb (3,465 kg)
1
= 16.31
( L / D )max =
2 ! C Do
$ L'
" MR = # cot #1 & ) = # cot #1 (16.3) = #3.5°
% D ( max
(CD )L/D
max
(CL )L/D
max
VL/Dmax
= 2C Do = 0.0326
C Do
= 0.531
!
76.49
=
m/s
*
=
4.68
m/s
h!L/Dmax = V sin " = #
*
Rho =10 km = (16.31) (10 ) = 163.1 km
Maximum Endurance Glide
Loaded Weight = 9,200 lb (3,465 kg)
S = 21.83 m 2
C DME = 4C Do = 4 ( 0.0163) = 0.0652
C LME =
3C Do
!
L
D
=
14.13
( )ME
=
3( 0.0163)
= 0.921
0.0576
2 $ W ' $ C DME '
4.11
m/s
h!ME = "
="
&% )( &
3/2 )
# S % C LME (
#
* ME = "4.05°
58.12
VME =
m/s
#
48