Fighter Aircraft Performance, Part II of two, describes the parameters that affect aircraft performance.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
14 fixed wing fighter aircraft- flight performance - ii
1. Fixed Wing Fighter Aircraft
Flight Performance
Part II
SOLO HERMELIN
Updated: 04.12.12
28.02.15
1
http://www.solohermelin.com
2. Table of Content
SOLO
2
Aerodynamics
Introduction to Fixed Wing Aircraft Performance
Earth Atmosphere
Mach Number
Shock & Expansion Waves
Reynolds Number and Boundary Layer
Knudsen Number
Flight Instruments
Aerodynamic Forces
Aerodynamic Drag
Lift and Drag Forces
Wing Parameters
Specific Stabilizer/Tail Configurations
F
i
x
e
d
W
i
n
g
P
a
r
t
I
Fixed Wing Fighter Aircraft Flight Performance
3. Table of Content (continue – 1)
SOLO
3
Specific Energy
Aircraft Propulsion Systems
Aircraft Propellers
Aircraft Turbo Engines
Afterburner
Thrust Reversal Operation
Aircraft Propulsion Summary
Vertical Take off and Landing - VTOL
Engine Control System
Aircraft Flight Control
Aircraft Equations of Motion
Aerodynamic Forces (Vectorial)
Three Degrees of Freedom Model in Earth Atmosphere
F
i
x
e
d
W
i
n
g
P
a
r
t
I
Fixed Wing Fighter Aircraft Flight Performance
4. Table of Content (continue – 2)
SOLO
Fixed Wing Fighter Aircraft Flight Performance
4
Parameters defining Aircraft Performance
Takeoff (no VSTOL capabilities)
Landing (no VSTOL capabilities)
Climbing Aircraft Performance
Gliding Flight
Level Flight
Steady Climb (V, γ = constant)
Optimum Climbing Trajectories using Energy State
Approximation (ESA)
Minimum Fuel-to- Climb Trajectories using Energy State
Approximation (ESA)
Maximum Range during Glide using Energy State
Approximation (ESA)
Aircraft Turn Performance
Maneuvering Envelope, V – n Diagram
5. Table of Content (continue – 3)
SOLO
Fixed Wing Fighter Aircraft Flight Performance
5
Air-to-Air Combat
Energy–Maneuverability Theory
Supermaneuverability
Constraint Analysis
References
Aircraft Combat Performance Comparison
6. SOLO
This Presentation is about Fixed Wing Aircraft Flight Performance.
The Fixed Wing Aircraft are
•Commercial/Transport Aircraft (Passenger and/or Cargo)
•Fighter Aircraft
Fixed Wing Fighter Aircraft Flight Performance
Continue from Part I
7. 7
Fixed Wing Fighter Aircraft Flight PerformanceSOLO
The Aircraft Flight Performance is defined by the following parameters
• Take-off distance
• Landing distance
• Maximum Endurance and Speed for Maximum Endurance
• Maximum Range and Speed for Maximum Range
• Ceiling(s)
• Climb Performance
• Turn Performance
• Combat Radius
• Maximum Payload
Parameters defining Aircraft Performance
8. 8
Performance of an Aircraft with Parabolic PolarSOLO
Assumptions:
•Point mass model.
•Flat earth with g = constant.
•Three-dimensional aircraft trajectory.
•Air density that varies with altitude ρ=ρ(h)
•Drag that varies with altitude, Mach
number and control effort D = D(h,M,n)
and is given by a Parabolic Polar.
•Thrust magnitude is controllable by the
throttle.
•No sideslip angle.
•No wind.
α
T
V
L
D
Bx
Wx
Bz
Wz
Wy
By
Aircraft Coordinate System
To understand how different parameters affect Aircraft Performance we start with a
Simplified Model, where Analytical Solutions can be obtained.
Results for real aircraft will then be presented.
Return to Table of Content
9. 9
SOLO
Aircraft Flight Performance
Takeoff
The Takeoff distance sTO
is divided as the sum of the
following distances:
sg – Ground Run
sr – Rotation Distance
st – Transition Distance
sc – Climb Distance to
reach Screen Height
ctrgTO sssss +++=
Ground Run
V = 0
sg
sTO
sr str
V TO
Rotation
Transition
sCL
θ CL
htr
hobs
R
Takeoff htransition < hobstacle
θ CL
Ground Run
V = 0
sg
sTO
sr sobs
V TO
Rotation
Transition
hobs
R
Takeoff htransition > hobstacle
We distinguish between two cases
of Takeoff
•The Aircraft must passes over an
obstacle at altitude hobs..
•The obstacle is cleared during
the transition phase.
Assume no Vertical Takeoff
Capability.
10. 10
Takeoff (continue – 1)
During the Ground Run there are additional
effects than in free flight, that must be considered:
-Friction between the tires and the ground
during rolling.
-Additional drag due to the landing gear
fully extended.
-Additional Lift Coefficient due to extended
flaps.
-Ground Effect due to proximity of the wings
to the ground, that reduces the Induced Drag
and the Lift.
Ground Run
SOLO
Aircraft Flight Performance
Ground run sg
Transition
distance st
Climb
distance sc
Stall safety
Take-off possible
with one engine
Continue take-off
if engine fails
after this point
Stop take-off if
engine fails before
this point
Acceleration at
full power
γ c
Total take-off if distance
VCRVMCG
VTVS
L
W
TD
R
μR
The Aircraft can leave the ground when the velocity
reaches the Stall Velocity where Lift equals Weight
max,
2
0
2
1
Lstallstall CSVLW ρ==
max,0
12
L
stall
CS
W
V
ρ
=
The Liftoff Velocity is 1.1 to 1.2 Vstall.
11. 11
ReactionGroundLWR
gW
RDT
td
Vd
V
V
td
xd
−=
−−
==
=
/
µ
( )
( )LWDT
gW
Vd
td
LWDT
gWV
Vd
sd xs
−−−
=
−−−
=
=
µ
µ
/
/
Takeoff (continue – 2)
Average Coefficient of Friction Values μ
Ground Run
SOLO
Aircraft Flight Performance
Ground run sg
Transition
distance st
Climb
distance sc
Stall safety
Take-off possible
with one engine
Continue take-off
if engine fails
after this point
Stop take-off if
engine fails before
this point
Acceleration at
full power
γ c
Total take-off if distance
VCRVMCG
VTVS
hc
L
W
R
μR
D
T
V
12. 12
Takeoff (continue – 3)
SOLO
Aircraft Flight Performance
Ground run sg
Transition
distance st
Climb
distance sc
Stall safety
Take-off possible
with one engine
Continue take-off
if engine fails
after this point
Stop take-off if
engine fails before
this point
Acceleration at
full power
γc
Total take-off if distance
VCRVMCG
VTVS
hc
L
W
R
μR
D
T
V
T (Jet)
Lift,Drag,Thrust,Resistance–lb
L,D,T,R
T (Prop)
D +μ R
Ground Speed – ft/s
Texcess(Prop)=T(Prop) -(D+μ R)
Texcess(Jet)=T(Jet) -(D+μ R)
Vground
( )
ReactionGroundLWR
RDT
g
WV
−=
+−= µ
Ground Run (continue -1)
13. 13
2
0 VCVBTT ++=
cVbVaVd
td
cVbVa
V
Vd
sd xs
++
=
++
=
=
2
2
1
Takeoff (continue – 4)
Ground Run (continue – 2)
To obtain an Analytic Solution assume that
during the Ground Run the Thrust can be
approximated by
Using
=
=
L
D
CSVL
CSVD
2
2
2
1
2
1
ρ
ρ
( )
−=
=
+−−=
µ
µ
ρ
W
T
gc
W
gB
b
W
gC
CC
W
Sg
a LD
0
:
2
:
2
:
where
SOLO
Aircraft Flight Performance
Ground run sg
Transition
distance st
Climb
distance sc
Stall safety
Take-off possible
with one engine
Continue take-off
if engine fails
after this point
Stop take-off if
engine fails before
this point
Acceleration at
full power
γ c
Total take-off if distance
VCRVMCG
VTVS
hc
L
W
R
μR
D
T
V
14. 14
cVbVaVd
td
cVbVa
V
Vd
sd xs
++
=
++
=
=
2
2
1
Takeoff (continue – 5)
Ground Run (continue – 3)
Integrating those equations between two
velocities V1 and V2 gives
−
−
⋅
+
+
−
+
++
++
=
2
1
1
2
2
1
2
1
2
2
2
1
1
1
1
ln
42
ln
2
1
a
a
a
a
caba
b
cVbVa
cVbVa
a
sg
−
−
⋅
+
+
−
=
1
2
2
1
2 1
1
1
1
ln
4
1
a
a
a
a
cab
tg
where
cab
bVa
a
cab
bVa
a
4
2
:
4
2
:
2
2
2
2
1
1
−
+
=
−
+
=
SOLO
Aircraft Flight Performance
Ground run sg
Transition
distance st
Climb
distance sc
Stall safety
Take-off possible
with one engine
Continue take-off
if engine fails
after this point
Stop take-off if
engine fails before
this point
Acceleration at
full power
γ c
Total take-off if distance
VCRVMCG
VTVS
hc
L
W
R
μR
D
T
V
2
0 VCVBTT ++=
15. 15
Takeoff (continue – 6)
Ground Run (continue – 4)
then
( )
( )
−
−
−−
=
+
=
TL
LDLD
g
CWT
CCCCg
SW
c
cVa
a
s
µ
µµρ
/
1
1
ln
/
ln
2
1
0
2
2
0,00 01 ==⇐== CBTTV
Assume
where
22
:&
/2
: VV
V
SW
C T
T
LT
==
ρ
A further simplification, using , givesZ
Z
Z 1
1
1
ln
<<
≈
−
−
=
µρ
W
T
Cg
SW
s
TL
g
0
/
SOLO
Aircraft Flight Performance
gL sCg
SW
W
T
T
ρ
/0
>
Ground run sg
Transition
distance st
Climb
distance sc
Stall safety
Take-off possible
with one engine
Continue take-off
if engine fails
after this point
Stop take-off if
engine fails before
this point
Acceleration at
full power
γc
Total take-off if distance
VCRVMCG
VTVS
hc
L
W
R
μR
D
T
V
16. 16
Takeoff (continue – 7)
Rotation Distance
At the ground roll and just prior to going into transition phase, most aircraft are
Rotated to achieve an Angle of Attack to obtain the desired Takeoff Lift Coefficient
CL. Since the rotation consumes a finite amount of time (1 – 4 seconds), the distance
traveled during rotation sr, must be accounted for by using
where Δt is usually taken as 3 seconds.
SOLO
Aircraft Flight Performance
tVs tr ∆=
Ground Run
V = 0
sg
sTO
sr str
V TO
Rotation
Transition
sCL
θ CL
htr
hobs
R
L
W
R
μR
D
T
V
17. 17
Takeoff (continue – 8)
Transition Distance
In the Transition Phase the Aircraft is in the Air (μ = 0) and turn to the Climb Angle.
The Equation of Motion are:
SOLO
Aircraft Flight Performance
Ta
Ta
t
Ta
t
VV
DT
VV
g
W
t
DT
VV
g
W
s
>
−
−
=
−
−
=
2
2
22
DT
gW
Vd
td
DT
gWV
Vd
sd xs
−
=
−
=
=
/
/
Assuming T – D = const., we can
Integrate the Equations of Motion
(assuming Va > VT)
Ground Run
V = 0
sg
sTO
sr str
V TO
Rotation
Transition
sCL
θ CL
htr
hobs
R
18. 18
Takeoff (continue – 9)
Climb Distance
The Climb Distance is evaluated from the following (see Figure):
SOLO
Aircraft Flight Performance
c
c
c
c
c
hh
s
c
γγ
γ 1
tan
<<
≈=
For small angles of Climb L = W.
We can write Ground Run
V = 0
sg
sTO
sr str
V TO
Rotation
Transition
sCL
θ CL
htr
hobs
R
cL
cLD
c
c
c
C
CkC
W
T
L
D
W
T
,
2
,0 +
−=−=γ
We have
cLcLD
c
c
CkCCWT
h
s
,,0 // −−
≈
19. 19
Takeoff (continue – 10)
SOLO
Aircraft Flight Performance
19
ctrgTO sssss +++=
sec41−=∆∆= ttVs tr
Ta
Ta
t
Ta
t
VV
DT
VV
g
W
t
DT
VV
g
W
s
>
−
−
=
−
−
=
2
2
22
−
−
⋅
+
+
−
+
++
++
=
2
1
1
2
2
1
2
1
2
2
2
1
1
1
1
ln
42
ln
2
1
a
a
a
a
caba
b
cVbVa
cVbVa
a
sg
cab
bVa
a
cab
bVa
a
4
2
:
4
2
:
2
2
2
2
1
1
−
+
=
−
+
=
−
−
⋅
+
+
−
=
1
2
2
1
2 1
1
1
1
ln
4
1
a
a
a
a
cab
tg
Ground Run
V = 0
sg
sTO
sr str
V TO
Rotation
Transition
sCL
θ CL
htr
hobs
R
Takeoff Summary
Rotation Phase
Climb Phase
Transition Phase
Ground Run
cLcLD
c
c
CkCCWT
h
s
,,0 // −−
≈
20. 20Minimum required takeoff runway lengths.
Summary of takeoff requirements
In order to establish the allowable
takeoff weight for a transport
category airplane, at any airfield,
the following must be considered:
•Airfield pressure altitude
•Temperature
•Headwind component
•Runway length
•Runway gradient or slope
•Obstacles in the flight path
Return to Table of Content
21. 21
Landing
Landing is similar to Takeoff, but in reverse.
We assume again that the Aircraft doesn’t have
VTOL capabilities.
The Landing Phase can be divide in the following
Phases:
1. The Final approach when the Aircraft
Glides toward the runway at a steady
speed and rate of descent.
2. The Flare, or Transition phase.
The Pilot attempts to rotate the Aircraft nose up and reduce the Rate of Sink to zero and the
forward speed to a minimum, that is larger than Vstall.
When entering this phase the velocity is less than 1.3Vstall and 1.15 Vstall at touchdown.
3. The Floating Phase, which is necessary if at the end of Flare phase, when the rate of
descent is zero, an additional speed reduction is necessary. The Float occurs when the
Aircraft is subjected to ground effect which requires speed reduction for touchdown.
4. The Ground Run after the Touchdown the Aircraft must reduce the speed to reach a
sufficient low one to be able to turn off the runway. For this it can use Thrust Reverse (if
available), spoilers or drag parachutes (like F-15 or MIG-21) and brakes are applied.
SOLO
Aircraft Flight Performance
Ground Run sgr
Transition
Airborne Phase
Total Landing Distance
Float
sf
Flare stGlide sg
γ
hg
hf
Touchdown
22. 22
Landing (continue – 1)
Descending Phase
SOLO
Aircraft Flight Performance
The Aircraft is aligned with the landing runaway at an altitude hg and a gliding angle γ.
The Aircraft Glides toward the runway at a steady speed and rate of descent, until it reaches
The altitude ht at which it goes to Transition Phase, turning with a Radius of Turn R. The
Descending Range on the ground is :
γγ
γ
γ
γ RhRhhh
s
ggtg
g
−
≈
−
=
−
=
<<1
tan
cos
tan
Ground Run sgr
Transition
Airborne Phase
Total Landing Distance
Float
sf
Flare stGlide sg
γ
hg
hf
Touchdown
23. 23
Landing (continue – 2)
Transition Phase
SOLO
Aircraft Flight Performance
If γ is the descent angle and R is the turn radius
then the Aircraft must start the Transition Phase
at an altitude ht, above the ground, given by:
( )γcos1−=Rht
The Transition Range on the ground is
γγ RRst ≈= sin
To calculate the turn radius we must use the flight velocity which varies between 1.3 Vstall
at the beginning to 1.1 Vstall at Touchdown. Let use an average velocity
3.11.1 −∈= tstalltt mVmV
If the Transition Turn Acceleration is nt = 1.15 – 1.25 g than the Turn Radius is
( ) gn
V
R
t
t
1
2
−
=
The Transition Turn time is
( ) gn
V
RV
t
t
t
t
t
1/ −
==
γγ
Ground Run sgr
Transition
Airborne Phase
Total Landing Distance
Float
sf
Flare stGlide sg
γ
hg
hf
Touchdown
24. 24
Landing (continue – 3)
Float Phase
SOLO
Aircraft Flight Performance
In this phase the Pilot brings the nose wheel to the ground at the touchdown velocity Vt:
tVs tf ∆=
where Δt is between 2 to 3 seconds.
Ground Run sgr
Transition
Airborne Phase
Total Landing Distance
Float
sf
Flare stGlide sg
γ
hg
hf
Touchdown
25. 25
Landing (continue – 4)
Ground Run Phase
SOLO
Aircraft Flight Performance
The equations of motion are the same as those
developed for Takeoff, but with different
parameters, adapted for Landing. Those equations
are:
cVbVaVd
td
cVbVa
V
Vd
sd xs
++
=
++
=
=
2
2
1
( )
−=
=
+−−=
µ
µ
ρ
W
T
gc
W
gB
b
W
gC
CC
W
Sg
a grLgrD
0
,,
:
2
:
2
:
where
−
−
⋅
+
+
−
+
++
++
=
2
1
1
2
2
1
2
1
2
2
2
1
1
1
1
ln
42
ln
2
1
a
a
a
a
caba
b
cVbVa
cVbVa
a
sg
−
−
⋅
+
+
−
=
1
2
2
1
2 1
1
1
1
ln
4
1
a
a
a
a
cab
tg
where
cab
bVa
a
cab
bVa
a
4
2
:
4
2
:
2
2
2
2
1
1
−
+
=
−
+
=
2
0 VCVBTT ++=
Assume a constant Thrust T = T0: B = 0, C = 0. V1 = Vtouchdown, V2 = final velocity
cVa
cVa
a
sg
+
+
−= 2
2
2
1
ln
2
1
−
−
⋅
+
+
−
=
1
2
2
1
1
1
1
1
ln
4
1
a
a
a
a
ca
tg ( )
−==−−= µµ
ρ
W
T
gcbCC
W
Sg
a LD
0
:,0,
2
:
Ground Run sgr
Transition
Airborne Phase
Total Landing Distance
Float
sf
Flare stGlide sg
γ
hg
hf
Touchdown
26. 26
Landing (continue – 5)
Ground Run Phase (continue – 1)
SOLO
Aircraft Flight Performance
where
( )
−=
−−=
µ
µ
ρ
W
T
gc
CC
W
Sg
a grLgrD
0
,,
:
2
:
cab
Va
a
ca
Va
a
touchdown
4
2
:
4
2
:
2
1
1
−
=
−
=
Assume a constant Thrust T = T0: B = 0, C = 0. V1 = Vtouchdown,
V2 = final velocity
cVa
cVa
a
sg
+
+
−= 2
2
2
1
ln
2
1
−
−
⋅
+
+
−
=
1
2
2
1
1
1
1
1
ln
4
1
a
a
a
a
ca
tg
For the Landing Ground Run Phase the following must included:
• if Thrust Reversal exists we must change T0 to – T0_reversal .
•The Drag Coefficient CD0,gr must consider:
- the landing gear fully extended.
- spoilers or drag parachutes (if exist)
•μ – the friction coefficient must be increased to describe the brakes effect.
Ground Run sgr
Transition
Airborne Phase
Total Landing Distance
Float
sf
Flare stGlide sg
γ
hg
hf
Touchdown
27. 27
Landing (continue – 6)
Summary
SOLO
Aircraft Flight Performance
where
( )
−=
−−=
µ
µ
ρ
W
T
gc
CC
W
Sg
a grLgrD
0
,,
:
2
:
cab
Va
a
ca
Va
a
touchdown
4
2
:
4
2
:
2
1
1
−
=
−
=cVa
cVa
a
sg
+
+
−= 2
2
2
1
ln
2
1
−
−
⋅
+
+
−
=
1
2
2
1
1
1
1
1
ln
4
1
a
a
a
a
ca
tg
Ground Run Phase
tVs tf ∆=
Float Phase
( ) gn
V
Rs
t
t
t
1
2
−
==
γ
γ
( ) gn
V
RV
t
t
t
t
t
1/ −
==
γγ
Transition Phase
Descent Phase
Ground Run sgr
Transition
Airborne Phase
Total Landing Distance
Float
sf
Flare stGlide sg
γ
hg
hf
Touchdown
( )
γγ
γ
γ
1/
tan
cos
tan
2
−−
=
−
=
−
=
ttggfg
g
nVhRhhh
s
28. 28
H.H. Hurt, Jr., “Aerodynamics for Naval Aviators “,NAVAIR 00=80T-80 1-1-1965, pg. 35
Return to Table of Content
29. 29
Fixed Wing Fighter Aircraft Flight Performance
SOLO
Level Flight
The forces acting on an airplane in Level Flight are
shown in Figure
0=
=
h
Vx
Lift and Drag Forces:
( ) TCkCSVCSVD
WCSVL
LDD
L
=+==
==
2
0
22
2
2
1
2
1
2
1
ρρ
ρ 2
2
VS
W
CL
ρ
=
+=
+=
SV
Wk
CSV
SV
Wk
CSVD DD 2
2
0
2
242
2
0
2 2
2
14
2
1
ρ
ρ
ρ
ρ
Lift
DragThrust
Weight
Equations of motion:
0
0
=−
=−
DT
WL
Quasi-Static
30. 30
Fixed Wing Fighter Aircraft Flight Performance
SOLO
Level Flight
DragInducedDragParasite
D
SV
Wk
CSVD 2
2
0
2 2
2
1
ρ
ρ +=
Because of opposite trends in
Parasite Drag and Induced Drag,
with changes in velocity, the Total
Drag assumes a minimum at a
certain velocity. If we ignore the
change in velocity of CD0 and k with
velocity we obtain
0
4
3
2
0 =−=
SV
Wk
CSV
Vd
Dd
D
ρ
ρ
The velocity of minimum Total
Drag is
*
4
0
2
V
C
k
S
W
V
D
==
ρ
We see that the velocity of minimum Total Drag is equal to the Reference Velocity.
0
2
2
1
DCSVρ
SV
Wk
2
2
2
ρ*
V
Lift
DragThrust
Weight
31. 31
Fixed Wing Fighter Aircraft Flight Performance
SOLO
Level Flight
For the velocity, V*, of minimum
Total Drag we have
02*
2
2
Di CkW
SV
Wk
D ==
ρ
DragInducedDragParasite
D
SV
Wk
CSVD 2
2
0
2 2
2
1
ρ
ρ +=
000min 2 DDD CkWCkWCkWD =+=
and
0
2
2
1
DCSVρ
SV
Wk
2
2
2
ρ
*
V
Lift
DragThrust
Weight
32. 32M. Corcoran, T. Matthewson, N. W. Lee, S. H. Wong, “Thrust Vectoring”
Comparison of Takeoff Weight and Empty Weight of different Aircraft
33. 33
Fixed Wing Fighter Aircraft Flight Performance
SOLO
Level Flight
The Power Required, PR, for Level Flight is
SV
Wk
CSVVDP DR
ρ
ρ
2
0
3 2
2
1
+=⋅=
The Power Required for Level Flight
assumes a minimum at a certain velocity Vmp.
If we ignore the change in velocity of CD0 and
k with velocity we obtain
0
2
2
3
2
2
0
2
=−=
SV
Wk
CSV
Vd
Pd
D
R
ρ
ρ
or
*
4
0 3
1
3
2
V
C
k
S
W
V
D
mp ==
ρ
*0
2, 3
32
L
D
mp
mpL C
k
C
VS
W
C ===
ρ
( )
*
000
0
2
,0
,
866.0
1
4
3
/3
/3
e
CkkCkC
kC
CkC
C
e
DDD
D
mpLD
mpL
mp ==
+
=
+
= *
2
min,
866.03
8
e
VW
SV
Wk
P mp
mp
R ==
ρ
0
3
2
1
DCSVρ
SV
Wk
ρ
2
2
*
3
1
V
min,RP
Lift
DragThrust
Weight
34. Fixed Wing Fighter Aircraft Flight Performance
SOLO
Available Aircraft Power and Thrust
• Throttle Effect
10 ≤≤= ηη ATT
• Propeller
airspeedwithvariationsmallVTP propellerA ≈⋅=,
V
Pa, propeller
Power
Propeller Aircraft Available Power
at Altitude (h)
At a given Altitude h
• Turbojet
airspeedwithvariationsmallT jetA ≈,
V
Ta, jet
Thrust
Jet Aircraft Available Power
at Altitude h
At a given Altitude h
Lift
DragThrust
Weight
Lift
DragThrust
Weight
Level Flight
35. 35
Fixed Wing Fighter Aircraft Flight Performance
SOLO
Vmin Vmax
Pa, propeller
PRPmin
BA
ηaPa, propeller
Propeller Aircraft
Vmin Vmax
Ta, jet
TR
Dmin
η Ta, jet
A
B
Jet Aircraft
Level Flight
To have a Level Flight the requirement must be satisfied by
the available propulsion performance.
•For a Propeller Aircraft, the available power Pa,propeller , at a
given altitude h, is almost insensitive with changes in velocity.
The Velocity in Level Flight is steady when the graph of
Required Power PR intersects the graph of Pa,propeller at points A
and B. We get two velocities Vmin (h) at A and Vmax (h) at B. By
controlling the Propeller Power ηa Pa,propeller (0< ηa <1) we can
reach any velocity between
Vmin (h) and Vmax (h).
•For a Jet Aircraft, the available Thrust Ta,jet , at a given
altitude h, is almost insensitive with changes in velocity. The
Velocity in Level Flight is steady when the graph of Required
Thrust TR intersects the graph of Ta,jet at points A and B. We
get two velocities Vmin (h) at A and Vmax (h) at B. By
controlling the Jet Thrust η Ta,jet (0< η<1) we can reach any
velocity between Vmin (h) and Vmax (h).
36. 36
Fixed Wing Fighter Aircraft Flight Performance
SOLO
Vmin Vmax
Ta, jet
TR
Dmin
η Ta, jet
A
B
Jet Aircraft
Level Flight
We have
Analytical Solution for Jet Aircraft
( ) SV
Wk
CSVCkCSVDT DLD 2
2
0
22
0
2 2
2
1
2
1
ρ
ρρ +=+==
Define
0
*
0
0
*
*
*
4
0
2
:
2*,*,:
2
:*,
*
:
D
DD
D
L
D
L
D
CkW
T
W
eT
z
CC
k
C
C
C
C
e
C
k
S
W
V
V
V
u
==
===
==
ρ
2
2
/1
2
0
0
2
2
0
2
2
2
u
D
u
Dz
D
V
C
k
S
W
C
k
S
W
V
T
CkW
ρ
ρ
+=
012 24
=+− uzu
Lift
DragThrust
Weight
37. 37
Fixed Wing Fighter Aircraft Flight Performance
SOLO
Vmin Vmax
Ta, jet
TR
Dmin
η Ta, jet
A
B
Jet Aircraft
Level Flight
Analytical Solution for Jet Aircraft
012 24
=+− uzu
Solving we obtain
1
1
2
max
2
min
−+=
−−=
zzu
zzu
4
0
maxmaxmax
4
0
minminmin
2
*
2
*
D
D
C
k
S
W
uVuV
C
k
S
W
uVuV
ρ
ρ
==
==
Lift
DragThrust
Weight
0
*
0
0
*
*
*
4
0
2
:
2*,*,:
2
:*,
*
:
D
DD
D
L
D
L
D
CkW
T
W
eT
z
CC
k
C
C
C
C
e
C
k
S
W
V
V
V
u
==
===
==
ρ
38. 38
Fixed Wing Fighter Aircraft Flight Performance
SOLO
Level Flight
Analytical Solution for Jet Aircraft
1
1
2
max
2
min
−+=
−−=
zzu
zzu
12
min −−= zzu
12
max −+= zzu
At the absolute Ceiling (when is only one possible velocity) we have umax = umin, therefore
z = 1.
max,
2
L
stall
CS
W
V
ρ
=
Lift
DragThrust
Weight
0
*
0
0
*
*
*
4
0
2
:
2*,*,:
2
:*,
*
:
D
DD
D
L
D
L
D
CkW
T
W
eT
z
CC
k
C
C
C
C
e
C
k
S
W
V
V
V
u
==
===
==
ρ
40. 40
Fixed Wing Fighter Aircraft Flight Performance
SOLO
Level Flight
Aircraft Range in Level Flight
Lift
DragThrust
Weight
Range in Level Flight of Jet Aircraft
Equations of motion:
0
0
=−
=−
DT
WL
0=
=
h
Vx
We add the equation of fuel consumption
TcW −=
c – specific fuel consumption
We assume that fuel consumption is constant for a given altitude.
V
td
Wd
Wd
xd
td
xd
==
Dc
V
Tc
V
W
V
Wd
xd DT
−=−==
=
41. 41
Fixed Wing Fighter Aircraft Flight Performance
SOLO
Level Flight
Aircraft Range in Level Flight
Lift
DragThrust
Weight
Range in Level Flight of Jet Aircraft
Dc
V
Wd
xd
−=
The quantity dx/dW is called the “Instantaneous Range”
and is equal to the Horizontal Range traveled per unit load
of fuel or the “Specific Range”.
Multiply and divide by L = W
Wc
V
C
C
Wc
V
D
L
Wd
xd
D
L
−=
−=
Integrating we obtain
∫
−=−=
f
i
W
W
D
L
if
W
Wd
V
cC
C
xxR
1
:
42. 42
Fixed Wing Fighter Aircraft Flight Performance
SOLO
Level Flight
Aircraft Range in Level Flight
Lift
DragThrust
Weight
Range in Level Flight of Jet Aircraft
To perform the integration we must specify the variation
of CL, CD and V. Let consider two cases:
∫
−=−=
f
i
W
W
D
L
if
W
Wd
V
cC
C
xxR
1
:
a. Range at Constant Altitude of Jet Aircraft
We have LCVSLW 2
2
1
ρ==
LCS
W
V
ρ
2
=
The velocity changes (decreases) since the weight W decreases due to fuel
consumption.
[ ]if
D
L
W
W
D
L
WW
C
C
cW
Wd
ScC
C
R
f
i
−
=
−= ∫
221
ρ
43. 43
Fixed Wing Fighter Aircraft Flight Performance
SOLO
Level Flight
Aircraft Range in Level Flight
Lift
DragThrust
Weight
a. Range at Constant Altitude of Jet Aircraft
[ ]if
D
L
W
W
D
L
WW
C
C
cW
Wd
ScC
C
R
f
i
−
=
−= ∫
221
ρ
The maximum range is obtained when
[ ]if
D
L
WW
C
C
c
R −
=
max
max
2
max
2
0max
+
=
LD
L
D
L
CkC
C
C
C
( )
030
2
2
1
2
022
0
2
0
2
0
=−⇒=
+
−
+
=
+
LD
LD
LL
L
LD
LD
L
L
CkC
CkC
CkC
C
CkC
CkC
C
Cd
d
The maximum range is obtained when
*0
3
1
3
1
L
D
L C
k
C
C ==
The Velocity at maximum range is ( ) ( ) ( ) ( )tV
CS
tW
CS
tW
tV
LL
*4
*
4
*
3
2
3
3/
2
===
ρρ
44. 44
Fixed Wing Fighter Aircraft Flight Performance
SOLO
Level Flight
Aircraft Range in Level Flight
Lift
DragThrust
Weight
b. Range at Constant Velocity of Jet Aircraft
=
−= ∫ f
i
D
L
W
W
D
L
W
W
C
C
c
V
W
Wd
V
cC
C
R
f
i
ln
1
The Velocity V is constant and equal to V* corresponding to initial weight Wi.
4
0
*
* 22
D
i
L
i
C
k
S
W
CS
W
V
ρρ
==
The maximum range is obtained when
=
=
f
i
f
i
D
L
W
W
e
c
V
W
W
C
C
V
c
R lnln
1 *
*
max
max
To keep Velocity V constant when weight W decreases, the air density ρ must
also decrease, hence the Aircraft will gain (qvasistatic) altitude
( ) Pc
td
Wd
td
hd
e
td
hd
p
hh
−==−= − 0/
0ρ
ρ
45. 45
Fixed Wing Fighter Aircraft Flight Performance
SOLO
Level Flight
Aircraft Range in Level Flight
Range in Level Flight of Propeller Aircraft
Lift
DragThrust
Weight
The equation of fuel consumption
PcW P−=
cp – specific fuel consumption (consumed per unit power developed by the engine per
unit time
We assume that fuel consumption is constant for a given altitude.
V
td
Wd
Wd
xd
td
xd
==
Pc
V
W
V
Wd
xd
p
−==
- Required PowerVDPR ⋅=
PP pA ⋅=η - Available Power
ηp – propulsive efficiency
AR PP =
p
VD
P
η
⋅
=
46. 46
Fixed Wing Fighter Aircraft Flight Performance
SOLO
Level Flight
Aircraft Range in Level Flight
Range in Level Flight of Propeller Aircraft
Lift
DragThrust
Weight
WcC
C
WcD
L
DcPc
V
Wd
xd
p
p
D
L
p
p
WL
p
p
p
ηηη
−=−=−=−=
=
Integration gives ∫
−=−=
f
i
W
W
p
p
D
L
ff
W
Wd
cC
C
xxR
η
:
We assume
• Angle of Attack is kept constant throughout cruise, therefore e = CL/CD is
constant
•ηp is independent on flight velocity
f
i
p
p
W
W
e
c
R ln
η
= Bréguet Range Equation
The maximum range of Propeller Aircraft in Level Flight is
f
i
Dp
p
f
i
p
p
W
W
CkcW
W
e
c
R ln
2
1
ln
0
*
max
ηη
==
47. 47
Louis Charles Bréguet
(1880 – 1955)
The Bréguet Range Equation
The Bréguet range equation determines the maximum flight
distance. The key assumptions are that SFC, L/D, and flight speed,
V are constant, and therefore take-off, climb, and descend portions
of flights are not well modeled (McCormick, 1979; Houghton,
1982).
( )
⋅
=
final
initial
W
W
SFCg
DLV
Range ln
/
Winitial = Wfuel + Wpayload + Wstructure + Wreserve
Wfinal = Wpayload + Wstructure + Wreserve
where
( )
++
+
⋅
=
reservestructurepayload
fuel
WWW
W
SFCg
DLV
Range 1ln
/
where SFC, L/D, and Wstructure are technology parameters while Wfuel, Wpayload, and Wreserve are
operability parameters.
Fixed Wing Fighter Aircraft Flight Performance
SOLO
48. 48
Fixed Wing Fighter Aircraft Flight Performance
SOLO
Level Flight
Aircraft Range in Level Flight
Range in Level Flight of Propeller Aircraft
Lift
DragThrust
Weight
Let assume that the flight to maximum range is
performed in one of two ways
1. Propeller Aircraft Flight at Constant Altitude
In Constant Altitude Flight the velocity changes with the decrease of weight such that
( ) ( ) 4
0
* 2
DC
k
S
tW
VtV
ρ
==
2. Propeller Aircraft Flight with Constant Velocity
In Constant Velocity Flight the velocity is the V* velocity based on the initial weight
of the Aircraft
.
2
4
0
*
const
C
k
S
W
VV
D
i
===
ρ
49. 49
Fixed Wing Fighter Aircraft Flight Performance
SOLO
Level Flight
Aircraft Endurance in Level Flight
Lift
DragThrust
Weight
The Endurance of an Airplane remains in the air and is
usually expressed in hours.
Endurance of Jet Aircraft in Level Flight
We have TcW −=
c – specific fuel consumption
W
Wd
c
e
W
Wd
D
L
cDc
Wd
Tc
Wd
td
WLDT
−=−=−=−=
== 1
Integrating we obtain
∫−=
f
i
W
W W
Wd
c
e
t
Assuming that the Angle of Attack is held constant throughout the flight, e =CL/CD is constant
f
i
W
W
c
e
t ln=
f
i
Df
i
W
W
CkcW
W
c
e
t ln
2
1
ln
0
*
max ==
The Maximum Endurance for Jet Aircraft occurs for e = e*, CL = CL*, V = V*, D = Dmin.
50. 50
Fixed Wing Fighter Aircraft Flight Performance
SOLO
Level Flight
Aircraft Endurance in Level Flight
The Endurance of an Airplane remains in the air and is
usually expressed in hours.
Endurance of Propeller Aircraft in Level Flight
We have ppp VDcPcW η/⋅−=−=
W
Wd
V
e
cW
Wd
VD
L
cVD
Wd
c
td
p
p
p
p
WL
p
p 11 ηηη
−=−=
⋅
−=
=
Assuming that the Angle of Attack is held constant throughout the flight, e =CL/CD is constant
Lift
DragThrust
Weight
cp – specific fuel consumption (consumed per unit power
developed by the engine per unit time.
ηp – propulsive efficiency
Integrating we obtain
∫−=
f
i
W
W
p
p
W
Wd
V
e
c
t
1η
The Endurance of Propeller Aircraft depends on Velocity, therefore we will assume two cases
1.Flight at Constant Altitude
2.Flight with Constant Velocity
51. 51
Fixed Wing Fighter Aircraft Flight Performance
SOLO
Level Flight
Endurance of Propeller Aircraft in Level Flight
Lift
DragThrust
Weight
The velocity will change to compensate for the decrease in weight
∫ =−=
f
i
W
W
D
L
p
p
C
C
e
W
Wd
V
e
c
t
1η
1. Propeller Aircraft Flight at Constant Altitude
We have LCVSLW 2
2
1
ρ==
LCS
W
V
ρ
2
=
−
=
ifD
L
p
p
WW
S
C
C
c
t
11
2
2 2/3
ρη
For Maximum Endurance Propeller Aircraft has to fly at that Angle of Attack such that
(CL
3/2
/CD) is maximum, which occurs when CL=√3 CL
*
and V = 0.76 V*
.
−
=
ifDp
p
WW
S
Ckc
t
11
2
27
4
12
0
3max
ρη
52. 52
Fixed Wing Fighter Aircraft Flight Performance
SOLO
Level Flight
Endurance of Propeller Aircraft in Level Flight
Lift
DragThrust
Weight
∫ =−=
f
i
W
W
D
L
p
p
C
C
e
W
Wd
V
e
c
t
1η
2. Propeller Aircraft Flight with Constant Velocity
f
i
p
p
W
W
V
e
c
t ln
1η
=
For Maximum Endurance Propeller Aircraft has to fly at a velocity such that e=(CL/CD) is
maximum, which occurs when CL=CL
*
and V = V*
, which is based on initial weight Wi
4
0
*
* 22
D
i
L
i
C
k
S
W
CS
W
V
ρρ
==
0
*
2
1
DCk
e =
f
i
D
i
p
p
f
i
D
i
Dp
p
f
i
p
p
W
W
CkS
W
cW
W
C
k
S
W
CkcW
W
V
e
c
t ln
1
2
ln
2
2
1
ln
1
4
3
0
4
00
*
*
max
ρ
η
ρ
ηη
===
53. 53
D=TR
V
V*
tmax
Slope min(PR/V)
Bréguet
Velocities for Maximum Range and Maximum
Endurance of Propeller Aircraft
Fixed Wing Fighter Aircraft Flight Performance
SOLO
Graphical Finding of Maximum Range and
Endurance of Jet Aircraft in Level Flight
=
⇔
V
D
D
V
R
VV
minmaxmax
Maximum Range
From Figure we can see that min (D/V) is
obtained by taking the tangent to D
graph that passes through origin.
The point of tangency will give D and V
for (D)min.
Maximum Endurance
∫∫
<
=
<
−=−=
00
111
Wd
Dc
Wd
Tc
t
DT
( )D
D
t
VV
min
1
maxmax =
⇔
From Figure we can see that min (PR) is
obtained by taking the PR and V for (PR)min.
Lift
DragThrust
Weight
∫∫
<
−==
0
Wd
Dc
V
xdR
54. 54
PR
V
V*
Rmax
0.866 V*
tmax
Slope min(PR/V)
Velocities for Maximum Range and Maximum
Endurance of Propeller Aircraft
Lift
DragThrust
Weight
Fixed Wing Fighter Aircraft Flight Performance
SOLO
Graphical Finding of Maximum Range and
Endurance of Propeller Aircraft in Level Flight
∫∫∫∫
>>>
⋅
−=−=−==
000
Wd
VD
V
c
Wd
P
V
c
Wd
Pc
V
xdR
p
p
Rp
p
p
ηη
D
V
P
P
V
R
V
R
V
R
V
minminmaxmax =
=
⇔
Maximum Range
From Figure we can see that min (PR/V)
is obtained by taking the tangent to PR
graph that passes through origin.
The point of tangency will give PR and V
for (PR/V)min.
Maximum Endurance
∫∫
<<
−=−=
00
11
Wd
Pc
Wd
Pc
t
Rp
p
p
η
( ) ( )VDP
P
t
V
R
V
R
V
⋅==
⇔ minmin
1
maxmax
From Figure we can see that min (PR) is
obtained by taking the PR and V for (PR)min.
58. 58
Flight Ceiling by the
available Climb Rate
- Absolute 0 ft/min
- Service 100 ft/min
- Performance 200 ft/min
True Airspeed
Altitude
Absolute Ceiling
Service Ceiling
Performance Ceiling
Excess Thrust
provides the ability
to accelerate or climb
True Airspeed
Thrust Available
Thrust
Required
Thrust
True Airspeed
Thrust
Available
Thrust
Required
Thrust
A AB B
C D
E
E
Thrust
True Airspeed
Available
Thrust
Required
Thrust
C D
Jet Aircraft Flight Envelope Determined by Available Thrust
Flight Envelope: Encompasses all Altitudes
and Airspeeds at which Aircraft can Fly
Stengel, MAE331, Lecture 7, Gliding, Climbing and Turning Performance
Fixed Wing Fighter Aircraft Flight Performance
SOLO
Lift
DragThrust
Weight
Changes in Jet Aircraft
Thrust with Altitude
59. 59
Propeller Aircraft Ceiling Determined by Available Power
To find graphically the maximum Flight Altitude (Ceiling) for a Propeller Aircraft we use
the PR (Power Required) versus V (Velocity) graph. The maximum Flight Altitude
corresponds to maximum Range Rmax.
We have shown that to find Rmax we draw the Tangent Line to PR Graph, passing trough
the origin.
Fixed Wing Fighter Aircraft Flight Performance
SOLO
Lift
DragThrust
Weight
Changes in Propeller Aircraft Power
and Thrust with Altitude
VC
Pa, propeller
PR
hcruise
A
h2
h1
h0
h0 < h1 <h2 < hcruise
The intersection point A with PR Graph defines the Ceiling Velocity VC, and the Pa
(Available Power – function of Altitude) with this point defines the Ceiling Altitude.
Return to Table of Content
60. 60
Fixed Wing Fighter Aircraft Flight Performance
SOLO
Gliding Flight
A Glider is an unpowered airplane.
0
sin
cos
=
=
=
W
Vh
Vx
γ
γ
1<<γ
0=+
=
γWD
WL
.constW
Vh
Vx
=
=
=
γ
Lift and Drag Forces:
( ) γρρ
ρ
WCkCSVCSVD
WCSVL
LDD
L
−=+==
==
2
0
22
2
2
1
2
1
2
1
LCS
W
V
ρ
2
=
eC
C
L
D
W
D
L
D
LW 1
−=−=−=−=
=
γ
Equations of motion:
0sin
0cos
=+
=−
γ
γ
WD
WLQuasi-Steady
Flight
61. 61
Fixed Wing Fighter Aircraft Flight Performance
SOLO
Gliding Flight
We found
LCS
W
V
ρ
2
= eC
C
L
D
W
D
L
D
LW 1
−=−=−=−=
=
γ
Flattest Glide” (γ = γmin)
The Flattest Glide (γ = γmin) is given by:
0
*
max
min
min 22
1
DL CkCk
eW
D
−=−=−=−=γ
e
LC*LC
*2
1
LCk
CL/CD as a function of CL
k
C
C D
L
0
* =
The flight velocity for the Flattest Glide is given by:
4
0
*..
2
*
2
DL
GF
C
k
S
W
V
CS
W
V
ρρ
===
The flight velocity for the Flattest Glide is equal to the reference velocity V*
or u = 1.
The Flattest Glide is conducted at constant dynamic pressure.
.
2
1
0
*
2
.. const
C
k
W
C
W
VSq
DL
GFG ==== ρ
62. 62
H.H. Hurt, Jr., “Aerodynamics for Naval Aviators “,NAVAIR 00=80T-80 1-1-1965, pg. 35
Gliding Flight
CLEAR CONFIGURATION
LANDING CONFIGURATION
LIFT to DRAG
RATIO
L/D
(L/D)max
LIFT COEFFICIENT, CL
CLEAR CONFIGURATION
LANDING CONFIGURATION
RATEOF
SINK
VELOCITY
(L/D)max
TANGENT TO RATE OF SINK
GRAPH AT THE ORIGIN
Gliding Performance
Fixed Wing Fighter Aircraft Flight Performance
SOLO
63. 63
Fixed Wing Fighter Aircraft Flight Performance
SOLO
Gliding Flight
We have:
Distance Covered with respect to Ground
The maximum Ground Range is covered for the Flattest Glide at the reference
velocity V*
or u = 1.
γV
td
hd
V
td
xd
=
=
D
L
e
V
V
hd
xd
−=−===
γγ
1
Assuming a constant Angle of Attack during Glide, e is constant and the Ground
Range R, to descend from altitude hi to altitude hf is given by:
( ) hehhehdexxR fi
h
h
if
f
i
∆=−=−=−= ∫:
and
0
maxmax
2 DCk
h
heR
∆
=∆=
e
LC*LC
*2
1
LCk
CL/CD as a function of CL
64. 64
Fixed Wing Fighter Aircraft Flight Performance
SOLO
Gliding Flight
Rate of sink is defined as:
Rate of Sink
==
⋅
=−=−=
=
=
2/3
2
22
L
D
L
D
L
L
D
W
D
CS
W
V
s
C
C
S
W
C
C
CS
W
W
VD
V
td
hd
h
L
ρρ
γ
ρ
The term DV = PR represents the Power Required to sustain the Gliding Flight.
Therefore the Rate of Sink is minimum when the Power Required is minimum, or
(CD/CL
3/2
) is minimum
( ) ( ) 0
2
3
2
342
3
2
2/5
0
2
2/5
2
0
2
3
2
0
2/12/3
2/3
2
0
2/3
=
−
=
+−
=
+−
=
+
=
L
DL
L
LDL
L
LDLLL
L
LD
LL
D
L C
CCk
C
CkCCk
C
CkCCCCk
C
CkC
Cd
d
C
C
Cd
d
Denote by CL,m the value of Lift Coefficient CL for which (CD/CL
3/2
) is minimum
*0
, 3
3
0*
L
k
C
C
D
mL C
k
C
C
D
L =
== 27
4
3
3
0
3
2/3
0
0
0
min
2/3
D
D
D
D
L
D Ck
k
C
k
C
kC
C
C
=
+
=
65. 65
Fixed Wing Fighter Aircraft Flight Performance
SOLO
Gliding Flight
Rate of Sink
*0
, 3
3
0*
L
k
C
C
D
mL C
k
C
C
D
L =
==
0
3
max
2/3
27
4
1
DD
L
CkC
C
=
We found:
The velocity Vm for glide with minimum
sink rate is given by:
*
4
0
76.0~
4
4
0,
76.0
2
3
1
3
22
*
V
C
k
S
W
C
k
S
W
CS
W
V
V
D
DmL
m
≈
=
==
ρ
ρρ
S
CkW
C
C
S
W
h D
L
D
s
ρρ 27
22 0
3
min
2/3min, =
=
The minimum sink rate is given by:
66. 66
Fixed Wing Fighter Aircraft Flight Performance
SOLO
Gliding Flight
Endurance
The Endurance is the total time the glider remains in the
air.
Minimum
Sink Rate
tmax
Flatest
Glide
Rmax
−== 2/3
2
L
D
C
C
S
W
V
td
hd
ρ
γ
−==
D
L
C
C
W
S
V
hd
td
2/3
2
ρ
γ
( )fi
D
L
h
h
D
L
hh
C
C
W
S
hd
C
C
W
S
t
f
i
−
=
−= ∫
2/32/3
22
ρρ
Assuming that the Angle of Attack is held constant during
the glide and ignoring the variation in density as function
of altitude, we have
For Maximum Endurance the Glider has to fly at that
Angle of Attack such that (CL
3/2
/CD) is maximum, which
occurs when CL=√3 CL
*
and V = 0.76 V*
.
−
=
4
27
2
4
0
3max
fi
D
hh
CkW
S
t
ρ
Return to Table of Content
67. 67
Performance of an Aircraft with Parabolic PolarSOLO
W
LT
n
+
=
αsin
:'
W
L
n =:
2
0
:
LD
L
D
L
CkC
C
CSq
CSq
D
L
e
+
===
We assume a Parabolic Drag Polar:
2
0 LDD CkCC +=
Let find the maximum of e as a function of CL
( ) ( )
0
2
22
0
2
0
22
0
22
0
=
+
−
=
+
−+
=
∂
∂
LD
LD
LD
LLD
L CkC
CkC
CkC
CkCkC
C
e e
LC*LC
*2
1
LCk
CL/CD as a function of CL
The maximum of e is obtained for
k
C
C D
L
0
* =
( ) 0
0
0
2
0 2** D
D
DLDD C
k
C
kCCkCC =+=+=
Start with
Load Factor
Total Load Number
Lift to Drag Ratio
Climbing Aircraft Performance
68. 68
Performance of an Aircraft with Parabolic PolarSOLO
e
LC*LC
*2
1
LCk
CL/CD as a function of CL
The maximum of e is obtained for
k
C
C D
L
0
* =
( ) 0
0
0
2
0 2** D
D
DLDD C
k
C
kCCkCC =+=+=
*2
1
*2
1
2
1
2*
*
*
22
00
0
LLDD
D
D
L
CkCkCkC
k
C
C
C
e =====
We have WnCSVCSqL LL === 2
2
1
ρ
Let define for n = 1
=
=
==
2
4
0
*
2
1
:*
*
:
2
*
2
1
:*
Vq
V
V
u
C
k
S
W
CS
W
V
D
L
ρ
ρρ
2
0
:
LD
L
D
L
CkC
C
CSq
CSq
D
L
e
+
===
Climbing Aircraft Performance
69. 69
Performance of an Aircraft with Parabolic PolarSOLO
Using those definitions we obtain
L
L
L
L
C
C
nqq
WCSq
WnCSqL *
*
**
=→
=
==
2
2
2
1
2
1
*
2
1
*
uV
V
n
q
q
==
ρ
ρ
2
*
*
*
u
C
nC
q
q
nC L
LL ==
( )
+=
+=
+=+=
=
2
2
2
04
02
0
2
*
4
2
2
0
22
0
**
*
*
0
2
u
n
uCSq
u
C
nCuSq
u
C
nkCuSqCkCSqD
D
D
D
CCk
L
DLD
DL
=
=
=
2
0
*
2
1
:*
*
:
2
:*
Vq
V
V
u
CS
kW
V
D
ρ
ρ
*2
1
*
*** 0
0
e
W
C
C
CSqCSq
L
D
LD ==
+= 2
2
2
*2 u
n
u
e
W
D
Therefore
Return to Table of Content
Climbing Aircraft Performance
70. 70
Performance of an Aircraft with Parabolic PolarSOLO
We obtained
+= 2
2
2
*2 u
n
u
e
W
D
u 0
- - - - 0 + + + + +
D ↓ min ↑
n
u
D
∂
∂
Let find the minimum of D as function of u.
nu
u
nu
e
W
u
n
u
e
W
u
D
=→
=
−
=
−=
∂
∂
2
3
24
3
2
0
*
22
*2
*
2min
e
Wn
DD nu
== =
Aircraft Drag
Climbing Aircraft Performance
=
=
=
2
0
*
2
1
:*
*
:
2
:*
Vq
V
V
u
CS
kW
V
D
ρ
ρ
71. 71
Performance of an Aircraft with Parabolic PolarSOLO
Aircraft Drag
( )
MAXn
W
VhL
n ≤=
,
+== 2
2
2
*2 u
n
u
e
W
D MAX
nn MAX
Maximum Lift Coefficient or Maximum Angle of Attack
( ) ( ) ( )VhorMCMC STALLMAXLL ,, _ ααα ≤≤
We have
u
C
C
u
n
u
C
nC
q
q
nC
L
MAXL
CC
L
LL
MAXLL
*
*
*
* _
2
_
=→==
=
2
2
_
2
2
_2
*
1
*2
**2_
u
C
C
e
W
u
C
C
u
e
W
D
L
MAXL
L
MAXL
CC MAXLL
+=
+==
Maximum dynamic pressure limit
( ) ( ) MAX
MAX
MAXMAX u
V
V
uhVVorqVhq =<→≤≤= :
*2
1 2
ρ
*e
W
D
MAXLC _
2
2
_
1
2
1
u
C
C
L
MAXL
+
+= 2
2
2
2
1
*
u
n
ue
W
D MAX
LIMIT
nn MAX=
2min
* ue
W
D
=
+= 2
2
2
2
1
*
u
n
ue
W
D
MAXuu =MAX
MAXL
L
CORNER n
C
C
u
_
*
=
n
LIMIT
u
MAXnu =
as a function of u*e
W
D
Maximum Load Factor
Climbing Aircraft Performance
=
=
=
2
0
*
2
1
:*
*
:
2
:*
Vq
V
V
u
CS
kW
V
D
ρ
ρ
72. 72
Performance of an Aircraft with Parabolic PolarSOLO
Energy per unit mass E
Let define Energy per unit mass E:
g
V
hE
2
:
2
+=
Let differentiate this equation:
( ) ( )
W
VDT
W
VDT
W
DT
g
g
V
V
g
VV
hEps
−
≈
−
=
−
−
+=+==
α
γ
α
γ
cos
sin
cos
sin:
*&
*2 2
2
2
VuV
u
n
u
e
W
D =
+=
Define *: e
W
T
z
=
We obtain ( )
+−=
+−
=
−
= 2
2
2
2
2
2
2
1
*
*
*
2
1
*
*
u
n
uzu
e
V
W
Vu
u
n
ue
W
T
e
W
W
VDT
ps
or ( )
u
nuzu
e
V
ps
224
2
*2
* −+−
=
020 224
=+−→==
nuzup constns
( ) ( )
2
224
2
2243
23
*
*244
*
*
u
nuzu
e
V
u
nuzuuuzu
e
V
u
p
constn
s ++−
=
−+−−+−
=
∂
∂
=
0=
∂
∂
=constn
s
u
p 2
21
2
uu
uu
MAX <<
+
nz >
Climbing Aircraft Performance
nz
nzzu
nzzu
>
−+=
−−=
22
2
22
1
3
3 22
nzz
uMAX
++
=
=
=
=
2
0
*
2
1
:*
*
:
2
:*
Vq
V
V
u
CS
kW
V
D
ρ
ρ
73. 73
Performance of an Aircraft with Parabolic PolarSOLO
Energy per unit mass E
sp
2u1u
MAXu
2
21 uu + u
MAXn
n
1=n
( )
u
nuzu
e
V
ps
224
2
*
* −+−
=
ps as a function of u
u
V
pe
uzunnuzuu
V
pe ss
*
*2
22
*
*2 242224
−+−=→−+−=
From which u
V
pe
uzun s
*
*2
2 24
−+−=
( )
*
*2
44 3
2
V
pe
uzu
u
n s
constps
−+−=
∂
∂
=
( )
3
0412 2
2
22
z
uzu
u
n
constps
=→=+−=
∂
∂
=
( )
u
nuzu
e
V
ps
224
2
*2
* −+−
=
Climbing Aircraft Performance
=
=
=
2
0
*
2
1
:*
*
:
2
:*
Vq
V
V
u
CS
kW
V
D
ρ
ρ
74. 74
Performance of an Aircraft with Parabolic PolarSOLO
Load Factor n
u
3
z z
z2z
3
z
u
2
n
0=sp
0>sp
0<sp
0<sp
0=sp
0>sp
( )
u
n
∂
∂ 2
( )
2
22
u
n
∂
∂
3
z
u
( ) ( ) 2
2
2
22
,, n
u
n
u
n
∂
∂
∂
∂ as a function of u
( )
3
0412 2
2
22
z
uzu
u
n
constps
=→=+−=
∂
∂
=
( )
*
*2
44 3
2
V
pe
uzu
u
n s
constps
−+−=
∂
∂
=
Integrating once
u
V
pe
uzun s
*
*2
2 24
−+−=
Integrating twice
Climbing Aircraft Performance
=
=
=
2
0
*
2
1
:*
*
:
2
:*
Vq
V
V
u
CS
kW
V
D
ρ
ρ
75. 75
Performance of an Aircraft with Parabolic PolarSOLO
Load Factor n
For ps = 0 we have
zuuzun 202 24
≤≤+−=
Let find the maximum of n as function of u.
0
22
44
24
3
=
+−
+−
=
∂
∂
uzu
uzu
u
n
Therefore the maximum value for n is
achieved for zu =
( ) zn
MAXps
==0
u 0 √z √2z
∂ n/∂u | + + + 0 - - - - | - -
n ↑ Max ↓
z2z
u
n
0=sp
0>sp
0<sp
MAXn
z
MAX
MAXL
L
n
C
C
_
*
n as a function of u
Climbing Aircraft Performance
=
=
=
2
0
*
2
1
:*
*
:
2
:*
Vq
V
V
u
CS
kW
V
D
ρ
ρ
76. Performance of an Aircraft with Parabolic PolarSOLO
Energy per unit mass E
g
V
hE
2
:
2
+=
Climbing Aircraft Performance
Energy Height versus Mach NumberEnergy Height versus True Airspeed
( )hV
V
M
sound
=:( )
00
:
T
T
V
T
T
MhVTAS sound ==
Return to Table of Content
77. 77
Performance of an Aircraft with Parabolic PolarSOLO
Steady Climb (V, γ = constant)
Climbing Aircraft Performance
0sin
0cos
==−−
==−
td
Vd
g
W
WDT
td
d
V
g
W
WL
γ
γ
γ
Equation of Motion for Steady Climb:
γ
γ
sin
cos
Vh
Vx
=
=
Define the Rate of Climb:
( )
s
Ra
C p
W
PP
W
DTV
Vh =
−
=
−⋅
== γsin
where
Pa = V T - available power
PR = V D - required power
ps - excess power per unit weight
Weight
ThrustExcess
W
DT
=
−
=γsin
C
C
WL
const
γ
γγ
cos
.
=
==
Lift
Drag
Thrust
Weight
78. 78
Performance of an Aircraft with Parabolic PolarSOLO
Climbing Aircraft Performance
LC CSVW 2
2
1
cos ργ =
( )
s
C
D
LD
C p
SV
W
kCSVVT
WW
CkCSVVT
h =
−−=
+−
=
ρ
γ
ρ
ρ
2
1
cos
2
112
1
22
0
3
2
0
3
Let find the velocity V for which the Rate of Climb is maximum, for the Propeller Aircraft:
0
cos2
2
31
2
22
0
2
=
+−==
SV
Wk
CSV
Wtd
pd
td
hd C
D
sC,Prop
ρ
γ
ρ
Steady Climb (V, γ = constant)
For a Propeller Aircraft we assume that Pa=T V= constant.
or **
4
4
0
4
76.0
3
12
3
1
VV
C
k
S
W
V
D
Climb.Prop ===
ρ
s
C
DaPropC p
SV
W
kCSVP
W
h =
−−=
ρ
γ
ρ
22
0
3
,
cos
2
2
11
We can see that the velocity at which the Rate of Climb of Propeller Aircraft is maximum
is the same as the velocity at which the Required Power in Level Flight is maximum.
Lift
Drag
Thrust
Weight
79. 79
Performance of an Aircraft with Parabolic PolarSOLO
Climbing Aircraft Performance
LC CSVW 2
2
1
cos ργ =
( )
−−=
+−
=
SV
W
kCSVVT
WW
CkCSVVT
h C
D
LD
C
ρ
γ
ρ
ρ
2
1
cos
2
112
1
22
0
3
2
0
3
Let find the velocity V for which the Rate of Climb is maximum, for the Jet Aircraft:
0
cos2
2
31
2
22
0
2
=
+−=
SV
Wk
CSVT
Wtd
hd C
D
C
ρ
γ
ρ
Steady Climb (V, γ = constant)
For a Jet Aircraft we assume that T = constant.
Define
0
*
0
0
*
*
*
4
0
2
:2*,*,:
2
:*,
*
:
D
DD
D
L
D
L
D CkW
T
W
eT
zCC
k
C
C
C
C
e
C
k
S
W
V
V
V
u =======
ρ
0cos
2
2
3
2 2
/1
2
0
0
2
2
0
2
2
=+− C
u
D
u
Dz
D
V
C
k
S
W
C
k
S
W
V
T
CkW
γ
ρ
ρ
0cos23 224
=−− Cuzu γ
Czzu γ22
cos3++=
80. 80
Performance of an Aircraft with Parabolic PolarSOLO
Climbing Aircraft Performance
Steady Climb (V, γ = constant)
ps versus the nondimensional velocity u
ps versus the velocity V
0sin ==−−
td
Vd
g
W
WDT γ
1
2
2
2
*2 =
+=
n
u
n
u
e
W
D
Define
0
*
0
0
*
*
*
4
0
2
:2*,*
,:
2
:*,
*
:
D
DD
D
L
D
L
D
CkW
T
W
eT
zCC
k
C
C
C
C
e
C
k
S
W
V
V
V
u
====
===
ρ
+−==
−
= 2
2
*
1
2
2
1
sin
u
uz
eV
p
W
DT s
γ
To find the maximum γ we must have
0
2
2
2
1sin
3*
=
−−=
u
u
eud
d γ
4
0
2
*max
DC
k
S
W
VV
ρ
γ ==
( ) ( )1
*
*2
*2
*
1
1
224
, max
−=
−+−
=
=
=
z
e
V
u
nuzu
e
V
p
u
n
s γ *
,
max
1
sin
max
max
e
z
V
ps −
==
γ
γ
γ
1max
=γu
81. SOLO
81
Aircraft Flight Performance
Construction of the Specific Excess Power contours ps in the
Altitude-Mach Number map for a Subsonic Aircraft below the
Drag-divergence Mach Number.
These contour are constructed for a fixed load factor W/S and
Thrust factor T/S, if the load or thrust factor change, the ps
contours will shift.
( ) ( )
W
VDT
W
VDT
W
DT
g
g
V
V
g
VV
hEps
−
≈
−
=
−
−
+=+==
α
γ
α
γ
cos
sin
cos
sin:
In Figure (a) is a graph of Specific Excess Power contours ps
versus Mach Number. Each curve is for a specific altitude h.
In Figure (b) each curve is for a given Specific Excess Power ps
in Altitude versus Mach Number coordinates.
The points a, b, c, d, e, f for ps = 0 in Figure (a) are plotted on
the curve for ps = 0 in Figure (b).
Similarly all points ps = 200 ft/sec in Figure (a) on the line AB
are projected on the curve ps = 200 ft/sec in Figure (b).
Specific Excess Power
contours ps for a Subsonic
Aircraft
Specific Excess Power contours ps
82. SOLO
82
Aircraft Flight Performance
Specific Excess Power contours ps for a Supersonic Aircraft
In the graphs of Specific Excess
Power ps versus Mach Number
Figure (a) for a Supersonic
Aircraft we see a “dent” in h
contour in the Transonic
Region. This is due to the
increase in Drag in this region.2
In Figure (b) the graphs of
Altitude versus Mach Number
we see a “closed” ps = 400 ft/sec
contour due to the increase in
Drag in this Transonic Region.
Specific Excess Power contours ps
( ) ( )
W
VDT
W
VDT
W
DT
g
g
V
V
g
VV
hEps
−
≈
−
=
−
−
+=+==
α
γ
α
γ
cos
sin
cos
sin:
Return to Table of Content
83. 83
Performance of an Aircraft with Parabolic PolarSOLO
Climbing Aircraft Performance
Optimum Climbing Trajectories using Energy State Approximation (ESA)
We defined the Energy per unit mass E (Specific Energy):
g
V
hE
2
:
2
+=
Differentiate this equation:
( ) ( )
W
VDT
W
VDT
W
DT
g
g
V
V
g
VV
h
td
Ed
ps
−
≈
−
=
−
−
+=+==
α
γ
α
γ
cos
sin
cos
sin:
Minimum Time-to-Climb
The time to reach a given Energy Height Ef is computed as follows
E
Ed
td
= ∫=
fE
E
f
E
Ed
t
0
The minimum time to reach the given Energy Height Ef is obtained by using
at each level.
( )∫=
fE
E
f
E
Ed
t
0
max
max,
( )maxE
84. 84
Performance of an Aircraft with Parabolic PolarSOLO
Climbing Aircraft Performance
Optimum Climbing Trajectories using Energy State Approximation (ESA)
Minimum Time Climb Profiles for Subsonic Speed
( ) ( )
W
VDT
W
VDT
W
DT
g
g
V
V
g
VV
hEps
−
≈
−
=
−
−
+=+==
α
γ
α
γ
cos
sin
cos
sin:
Stengel, MAE331, Lecture 7, Gliding,
Climbing and Turning Performance
The minimum time to reach the given Energy Height Ef is obtained by using at
each level.
( )maxE
Energy can be converted from potential to kinetic or vice versa along lines of constant
energy in zero time with zero fuel expended. This is physically not possible so the
method gives only an approximation of real paths.
85. SOLO
85
Aircraft Flight Performance
Stengel, MAE331, Lecture 7, Gliding,
Climbing and Turning Performance
Minimum Time Climb Profiles for Supersonic Speed
( ) ( )
W
VDT
W
VDT
W
DT
g
g
V
V
g
VV
hEps
−
≈
−
=
−
−
+=+==
α
γ
α
γ
cos
sin
cos
sin:
The minimum time to reach the given Energy Height Ef is obtained by using at
each level.
( )maxE
The optimum flight profile for the fastest time to altitude or time to speed involves climbing to
maximal altitude at subsonic speed, then diving in order to get through the transonic speed
range as quickly as possible, and than climbing at supersonic speeds again using .( )maxE
86. 86
SOLO
Climbing Aircraft Performance
Optimum Climbing Trajectories using Energy State Approximation (ESA)
Shaw, “Fighter Combats – Tactics and Maneuvering”
Minimum Time Climb Profiles
Aircraft Flight Performance
The minimum time to reach the given Energy Height Ef is obtained by using at
each level .
( )maxE
87. 87
SOLO
Climbing Aircraft Performance
Optimum Climbing Trajectories using Energy State Approximation (ESA)
A.E. Bryson, Course “Performance Analysis of Flight Vehicles”, AA200, Stanford University, Winter 1977-1978
Aircraft Flight Performance
A.E. Bryson, Jr., “Energy-State Approximation in Performance Optimization of Supersonic Aircraft”, Journal of Aircraft, Vol.6,
No. 5, Nov-Dec 1969, pp. 481-488
Approximate (ESA) Solutions.
Implicit to ESA Approximation
is the possibility of
instantaneous jump between
kinetic to potential energy
(from A to B ).
This non physical situation is
called a “zoom climb” or
“zoom dive”.
A
B
The minimum time to reach the given Energy Height Ef is obtained by using at
each level.
( )maxE
88. SOLO
Climbing Aircraft Performance
Optimum Climbing Trajectories using Energy State Approximation (ESA)
“Exact” calculated using
Optimization Methods
Computations
Aircraft Flight Performance
Comparison between
“Exact” and Approximate
(ESA) Solutions.
Implicit to ESA
Approximation is the
possibility of instantaneous
jump between kinetic to
potential energy (from
A to B , and from C to D).
This non physical situation
is called a “zoom climb”
or “zoom dive”. We can see
the “exact” solution in
those cases.
A
B
C
D
The minimum time to reach the given Energy Height Ef is obtained by using at
each level.
( )maxE
88
A.E. Bryson, Course “Performance Analysis of Flight Vehicles”, AA200, Stanford University, Winter 1977-1978
A.E. Bryson, Jr., “Energy-State Approximation in Performance Optimization of Supersonic Aircraft”, Journal of Aircraft, Vol.6,
No. 5, Nov-Dec 1969, pp. 481-488
89. 89http://msflights.net/forum/showthread.php?1184-Supersonic-Level-Flight-Envelopes-in-FSX
F-15 Streak Eagle Time to Climb Records, which follow the ideal path to reach set altitudes in a
minimal amount of time. The Streak Eagle could break the sound barrier in a vertical climb, so
the ideal flightpath to 30000m involved a large Immelmann.
https://www.youtube.com/watch?v=HLka4GoUbLo https://www.youtube.com/watch?v=S7YAN9--3MA
F-15 Streak Eagle Record Flights part 2F-15 Streak Eagle Record Flights part 1
SOLO
Climbing Aircraft Performance
Optimum Climbing Trajectories using Energy State Approximation (ESA)
Aircraft Flight Performance
90. 90
How to climb as fast as possible
Takeoff and pull up: You want to build energy (kinetic or potential) as quickly as you
can. Peak acceleration is at mach 0.9, which is the speed that energy is gained the
fastest. You should first accelerate to near that speed. Avoid bleeding off energy in a
high-g pull up. Start a smooth pull up before at mach 0.7-0.8 and accelerate to mach 0.9
during the pull.
http://msflights.net/forum/showthread.php?1184-Supersonic-Level-Flight-Envelopes-in-FSX
SOLO
Climbing Aircraft Performance
Optimum Climbing Trajectories using Energy State Approximation (ESA)
F-15 Streak Eagle Time to Climb Records, which follow the ideal path to reach set altitudes in a
minimal amount of time. The Streak Eagle could break the sound barrier in a vertical climb, so
the ideal flightpath to 30000m involved a large Immelmann.
Aircraft Flight Performance
Climb again: to 36000ft for maximum speed, or higher as to not exceed design limits or
to save fuel for a longer run
Climb: Adjust your climb angle to maintain mach 0.9. In a modern fighter, the climb angle
may be 45-60 degrees. If you need a heading change, during the pull and climb is a good time
to make it.
Level off: between 25000 and 36000ft by rolling inverted. Maximum speed is reached at
36000, but remember the engines produce more thrust at higher KIAS, so slightly denser air
may not hurt acceleration through the sound barrier.
Break the mach barrier: Accelerate to mach 1.25 with minimal wing loading (don't turn, try to
set 0AoA)
Return to Table of Content
91. 91
SOLO
Climbing Aircraft Performance
Optimum Climbing Trajectories using Energy State Approximation (ESA)
Aircraft Flight Performance
Minimum Fuel-to- Climb Trajectories using Energy State Approximation (ESA)
The Rate of Fuel consumed by the Aircraft is given by:
=−=
AircraftJetTc
AircraftPropellerPc
td
Wd
td
fd
T
p
We can write
( )DTV
EdW
E
Ed
td
−
==
The fuel consumed in a flight time , tf for a Jet Aircraft is:
( )∫∫∫ −
===
fff t
T
t
T
t
f Ed
TDV
Wc
E
Ed
Tctd
td
fd
f
000 /1
The minimum fuel consumed in a flight time tf is obtained when using
Maximum Thrust and the Mach Number that minimize the integrand:
( )∫ −
=
ft
T
M
f Ed
TDV
Wc
f
0
max
min,
/1
minarg
for each level of E.
92. 92
SOLO
Climbing Aircraft Performance
Optimum Climbing Trajectories using Energy State Approximation (ESA)
Aircraft Flight Performance
Minimum Fuel-to- Climb Trajectories using Energy State Approximation (ESA)
Assuming W nearly constant, during the climb period, contours of constant
( )
max
max
Tc
DTV
T
−
can be computed, as we see in the Figure
A.E. Bryson, Course “Performance Analysis of Flight Vehicles”, AA200, Stanford University, Winter 1977-1978
A.E. Bryson, Jr., “Energy-State Approximation in Performance Optimization of Supersonic Aircraft”, Journal of Aircraft, Vol.6,
No. 5, Nov-Dec 1969, pp. 481-488
The Minimum Fuel-to- Climb
Trajectory is obtained by choosing
at each state.
( )
max
max
Tc
DTV
T
−
The Minimum Time-to- Climb
Path is also displayed.
Implicit to ESA Approximation is the
possibility of instantaneous jump
between kinetic to potential energy
(from A to B) where the Total Energy
is constant.
A
B
Return to Table of Content
93. 93
SOLO
Climbing Aircraft Performance
Optimum Climbing Trajectories using Energy State Approximation (ESA)
Aircraft Flight Performance
Maximum Range during Glide using Energy State Approximation (ESA)
Equations of motion
A.E. Bryson, Jr., “Energy-State Approximation in Performance Optimization of Supersonic Aircraft”, Journal of Aircraft, Vol.6,
No. 5, Nov-Dec 1969, pp. 481-488
γ
γ
γ
sin
0cos
WDTV
g
W
WL
td
d
V
g
W
−−=
≈−=
( )
W
VDT
Eps
−
== :
g
V
W
DT
−
−
=γsin
γ
γ
sin
cos
Vh
Vx
=
=
−
−
=
g
V
W
DT
Vh
γγ cos
1
cos
−
−
===
g
V
W
DT
V
h
x
h
xd
hd
During Glide we have: T = 0, W = constant, dE≤0, |γ| <<1, therefore
+−=
g
V
W
D
xd
hd
( )
γcos
1
VW
DT
xd
Ed −
=
2
2
1
: VhE +=
( ) ( )
( )EL
VED
W
VED
td
Ed
−≈−=
V
td
xd
= ( )
( )
( )ED
EL
ED
W
Ed
xd
−≈−=
( )
( )
( )∫∫∫ −≈−== Ed
ED
EL
Ed
ED
W
xdR
94. 94
SOLO
Climbing Aircraft Performance
Optimum Climbing Trajectories using Energy State Approximation (ESA)
Aircraft Flight Performance
Maximum Range during Glide using Energy State Approximation (ESA)
We found
A.E. Bryson, Jr., “Energy-State Approximation in Performance Optimization of Supersonic Aircraft”, Journal of Aircraft, Vol.6,
No. 5, Nov-Dec 1969, pp. 481-488
( )
( )
( )∫∫ −≈−= Ed
ED
EL
Ed
ED
W
R
Using the first integral we see that to maximize R we must choose the path that
minimizes the drag D (E). The approximate optimal trajectory can be divided in:
1.If the initial conditions are not on the maximum range glide path the Aircraft
shall either “zoom dive” or “zoom climb” at constant E0, A to B path in Figure .
2.The Aircraft will dive on the min D (E)
until it reaches the altitude h = 0 at a
velocity V and Specific Energy E1=V2
/2,
B to C in the Figure.
3.Since h=0 no optimization is possible and
to stay airborne one must keep the drag
such that L = W, by increasing the Angle of
Attack and decreasing velocity until it
reaches Vstall and Es=Vstall
2
/2, C to D in Figure
Since h=0, d E=V dV.
( ) ( ) ( )[ ]∫ ∫∫ =
=
−−−=
1
0 1
0
0
0min
10
max
E
E
E
E
h
pathon
E
E
s
Vd
VD
VW
Ed
ED
W
Ed
ED
W
R
Return to Table of Content
95. 95
Performance of an Aircraft with Parabolic PolarSOLO
−
+
=
=
γσ
α
γσ
coscos
sin
cossin
V
g
Vm
LT
q
V
g
r
W
W
n
W
L
W
LT
n =≈
+
=
αsin
:'
Therefore
( )
−=
=
γσ
γσ
coscos'
cossin
n
V
g
q
V
g
r
W
W
γσγσγσω 2222222
coscoscoscos'2'cossin +−+=+= nn
V
g
qr WW
or
γγσω 22
coscoscos'2' +−= nn
V
g
γγσω 22
2
coscoscos'2'
1
+−
==
nng
VV
R
Aircraft Turn Performance
96. 96
Performance of an Aircraft with Parabolic PolarSOLO
( ) ( )
( )
γ
σ
σ
γ
α
χ
γσγσ
α
γ
cos
sin
sin
cos
sin
coscos'coscos
sin
V
gLT
n
V
g
V
g
Vm
LT
=
+
=
−=−
+
=
2. Horizontal Plan Trajectory ( )0,0 == γγ
( )
1'
1
1'
'
1
1'sin'
cos
1
'01cos'
2
2
2
2
−
=
−=
−==
=→=−=
ng
V
R
n
V
g
n
n
V
g
n
V
g
nn
V
g
σχ
σ
σγ
Aircraft Turn Performance
1. Vertical Plan Trajectory (σ = 0)
( )
γ
γγ
χ
cos'
1
cos'
0
2
−
=
−=
=
ng
V
R
n
V
g
98. 98
Vertical Plan Trajectory (σ = 0)
SOLO
Prof. Earll Murman, “Introduction to Aircraft Performance and Static Stability”, September 18, 2003
99. 99
R
V
=:χ1'2
−= n
V
g
χ
Contours of Constant n and Contours of Constant Turn Radius
in Turn-Rate in Horizontal Plan versus Mach coordinates
Horizontal Plan TrajectorySOLO
102. 102
Performance of an Aircraft with Parabolic PolarSOLO
2. Horizontal Plan Trajectory ( )0,0 == γγ
We can see that for n > 1
We found that
2
2
*
*
u
C
C
n
u
C
nC
L
LL
L =→=
n
1n
2n
MAXn
u u
LC
MAXLC _
1
_
n
C
C
MAXL
L
MAX
MAXL
L
corner n
C
C
u
_
*
=
*2 L
MAX
L C
u
n
C =
MAX
MAXL
L
corner n
C
C
u
_
*
= MAX
L
L
n
C
C
1
*
MAXLC _
2LC
1LC
2
*
1
u
C
C
n
L
L
=
MAXn
n, CL as a function of u
Aircraft Turn Performance
=
=
=
2
0
*
2
1
:*
*
:
2
:*
Vq
V
V
u
CS
kW
V
D
ρ
ρ
1
1
1'
1
11'
2
2
2
2
22
−
≈
−
=
−≈−=
ng
V
ng
V
R
n
V
g
n
V
g
χ Horizontal Turn Rate
Horizontal Turn Radius
103. 103
Performance of an Aircraft with Parabolic PolarSOLO
MAX
MAX
L
MAXL
n
n
C
C
V
g 1
**
2
_ −
MAX
MAXL
L
corner n
C
C
V
g
u
_
*
*
=
MAXL
L
C
C
V
g
u
_
1
*
*
=
MAXn
2n
1n
MAXLC _
2LC
1LC
u
χ
MAXu
Horizontal Turn Rate as function of u, with n and CL as parametersχ
We defined 2
*
&
*
: u
C
C
n
V
V
u
L
L
==
We found 2
2
2
22 1
**
1
*
1
u
u
C
C
V
g
n
Vu
g
n
V
g
L
L
−
=−=−=χ
This is defined for 1:
**
1
__
<=≥≥= u
C
C
un
C
C
u
MAXL
L
MAX
MAXL
L
corner
2. Horizontal Plan Trajectory ( )0,0 == γγ
Aircraft Turn Performance
=
=
=
2
0
*
2
1
:*
*
:
2
:*
Vq
V
V
u
CS
kW
V
D
ρ
ρ
104. 104
Performance of an Aircraft with Parabolic PolarSOLO
From
2
2
2
22 1
**
1
*
1
u
u
C
C
V
g
n
Vu
g
n
V
g
L
L
−
=−=−=χ
4
2
2
2
22
1
*
1*
1
*
:
uC
Cg
V
n
u
g
VV
R
L
L
−
=
−
==
χ
Therefore
cornerMAX
MAXL
L
MAXL
L
L
MAXL
C
un
C
C
u
C
C
u
uC
Cg
V
R
MAXL
=≤≤=
−
=
__
1
4
2
_
2
**
1
*
1*
_
cornerMAX
MAXL
L
MAX
n
un
C
C
u
n
u
g
V
R
MAX
=≥
−
=
_
2
22
*
1
*
MAX
L
L
L
L
L
L
C
n
C
C
u
C
C
u
uC
Cg
V
R
L
**
1
*
1*
1
4
2
2
≤≤=
−
=
n
C
C
u
n
u
g
V
R
MAXL
L
n
_
2
22
*
1
*
≥
−
=
2. Horizontal Plan Trajectory ( )0,0 == γγ
Aircraft Turn Performance
=
=
=
2
0
*
2
1
:*
*
:
2
:*
Vq
V
V
u
CS
kW
V
D
ρ
ρ
105. 105
Performance of an Aircraft with Parabolic PolarSOLO
R (Radius of Turn) a function of u, with n and CL as parameters
1
**
2
_
2
−MAX
MAX
MAXL
L
n
n
C
C
g
V
MAX
MAXL
L
corner n
C
C
V
g
u
_
*
*
=
MAXL
L
C
C
V
g
u
_
1
*
*
=
MAXn
2n
1nMAXLC _
2LC 1LC
u
R
4
2
2
2
22
1
*
1*
1
*
:
uC
Cg
V
n
u
g
VV
R
L
L
−
=
−
==
χ
2. Horizontal Plan Trajectory ( )0,0 == γγ
Return to Table of Content
Aircraft Turn Performance
=
=
=
2
0
*
2
1
:*
*
:
2
:*
Vq
V
V
u
CS
kW
V
D
ρ
ρ
106. 106
Performance of an Aircraft with Parabolic PolarSOLO
Horizontal Turn Rate as Function of ps, n
( )
u
nuzu
e
V
ps
224
2
*2
* −+−
=
up
V
e
uzun s
*
*2
2 242
−+−=
2
24
2
2 1
*
*2
2
*
1
* u
up
V
e
uzu
V
g
u
n
V
g s −−+−
=
−
=χ
2
24
4
2423
1
*
*2
2
2
1
*
*2
22
*
*2
44
*
u
up
V
e
uzu
u
up
V
e
uzuuup
V
e
uzu
V
g
u
s
ss
−−+−
−−+−−
−+−
=
∂
∂ χ
Therefore
−−+−
++−
=
∂
∂
1
*
*2
2
1
*
*
* 244
4
up
V
e
uzuu
up
V
e
u
V
g
u
s
s
χ
For ps = 0
2
22
12
24
0
11
12
*
uzzuzzu
u
uzu
V
g
sp
=−+<<−−=
−+−
==
χ
( ) 2
22
1
244
4
0
11
12
1
*
uzzuzzu
uzuu
u
V
g
u
sp
=−+<<−−=
−+−
+−
=
∂
∂
=
χ
Aircraft Turn Performance
=
=
=
2
0
*
2
1
:*
*
:
2
:*
Vq
V
V
u
CS
kW
V
D
ρ
ρ
107. 107
Performance of an Aircraft with Parabolic PolarSOLO
Horizontal Turn Rate as Function of ps, n
For ps = 0
2
22
12
24
0
11
12
*
uzzuzzu
u
uzu
V
g
sp
=−+<<−−=
−+−
==
χ
( ) 2
22
1
244
4
0
11
12
1
*
uzzuzzu
uzuu
u
V
g
u
sp
=−+<<−−=
−+−
+−
=
∂
∂
=
χ
Let find the maximum of as a function of uχ
( )12
1
* 244
4
0 −+−
+−
=
∂
∂
= uzuu
u
V
g
u
sp
χ
( ) ( )12
*
1 00
−=== ==
z
V
g
u
ss ppMAX χχ
u 0 u1 1 (u1+u2)/2 u2
∞ + + 0 - - - - - - -∞
↑ Max ↓
u∂
∂ χ
χ
From
2
24
1
*
*2
2
* u
up
V
e
uzu
V
g s −−+−
=χ
−−+−
++−
=
∂
∂
1
*
*2
2
1
*
*
* 244
4
up
V
e
uzuu
up
V
e
u
V
g
u
s
sχ
Aircraft Turn Performance
=
=
=
2
0
*
2
1
:*
*
:
2
:*
Vq
V
V
u
CS
kW
V
D
ρ
ρ
108. 108
Performance of an Aircraft with Parabolic PolarSOLO
Horizontal Turn Rate as Function of ps, n
u
u
0<sp
0<sp
0=sp
0=sp 0>sp
0>sp
χ
u∂
∂ χ
( )12
*
−z
V
g
1=u1u
2u
as a function of u with ps as
parameter
u∂
∂ χ
χ
,
−−+−
++−
=
∂
∂
1
*
*2
2
1
*
*
* 244
4
up
V
e
uzuu
up
V
e
u
V
g
u
s
sχ
2
24
1
*
*2
2
* u
up
V
e
uzu
V
g s −−+−
=χ
Because ,we have0
*
*
>u
V
e
000 >=<
>>
sss ppp
χχχ
0
1
0
1
0
1
0
>
=
=
=
<
= ∂
∂
<=
∂
∂
<
∂
∂
sss p
u
p
u
p
u uuu
χχχ
Aircraft Turn Performance
=
=
=
2
0
*
2
1
:*
*
:
2
:*
Vq
V
V
u
CS
kW
V
D
ρ
ρ
109. 109
Performance of an Aircraft with Parabolic PolarSOLO
Horizontal Turn Rate as Function of ps, n
a function of u, with ps
as parameter
χ
2
24
1
*
*2
2
* u
up
V
e
uzu
V
g s −−+−
=χ
Aircraft Turn Performance
=
=
=
2
0
*
2
1
:*
*
:
2
:*
Vq
V
V
u
CS
kW
V
D
ρ
ρ
Sustained
Turn
Instantaneous
Turn
110. 110
Performance of an Aircraft with Parabolic PolarSOLO
Horizontal Turn Rate as Function of ps, n
2
24
1
*
*2
2
* u
up
V
e
uzu
V
g s −−+−
=χ
( ) ( )ss
s
puupu
up
V
e
uzu
u
g
VV
R 21
24
42
1
*
*2
2
*
<<
−−+−
==
χ
3
242
23
2
24
4
2
24
34243
2
1
*
*2
22
2
*
*3
22
*
1
*
*2
2
2
1
*
*2
2
*
*2
441
*
*2
24
*
−−+−
−−
=
−−+−
−−+−
−+−−
−−+−
=
∂
∂
up
V
e
uzuu
up
V
e
uzu
g
V
up
V
e
uzu
u
up
V
e
uzu
p
V
e
uzuuup
V
e
uzuu
g
V
u
R
s
s
s
s
ss
Aircraft Turn Performance
=
=
=
2
0
*
2
1
:*
*
:
2
:*
Vq
V
V
u
CS
kW
V
D
ρ
ρ
111. 111
Performance of an Aircraft with Parabolic PolarSOLO
Horizontal Turn Rate as Function of ps, n
3
24
2
2
1
*
*2
2
2
*
*3
2
*
−−+−
−−
=
∂
∂
up
V
e
uzu
up
V
e
uzu
g
V
u
R
s
s
or
We have
>
+
+
=
<
+
−
=
→=
∂
∂
0
4
16
*
*
9
*
*3
0
4
16
*
*
9
*
*3
0
2
2
2
1
z
zp
V
e
up
V
e
u
z
zp
V
e
up
V
e
u
u
R
ss
R
ss
R
u 0 u1 uR2 u2
∞ - - - 0 + + ∞
↓ min ↑
u
R
∂
∂
R
2
22
124
42
0
11
12
*
uzzuzzu
uzu
u
g
V
R
sp
=−+<<−−=
−+−
==
( )
( )
2
22
1
324
22
0
11
12
1*2
uzzuzzu
uzu
uzu
g
V
u
R
sp
=−+<<−−=
−+−
−
=
∂
∂
=
Aircraft Turn Performance
=
=
=
2
0
*
2
1
:*
*
:
2
:*
Vq
V
V
u
CS
kW
V
D
ρ
ρ
112. 112
Performance of an Aircraft with Parabolic PolarSOLO
Horizontal Turn Rate as Function of ps, n
u
R
0>sp
0=sp
0<sp
MAXL
L
C
C
_
*
1
**
2
_ −MAX
MAX
MAXL
L
n
n
C
C
g
V
1
1*
2
−zg
V
4
2
_
1*
1*
uC
C
g
V
MAXL
L
−
1
*
2
22
−MAXn
u
g
V
MAX
MAXL
L
n
C
C
_
*
LIMIT
C MAXL_
LIMIT
nMAX
z
1
12
−− zz 12
−+ zz
1
*
*2
2
*
24
42
−−+−
=
up
V
e
uzu
u
g
V
R
s
Minimum Radius of Turn R is obtained for zu /1=
1
1*
2
2
0
−
==
zg
V
R
sp
R (Radius of Turn) a function
of u, with ps as parameter
( ) ( )ss
s
puupu
up
V
e
uzu
u
g
VV
R
21
24
42
1
*
*2
2
*
<<
−−+−
==
χ
Return to Table of Content
Because ,we have0
*
*
>u
V
e
000 >=<
<<
sss ppp
RRR 000 minminmin >=<
<<
sss pRpRpR uuu
Aircraft Turn Performance
=
=
=
2
0
*
2
1
:*
*
:
2
:*
Vq
V
V
u
CS
kW
V
D
ρ
ρ
113. 113
Performance of an Aircraft with Parabolic PolarSOLO
Horizontal Turn Rate as Function of nV,
( )
W
VDT
g
VV
hEps
−
≈+==
:
For an horizontal turn 0=h
V
g
Vu
g
VV
ps
*
==
We found
2
24
1
*
*2
2
* u
up
V
e
uzu
V
g s −−+−
=χ
from which
2
24
1*2
* u
ue
g
V
zu
V
g
−
−+−
=
χ
defined for
2
22
1 :1**1**: ue
g
V
ze
g
V
zue
g
V
ze
g
V
zu =−
−+
−≤≤−
−−
−=
Aircraft Turn Performance
=
=
=
2
0
*
2
1
:*
*
:
2
:*
Vq
V
V
u
CS
kW
V
D
ρ
ρ
114. 114
Performance of an Aircraft with Parabolic PolarSOLO
Horizontal Turn Rate as Function of nV,
Let compute
2
24
4
2423
1*2
2
1*22*44
*
u
ue
g
V
zu
u
ue
g
V
zuuuue
g
V
zu
V
g
u
−
−+−
−
−+−−
−+−
=
∂
∂
χ
−
−+−
+−
=
∂
∂
1*2
1
*
244
4
ue
g
V
zuu
u
V
g
u
χ
or
u 0 u1 1 (u1+u2)/2 u2
∞ + + 0 - - - - - - -∞
↑ Max ↓
u∂
∂ χ
χ
−−= 1*2
*
e
g
V
z
V
g
MAX
χ
Aircraft Turn Performance
=
=
=
2
0
*
2
1
:*
*
:
2
:*
Vq
V
V
u
CS
kW
V
D
ρ
ρ
115. 115
Performance of an Aircraft with Parabolic PolarSOLO
Horizontal Turn Rate as Function of nV,
u
0<V
0=V
0>V
χ
( )12
*
−z
V
g
1=u1u
2u
2
24
1*2
* u
ue
g
V
zu
V
g
−
−+−
=
χ
1
*
2
−MAXn
uV
g
2
2
2
_ 1
** u
u
C
C
V
g
L
MAXL
−
MAXL
L
C
C
_
*
MAX
MAXL
L
n
C
C
_
*
LIMIT
nMAXLIMIT
C MAXL _
MAX
MAX
L
MAXL
n
n
C
C
V
g 1
**
2
_ −
as function of u
and as parameter
χ
V
Return to Table of Content
Aircraft Turn Performance
=
=
=
2
0
*
2
1
:*
*
:
2
:*
Vq
V
V
u
CS
kW
V
D
ρ
ρ
117. 117
Mirage 2000 at 15000ft.
http://forums.eagle.ru/showthread.php?t=98497
Max sustained rate
(at around 6.5G on
the 0 Ps line)
occurring at around
0.9M/450KCAS
looking at around
12.5 deg sec
9G Vc (Max instant.
Rate) is around
0.65M/320KCAS
looking at 23.5 deg
sec
SOLO
Aircraft Flight Performance
121. 121
The black lines are the F-4D, the dark orange
lines are the heavy F-4E, and the blue lines are
the lightweight F-4E (same weight as F-4D). Up
to low transonic mach numbers and up to
medium altitudes, the F-4E is about 7% better
than the F-4D (15% better with the same weight).
At higher mach numbers, the F-4 doesn't have to
pull as much AoA to get the same lift, so the slats
actually cause a drag penalty that allows the F-
4D to perform better. For reference, the F-14 is
known to turn about 20% better than the
unslatted F-4J. So, if the slats made the F-4S
turn about 15% better, sustained turn rates would
almost be pretty close between the F-14 and F-
4S. The F-4E, being heavier, would still be
significantly under the F-14. However, with
numbers this close, pilot quality is everything
rather than precise performance figures.
http://combatace.com/topic/71161-beating-a-dead-horse-us-fighter-turn-performance/
F-4
SOLO
Aircraft Flight Performance
124. 124
Corner Speed
Maximum
Positive
Capability
(CL) max
Maximum
Negative
Capability
(CL) min
LoadFactor-n
Structural
Limit
Structural
Limit
Limit
Airspeed
Area of
Structural
Damage of
Failure
Vmin V
n
Operational
Load Limit
Operational
Load Limit
Structural
Load Limit
Structural
Load Limit
Typical Maneuvering Envelope
V – n Diagram
Maneuvering Envelope:
Limits on Normal Load Factor and
Allowable Equivalent Airspeed
-Structural Factor
-Maximum and Minimum
allowable Lift Coefficient
-Maximum and Minimum
Airspeeds
-Corner Velocity: Intersection of
Maximum Lift Coefficient and
Maximum Load Factor
SOLO
Aircraft Flight Performance
126. 126R.W. Pratt, Ed., “Flight Control Systems, Practical issues in design and implementation”,
AIAA Publication, 2000
SOLO
Aircraft Flight Performance
Return to Table of Content
127. 127
Air-to-Air Combat
Destroy Enemy Aircraft to achieve Air Supremacy in order to prevent the enemy
to perform their missions and enable to achieve tactical goals.
SOLO
See S. Hermelin, “Air Combat”, Presentation, http://www.solohermelin.com
129. SOLO
129
Energy–Maneuverability Theory
Aircraft Flight Performance
Energy–maneuverability theory is a model of aircraft performance. It was
promulgated by Col. John Boyd, and is useful in describing an aircraft's
performance as the total of kinetic and potential energies or aircraft specific
energy. It relates the thrust, weight, drag, wing area, and other flight
characteristics of an aircraft into a quantitative model. This allows combat
capabilities of various aircraft or prospective design trade-offs to be predicted and
compared.
Colonel John Richard Boyd
(1927 –1997)
Boyd, a skilled U.S. jet fighter pilot in the Korean War, began
developing the theory in the early 1960s. He teamed with
mathematician Thomas Christie at Eglin Air Force Base to
use the base's high-speed computer to compare the
performance envelopes of U.S. and Soviet aircraft from the
Korean and Vietnam Wars. They completed a two-volume
report on their studies in 1964. Energy Maneuverability came
to be accepted within the U.S. Air Force and brought about
improvements in the requirements for the F-15 Eagle and
later the F-16 Fighting Falcon fighters
136. SOLO
136
Aircraft Flight Performance
In combat, a pilot is faced with a variety of limiting factors. Some limitations are
constant, such as gravity, drag, and thrust-to-weight ratio. Other limitations vary
with speed and altitude, such as turn radius, turn rate, and the specific energy of
the aircraft. The fighter pilot uses Basic Fighter Maneuvers (BFM) to turn these
limitations into tactical advantages. A faster, heavier aircraft may not be able to
evade a more maneuverable aircraft in a turning battle, but can often choose to
break off the fight and escape by diving or using its thrust to provide a speed
advantage. A lighter, more maneuverable aircraft can not usually choose to
escape, but must use its smaller turning radius at higher speeds to evade the
attacker's guns, and to try to circle around behind the attacker.[13]
BFM are a constant series of trade-offs between these limitations to conserve
the specific energy state of the aircraft. Even if there is no great difference
between the energy states of combating aircraft, there will be as soon as the
attacker accelerates to catch up with the defender. Instead of applying thrust, a
pilot may use gravity to provide a sudden increase in kinetic energy (speed), by
diving, at a cost in the potential energy that was stored in the form of altitude.
Similarly, by climbing the pilot can use gravity to provide a decrease in speed,
conserving the aircraft's kinetic energy by changing it into altitude. This can help
an attacker to prevent an overshoot, while keeping the energy available in case
one does occur
Energy Management
137. SOLO
137
Aircraft Flight Performance
Energy Management
Colonel J. R. Boyd:
In an air-to-air battle offensive maneuvering advantage will belong to the pilot
who can enter an engagement at a higher energy level and maintain more energy than his
opponent while locked into a maneuver and counter-maneuver duel. Maneuvering
advantage will also belong to the pilot who enters an air-to-air battle at a lower energy
level, but can gain more energy than his opponent during the course of the battle, From a
performance standpoint, such an advantage is clear because the pilot with the most energy
has a better opportunity to engage or disengage at his own choosing. On the other hand,
energy-loss maneuvers can be employed defensively to nullify an attack or to gain a
temporary offensive maneuvering position.
http://www.ausairpower.net/JRB/fast_transients.pdf
“New Conception for Air-to-Air Combat”, J. Boyd, 4 Aug. 1976
141. 141
Comparison of Turn Performance of two WWII Fighter Aircraft:
Russian Lavockin La5 vs German Messershmitt Bf 109
http://en.wikipedia.org/wiki/Lavochkin_La-5http://en.wikipedia.org/wiki/Messerschmitt_Bf_109
http://www.simhq.com/_air/air_065a.html
SOLO
Aircraft Flight Performance
142. 142
F-86F Sabre and MiG-15 performance comparison
North American
F-86 Sabre
MiG-15
SOLO
Aircraft Flight Performance
143. 143
Falcon F-16C versus Fulcrum MIG 29,
left is w/o afterburner, right is with it, fuel reserves 50%
http://forum.keypublishing.com/showthread.php?47529-MiG-29-kontra-F-16-(aerodynamics-)
FulcrumMiG-29F-16
SOLO
Aircraft Flight Performance
144. 144
Comparison of Turn Performance of two Modern Fighter Aircraft:
Russian MiG-29 vs USA F-16
FulcrumMiG-29
F-16
http://www.simhq.com/_air/air_012a.html
http://www.evac-fr.net/forums/lofiversion/index.php?t984.html
145. 145
Comparison of Turn Performance of two Modern Fighter Aircraft:
Russian MiG-29 vs USA F-16
FulcrumMiG-29
F-16
http://www.simhq.com/_air/air_012a.html
http://www.evac-fr.net/forums/lofiversion/index.php?t984.html
147. 147
http://www.simhq.com/_air3/air_117e.html
While the turn radius of both aircraft is very similar, the MiG-29
has gained a significant angular advantage.
Comparison of Turn Performance of two Modern Fighter Aircraft:
Russian MiG-29 vs USA F-16
MiG-29
F-16
SOLO Aircraft Flight Performance
150. 150
An assessment is made of the applicability of Energy Maneuverability techniques (EM)
to flight path optimization. A series of minimum time and fuel maneuvers using the F-4C
aircraft were established to progressively violate the assumptions inherent in the EM program
and comparisons were made with the Air Force Flight Dynamics Laboratory's (AFFDL)
Three-Degree-of-Freedom Trajectory Optimization Program and a point mass option of the
Six-Degree-of-Freedom flight path program. It was found the EM results were always optimistic
in the value of the payoff functions with the optimism increasing as the percentage
of the maneuver involving constant energy transitions Increases. For the minimum time
paths the resulting optimism was less than 27%f1o r the maneuvers where the constant energy
percentage was less than 35.',", followed by a rather steeply rising curve approaching in the
limit 100% error for paths which are comprised entirely of constant energy transitions. Two
new extensions are developed in the report; the first is a varying throttle technique for use
on minimum fuel paths and the second a turning analysis that can be applied in conjunction
with a Rutowski path. Both extensions were applied to F-4C maneuvers in conjunction with
'Rutowski’s paths generated from the Air Force Armament Laboratory's Energy Maneuverability
program. The study findings are that energy methods offer a tool especially useful in
the early stages of preliminary design and functional performance studies where rapid
results with reasonable accuracy are adequate. If the analyst uses good judgment in its applications
to maneuvers the results provide a good qualitative insight for comparative purposes.
The paths should not, however, be used as a source of maneuver design or flight
schedule without verification especially on relatively dynamic maneuvers where the accuracy
and optimality of the method decreases.
David T. Johnson, “Evaluation of Energy Maneuverability Procedures
in Aircraft Flight Path Optimization and Performance Estimation”,
November 1972, AFFDL-TR-72-53
SOLO Aircraft Flight Performance
159. 159
Supermaneuverability is defined as the ability of an aircraft to perform high
alpha maneuvers that are impossible for most aircraft is evidence of the
aircraft's supermaneuverability. Such maneuvers include Pugachev's Cobra
and the Herbst maneuver (also known as the "J-turn").
Some aircraft are capable of performing Pugachev's Cobra without the aid
of features that normally provide post-stall maneuvering such as thrust
vectoring. Advanced fourth generation fighters such as the Su-27, MiG-29
along with their variants have been documented as capable of performing
this maneuver using normal, non-thrust vectoring engines. The ability of
these aircraft to perform this maneuver is based in inherent instability like
that of the F-16; the MiG-29 and Su-27 families of jets are designed for
desirable post-stall behavior. Thus, when performing a maneuver like
Pugachev's Cobra the aircraft will stall as the nose pitches up and the
airflow over the wing becomes separated, but naturally nose down even from
a partially inverted position, allowing the pilot to recover complete control.
http://en.wikipedia.org/wiki/Supermaneuverability
Supermaneuverability
SOLO
Aircraft Flight Performance
162. 162
SOLO
Aircraft Flight Performance
The Herbst maneuver or "J-Turn" named after Wolfgang Herbst is the only thrust
vector post stall maneuver that can be used in actual combat but very few air frames
can sustain the stress of this violent maneuver.
Herbst Maneuver
http://en.wikipedia.org/wiki/Herbst_maneuver
Return to Table of Content
163. 163
Constraint Analysis
SOLO Aircraft Flight Performance
The Performance Requirements can be translated
into functional relationship between the Thrust-to-
Weight or Thrust Loading at Sea Level Takeoff
(TSL/WTO) and the Wing Loading at Takeoff (WTO/S).
The keys to the development are
•Reasonable assumption hor Aircraft Lift-to-Drag
Polar.
•The low sensibility of Engine Thrust with Flight
Altitude and Mach Number.
The minimum of TSL/WTO as functions of WTO/S are required for:
•Takeoff from a Runway of a specified length.
•Flight at a given Altitude and Required Speed.
•Climb at a Required Speed.
•Turn at a given Altitude, Speed and a required Rate.
•Acceleration capability at constant Altitude.
•Landing without reverse Thrust on a Runway of a given length.
164. 164
Energy per unit mass E
Let define Energy per unit mass E:
g
V
hE
2
:
2
+=
Let differentiate this equation:
( ) ( )
W
VDT
W
VDT
W
DT
g
g
V
V
g
VV
hEps
−
≈
−
=
−
−
+=+==
α
γ
α
γ
cos
sin
cos
sin:
define
10 ≤<= ββ TOWW WTO – Take-off Weight
( ) ( ) ( ) SLThThhT αα === 0 TSL – Sea Level Thrust
V
p
W
D
W
T s
+=
Load Factor
W
CSq
W
L
n L
==:
SOLO Aircraft Flight Performance
TOL W
Sq
n
W
Sq
n
C β==
+=
V
p
W
D
W
T s
TO
SL
α
β
Constraint Analysis
165. 165
SOLO Aircraft Flight Performance
General Mission Description of a Typical Fighter Aircraft
10: ≤<= ββ
TOW
W
WTO – Take-off Weight
W – Aircraft Weight during Flight
Constraint Analysis
166. 166
Assume a General Lift-to-Drag Polar Relationship
Total DragRD CSqCSqRD +=+
D, CD - Clean Aircraft Drag and Drag Coefficient
R, CR – Additional Aircraft Drag and Additional Drag Coefficient caused by
External Stores, Bracking Parachute, Flaps, External Hardware
02
2
102
2
1 D
TOTO
DLLD C
S
W
q
n
K
S
W
q
n
KCCKCKC +
+
=++=
ββ
TOL W
Sq
n
W
Sq
n
C β==
( )
++=
V
p
CC
W
Sq
W
T s
RD
TOTO
SL
βα
β
+
++
+
=
V
p
CC
S
W
q
n
K
S
W
q
n
K
W
Sq
W
T s
DRD
TOTO
TOTO
SL
02
2
1
ββ
βα
β
SOLO Aircraft Flight Performance
Constraint Analysis
167. 167
( )WLn
td
Vd
td
hd
==== ,1,0,0
Case 1: Constant Altitude/Speed Cruise (ps = 0)
Given:
+
++
=
S
W
q
CC
K
S
W
q
K
W
T
TO
DRDTO
TO
SL
β
β
α
β 0
21
We obtain:
We can see that TSL/WTO → ∞ for WTO/S → 0 and WTO/S→∞, therefore a
minimum exist. By differentiating TSL/WTO with respect to WTO/S and
setting the result equal to zero, we obtain:
1
0
/min K
CCq
S
W DRD
WT
TO +
=
β
( )[ ]210
min
2 KKCC
W
T
DRD
TO
SL
++=
α
β
Lift
DragThrust
Weight
SOLO Aircraft Flight Performance
Constraint Analysis
168. 168M. Corcoran, T. Matthewson, N. W. Lee, S. H. Wong, “Thrust Vectoring”
Case 1: Constant Altitude/Speed Cruise (ps = 0)
SOLO Aircraft Flight Performance
Constraint Analysis
169. 169
( )WLn
td
hd
≈≈= ,1,0
Case 2: Constant Speed Climb (ps = dh/dt)
Given:
We can see that TSL/WTO → ∞ for WTO/S → 0 and WTO/S→∞, therefore a
minimum exist. By differentiating TSL/WTO with respect to WTO/S and
setting the result equal to zero, we obtain:
1
0
/min K
CCq
S
W DRD
WT
TO +
=
β
( )
+++=
td
hd
V
KKCC
W
T
DRD
TO
SL 1
2 210
min
α
β
We obtain:
+
+
++
=
td
hd
V
S
W
q
CC
K
S
W
q
K
W
T
TO
DRDTO
TO
SL 10
21
β
β
α
β
SOLO Aircraft Flight Performance
170. 170
,1,0,0
,,
>== n
td
hd
td
Vd
givenhVgivenhV
Case 3: Constant Altitude/Speed Turn (ps = 0)
Given:
We can see that TSL/WTO → ∞ for WTO/S → 0 and WTO/S→∞, therefore a
minimum exist. By differentiating TSL/WTO with respect to WTO/S and
setting the result equal to zero, we obtain:
1
0
/min K
CC
n
q
S
W DRD
WT
TO +
=
β
( )[ ]210
min
2 KKCC
n
W
T
DRD
TO
SL
++=
α
β
We obtain:
+
+
++
=
td
hd
V
S
W
q
CC
nK
S
W
q
nK
W
T
TO
DRDTO
TO
SL 10
2
2
1
β
β
α
β
2
0
2
2
0
11
+=
Ω
+=
cRg
V
g
V
n
SOLO Aircraft Flight Performance
Constraint Analysis
171. 171
( )WLn
td
hd
givenh
=== ,1,0
Case 4: Horizontal Acceleration (ps = (V/g0) (dV/dt) )
Given:
We obtain:
+
+
++
=
td
Vd
g
S
W
q
CC
K
S
W
q
K
W
T
TO
DRDTO
TO
SL
0
0
21
1
β
β
α
β
SOLO Aircraft Flight Performance
Lift
DragThrust
Weight
This can be rearranged to give:
+
++
=
S
W
q
CC
K
S
W
q
K
W
T
td
Vd
g TO
DRDTO
TO
SL
β
β
β
α 0
21
0
1
Constraint Analysis
172. 172
( )WLn
td
hd
givenh
=== ,1,0
Case 4: Horizontal Acceleration (ps = (V/g0) (dV/dt) ) (continue – 1)
Given:
SOLO Aircraft Flight Performance
Lift
DragThrust
Weight
We obtain:
+
++
=
S
W
q
CC
K
S
W
q
K
W
T
td
Vd
g TO
DRDTO
TO
SL
β
β
β
α 0
21
0
1
This equation can be integrated from initial velocity V0 to final velocity Vf,
from initial t0 to final tf times.
( )∫=−
fV
V
s
f
Vp
VdV
g
tt
0
0
0
1
where
+
++
−=
S
W
q
CC
K
S
W
q
K
W
T
Vp
TO
DRDTO
TO
SL
s
β
β
β
α 0
21
The solutions of TSL/WTO for different WTO/S are obtained iteratively.
Constraint Analysis
174. 174
0=
givenh
td
hd
Case 5: Takeoff (sg given and TSL >> (D+R) )
Given:
SOLO Aircraft Flight Performance
Ground Run
V = 0
sg
sTO
sr str
V TO
Rotation
Transition
sCL
θ CL
htr
hobs
R
Start from:
( )
TO
T
SL
s
W
VRDT
td
Vd
g
V
td
hd
p
SL
β
α
α
+−
≈+=
≈
0
==
TO
SL
V
W
Tg
td
sd
sd
Vd
td
Vd
β
α 0
/1
VdV
T
W
g
sd
SL
TO
=
0α
β
max,2
2
0max,
2
0
2
1
2
1
L
TO
TO
LstallstallTO CS
k
V
CSVLW ρρβ ===
The take-off velocity VTO is
VTO = kTO Vstall
Where Vstall is the minimum velocity at at which Lift equals weight and
kTO ≈ 1.1 to 1.2:
==
S
W
C
kV
k
V TO
L
TOstall
TO
TO
max,0
22
2
2
22 ρ
β
Integration from:
s = 0 to s = sg
V = 0 to V = VTO
2
2
0
TO
SL
TO
g
V
T
W
g
s
=
α
β
sg – Ground Run
Constraint Analysis
175. 175
Case 5: Takeoff (sg given and TSL >> (D+R) ) (continue – 1)
SOLO Aircraft Flight Performance
Ground Run
V = 0
sg
sTO
sr str
V TO
Rotation
Transition
sCL
θ CL
htr
hobs
R
2
2
0
TO
SL
TO
g
V
T
W
g
s
=
α
β
==
S
W
C
kV
k
V TO
L
TOstall
TO
TO
max,0
22
2
2
22 ρ
β
=
S
W
Cgs
k
W
T TO
Lg
TO
TO
SL
max,00
22
ρ
β
α
β
We obtained:
from which:
=
S
W
C
k
T
W
g
s TO
L
TO
SL
TO
g
max,0
2
0 ρ
β
α
β
We have a Linear Relation between TSL/WTO and WTO/S
Constraint Analysis
176. 176
Case 6: Landing
SOLO Aircraft Flight Performance
where ( )
( )
−=
−−=
µ
µ
β
ρ
W
T
gc
CC
SW
g
a grLgrD
TO
0
,,
:
/2
:
cab
Va
a
ca
Va
a
touchdown
4
2
:
4
2
:
2
1
1
−
=
−
=cVa
cVa
a
sg
+
+
−= 2
2
2
1
ln
2
1
−
−
⋅
+
+
−
=
1
2
2
1
1
1
1
1
ln
4
1
a
a
a
a
ca
tg
Ground Run Phase
We found
Ground Run sgr
Transition
Airborne Phase
Total Landing Distance
Float
sf
Flare stGlide sg
γ
hg
hf
Touchdown
2
0 VCVBTT ++=
For a given value of sg , there is only one value of WTO/S that satisfies this equation.
( )gTO sfSW =/
This constraint is represented in the TSL/WTO versus WTO/S plane as a vertical line, at
WTO/S corresponding to the required sg.
Constraint Analysis
177. 177Constraint Diagram
SOLO Aircraft Flight Performance
+
++
=
S
W
q
CC
K
S
W
q
K
W
T
TO
DRDTO
TO
SL
β
β
α
β 0
21
+
+
++
=
td
hd
V
S
W
q
CC
nK
S
W
q
nK
W
T
TO
DRDTO
TO
SL 10
2
2
1
β
β
α
β
=
S
W
Cgs
k
W
T TO
Lg
TO
TO
SL
max,00
22
ρ
β
α
β
( )gTO sfSW =/
Constraint Analysis
M. Saarlas, “Aircraft Performance”, John Wiley & Sons, 2007
M. Saarlas, “Aircraft Performance”, John Wiley & Sons, 2007
J. D. Anderson, Jr., “Aircraft Performance and Design”, McGraw-Hill, 1999
M. Saarlas, “Aircraft Performance”, John Wiley & Sons, 2007
J. D. Anderson, Jr., “Aircraft Performance and Design”, McGraw-Hill, 1999
M. Saarlas, “Aircraft Performance”, John Wiley & Sons, 2007
J. D. Anderson, Jr., “Aircraft Performance and Design”, McGraw-Hill, 1999
M. Saarlas, “Aircraft Performance”, John Wiley & Sons, 2007
J. D. Anderson, Jr., “Aircraft Performance and Design”, McGraw-Hill, 1999
M. Saarlas, “Aircraft Performance”, John Wiley & Sons, 2007
J. D. Anderson, Jr., “Aircraft Performance and Design”, McGraw-Hill, 1999
M. Saarlas, “Aircraft Performance”, John Wiley & Sons, 2007
J. D. Anderson, Jr., “Aircraft Performance and Design”, McGraw-Hill, 1999
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
Stengel, MAE331, Lecture 6
Stengel, MAE331, Lecture 6
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
Stengel, MAE331, Lecture 6
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
Stengel, MAE331, Lecture 6
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962, Ch.9
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998, Ch.2
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
J.D. Anderson, Jr., “Aircraft Performance and Design”, McGraw Hill, 1999
J.D. Anderson, Jr., “Aircraft Performance and Design”, McGraw Hill, 1999
J.D. Anderson, Jr., “Aircraft Performance and Design”, McGraw Hill, 1999
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
Prof. Earll Murman, “Introduction to Aircraft Performance and Static Stability”, September 18, 2003
Prof. Earll Murman, “Introduction to Aircraft Performance and Static Stability”, September 18, 2003
http://scilib.narod.ru/Avia/Bonanni/bonanni.htm
Brandt, “Introduction to Aerodynamics – A Design Perspective”, Ch. 5 , Performance and Constraint Analysis
Randal K. Liefer, John Valasek, David P. Eggold, “Fighter Aircraft Metrics, Research , and Test”,
Phase I Report, KU-FRL-831-2
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
“Climb and Turn Performance”, G. Leng, NUS
Klaus Huenecke, “Modern Combat Aircraft Design”, Airlife, 1984, 1987
“Climb and Turn Performance”, G. Leng, NUS
Chapter 4 Aerodynamics of Flight
http://www.gov/library/manuals/aviation/pilot_handbook/media/PHAK%20-%20Chapter%2004.pdf
http://en.wikipedia.org/wiki/Energy%E2%80%93maneuverability_theory
J.R. Boyd, “Energy Maneuvrability”, AD 372 287, March 1966
“Climb and Turn Performance”, G. Leng, NUS
“Certain to Win, John Boyd ‘s Strategy in the 21st Strategy”, Chet Richards,
“Certain to Win, John Boyd ‘s Strategy in the 21st Strategy”, Chet Richards,
Asselin, M., “Introduction to Aircraft Performance”, AIAA Education Series, 1997
Klaus Huenecke, “Modern Combat Aircraft Design”, Airlife, 1984, 1987
A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962
B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
“Supplemental FLIGHT MANUAL F-16C/D Blocks 50 and 52+”, Lockheed Martin Corporation,
T.O. GR1F-16CJ-1-1
“Supplemental FLIGHT MANUAL F-16C/D Blocks 50 and 52+”, Lockheed Martin Corporation,
T.O. GR1F-16CJ-1-1
“Supplemental FLIGHT MANUAL F-16C/D Blocks 50 and 52+”, Lockheed Martin Corporation,
T.O. GR1F-16CJ-1-1
“Supplemental FLIGHT MANUAL F-16C/D Blocks 50 and 52+”, Lockheed Martin Corporation,
T.O. GR1F-16CJ-1-1
Klaus Huenecke, “Modern Combat Aircraft Design”, Airlife, 1984, 1987
Klaus Huenecke, “Modern Combat Aircraft Design”, Airlife, 1984, 1987
Klaus Huenecke, “Modern Combat Aircraft Design”, Airlife, 1984, 1987
R.M. Engelbeck, “Investigation Into the Impact of Agility in Conceptual Fighter Design”, NASA Contractor Report 195079, Boeing Defense & Space Group, Seattle Washington, Contract NAS1=18762, June 1995
Brandt, “Introduction to Aerodynamics – A Design Perspective”, Ch. 5 , Performance and Constraint Analysis
Brandt, “Introduction to Aerodynamics – A Design Perspective”, Ch. 5 , Performance and Constraint Analysis