Fighter Aircraft Performance, Part I of two, describes the parameters that affect aircraft performance.
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Fighter Aircraft Performance, Part II of two, describes the parameters that affect aircraft performance.
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Describes Radar Tracking Loops in Range, Doppler and Angles.
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Fighter Aircraft Performance, Part I 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.
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.
Describes Radar Tracking Loops in Range, Doppler and Angles.
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For more presentations on different subjects visit my website at http://solohermelin.com.
Abstract This paper presents the design and implementation of a quadcopter capable of payload delivery. A quadcopter is a unique unmanned aerial vehicle which has the capability of vertical take-off and landing. In this design, the quadcopter was controlled wirelessly from a ground control station using radio frequency. It was modeled mathematically considering its attitude and altitude, and a simulation carried out in MATLAB by designing a proportional Integral Derivative (PID) controller was applied to a mathematical model. The PID controller parameters were then applied to the real system. Finally, the output of the simulation and the prototype were compared both in the presence and absence of disturbances. The results showed that the quadcopter was stable and able to compensate for the external disturbances.
Abstract This paper presents the design and implementation of a quadcopter capable of payload delivery. A quadcopter is a unique unmanned aerial vehicle which has the capability of vertical take-off and landing. In this design, the quadcopter was controlled wirelessly from a ground control station using radio frequency. It was modeled mathematically considering its attitude and altitude, and a simulation carried out in MATLAB by designing a proportional Integral Derivative (PID) controller was applied to a mathematical model. The PID controller parameters were then applied to the real system. Finally, the output of the simulation and the prototype were compared both in the presence and absence of disturbances. The results showed that the quadcopter was stable and able to compensate for the external disturbances.
Aerodynamic and Acoustic Parameters of a Coandã Flow – a Numerical Investigationdrboon
Coandã flows have been the study of aircraft designers primarily for the prospect of achieving higher lift coefficient wings. Recently the environmental problem of noise pollution attracted further interest on the matter. The approach used is numerical; the computations were made using a large eddy simulation (LES) technique coupled with a Ffowcs-Williams-Hawkings (FWH) acoustic analysis. The spectrum of the flow was measured at three locations in the vicinity of the ramp showing that the low frequency region is dominant. The findings may be used as reference for the development of quiet aircraft that use super-circulation, as it is the case with the Upper Surface Blown (USB) configurations.
Calibrating a CFD canopy model with the EC1 vertical profiles of mean wind sp...Stephane Meteodyn
For some projects, applying the basic rules of EC1 is not sufficient, and it is required to get a more accurate estimation of the wind speed on the construction site. This can be done by using computational fluid dynamics codes which have the advantage, both to take into account of the terrain inhomogeneity and to calculate 3D orographic effects. In this way, the orography and roughness effects are coupled as they are in the real world. However, applying CFD computations must be in coherence with EC1 code. Then it is necessary to calibrate the ground friction for low roughness terrains as well as the drag force and turbulence production in case of high roughness lengths due to the presence of a canopy (forests or built areas). That is the condition for such methods to be commonly used and agreed by Building Control Officers. In this mind, TopoWind has been developed especially for wind design applications and can be a very useful, practical and objective tool for wind design engineers. The canopy model implemented in TopoWind has been calibrated in order to get the mean wind and turbulence profiles as defined in the EC1 for standard terrains. In this way, TopoWind computations satisfy the continuity between the EC1 values for homogeneous terrains and the more complex cases involving inhomogeneous roughness or orographic effects
Specific energy and curve, criterion for critical flow,free over fall, determination of velocity head,Local phenomenon-hydraulic jump, examples, determination of specific energy.
Abstract This paper presents the design and implementation of a quadcopter capable of payload delivery. A quadcopter is a unique unmanned aerial vehicle which has the capability of vertical take-off and landing. In this design, the quadcopter was controlled wirelessly from a ground control station using radio frequency. It was modeled mathematically considering its attitude and altitude, and a simulation carried out in MATLAB by designing a proportional Integral Derivative (PID) controller was applied to a mathematical model. The PID controller parameters were then applied to the real system. Finally, the output of the simulation and the prototype were compared both in the presence and absence of disturbances. The results showed that the quadcopter was stable and able to compensate for the external disturbances.
Abstract This paper presents the design and implementation of a quadcopter capable of payload delivery. A quadcopter is a unique unmanned aerial vehicle which has the capability of vertical take-off and landing. In this design, the quadcopter was controlled wirelessly from a ground control station using radio frequency. It was modeled mathematically considering its attitude and altitude, and a simulation carried out in MATLAB by designing a proportional Integral Derivative (PID) controller was applied to a mathematical model. The PID controller parameters were then applied to the real system. Finally, the output of the simulation and the prototype were compared both in the presence and absence of disturbances. The results showed that the quadcopter was stable and able to compensate for the external disturbances.
Aerodynamic and Acoustic Parameters of a Coandã Flow – a Numerical Investigationdrboon
Coandã flows have been the study of aircraft designers primarily for the prospect of achieving higher lift coefficient wings. Recently the environmental problem of noise pollution attracted further interest on the matter. The approach used is numerical; the computations were made using a large eddy simulation (LES) technique coupled with a Ffowcs-Williams-Hawkings (FWH) acoustic analysis. The spectrum of the flow was measured at three locations in the vicinity of the ramp showing that the low frequency region is dominant. The findings may be used as reference for the development of quiet aircraft that use super-circulation, as it is the case with the Upper Surface Blown (USB) configurations.
Calibrating a CFD canopy model with the EC1 vertical profiles of mean wind sp...Stephane Meteodyn
For some projects, applying the basic rules of EC1 is not sufficient, and it is required to get a more accurate estimation of the wind speed on the construction site. This can be done by using computational fluid dynamics codes which have the advantage, both to take into account of the terrain inhomogeneity and to calculate 3D orographic effects. In this way, the orography and roughness effects are coupled as they are in the real world. However, applying CFD computations must be in coherence with EC1 code. Then it is necessary to calibrate the ground friction for low roughness terrains as well as the drag force and turbulence production in case of high roughness lengths due to the presence of a canopy (forests or built areas). That is the condition for such methods to be commonly used and agreed by Building Control Officers. In this mind, TopoWind has been developed especially for wind design applications and can be a very useful, practical and objective tool for wind design engineers. The canopy model implemented in TopoWind has been calibrated in order to get the mean wind and turbulence profiles as defined in the EC1 for standard terrains. In this way, TopoWind computations satisfy the continuity between the EC1 values for homogeneous terrains and the more complex cases involving inhomogeneous roughness or orographic effects
Specific energy and curve, criterion for critical flow,free over fall, determination of velocity head,Local phenomenon-hydraulic jump, examples, determination of specific energy.
Air Combat History describes the main air combats and fighter aircraft, from the beginning of aviation. The additional Youtube links are an important part of the presentation. A list of Air-to-Air Missile from different countries. is also given
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RADAR - RAdio Detection And Ranging
This is the Part 1 of 2 of RADAR Introduction.
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Part of the Figures were not properly downloaded. I recommend viewing the presentation on my website under RADAR Folder.
RADAR - RAdio Detection And Ranging
This is the Part 2 of 2 of RADAR Introduction.
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Part of the Figures were not properly downloaded. I recommend viewing the presentation on my website under RADAR Folder.
Estimate the hidden States, Parameters, Signals of a Linear Dynamic Stochastic System from Noisy Measurements. It requires knowledge of probability theory. Presentation at graduate level in math., engineering
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For more presentations on different subjects visit my website at http://solohermelin.com. Since a few Figure were not downloaded I recommend to see the presentation on my website at RADAR Folder, Tracking subfolder.
In 2001 Euroavia Toulouse organized a symposium on ground effect. We invited most of the Russian and German actors, and some experts from Holland, UK or France for a week of science around the subject of ekranoplans / flying boats. This was dedicated to students. A book was issued... and now that all copies have been sold for a while I am sharing this on LinkedIn for everyone.
Enjoy.
Stéphan AUBIN
We present a novel modeling
methodology to derive a nonlinear dynamical model which
adequately describes the effect of fuel sloshing on the attitude dynamics of a spacecraft. We model the impulsive thrusters using mixed logic and dynamics leading to a hybrid formulation.
We design a hybrid model predictive control scheme for the
attitude control of a launcher during its long coasting period,
aiming at minimising the actuation count of the thrusters.
In this relative motion and relative speed concept is demonstrated with help of examples, graphically and mathematically. The concepts of Einstein and Galileo
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Theoretical work submitted to the Journal should be original in its motivation or modeling structure. Empirical analysis should be based on a theoretical framework and should be capable of replication. It is expected that all materials required for replication (including computer programs and data sets) should be available upon request to the authors.
This paper is my work on Einstein's Relativity theory for speed greater than light.The paper mainly focus on to generalize the Lorentz factor , equation of negative energy and negative mass thus proving the existence of wormholes and the derivation of new space metric.
Platoon Control of Nonholonomic Robots using Quintic Bezier SplinesKaustav Mondal
In this project, quintic polynomials were used to perform platooning in nonholonomic robots. Both hardware and simulations results have been presented.
Aircraft Susceptibility and Vulnerability.
This is from the last presentations from my side. Medical Problems prevent me to continue with new presentations.Please do not contact me.
Describes concepts and development of flying cars and other flying vehicles. Reference are given including to YouTube movies. At the end my view of Main Requirements and the related Design Requirements for a SkyCar are given. The main conclusion is that technologically we are ready to develop and product such a SkyCar in a few years.
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This presentation gives example of "Calculus of Variations" problems that can be solved analytical. "Calculus of Variations" presentation is prerequisite to this one.
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Describes the mathematics of the Calculus of Variations.
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Aerodynamics Part III of 3 describes aerodynamics of wings in supersonic flight.
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Aerodynamics Part II of 3 describes aerodynamics of bodies in supersonic flight.
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Aerodynamics Part I of 3 describes aerodynamics of wings and bodies in subsonic flight.
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Richard's aventures in two entangled wonderlandsRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
Slide 1: Title Slide
Extrachromosomal Inheritance
Slide 2: Introduction to Extrachromosomal Inheritance
Definition: Extrachromosomal inheritance refers to the transmission of genetic material that is not found within the nucleus.
Key Components: Involves genes located in mitochondria, chloroplasts, and plasmids.
Slide 3: Mitochondrial Inheritance
Mitochondria: Organelles responsible for energy production.
Mitochondrial DNA (mtDNA): Circular DNA molecule found in mitochondria.
Inheritance Pattern: Maternally inherited, meaning it is passed from mothers to all their offspring.
Diseases: Examples include Leber’s hereditary optic neuropathy (LHON) and mitochondrial myopathy.
Slide 4: Chloroplast Inheritance
Chloroplasts: Organelles responsible for photosynthesis in plants.
Chloroplast DNA (cpDNA): Circular DNA molecule found in chloroplasts.
Inheritance Pattern: Often maternally inherited in most plants, but can vary in some species.
Examples: Variegation in plants, where leaf color patterns are determined by chloroplast DNA.
Slide 5: Plasmid Inheritance
Plasmids: Small, circular DNA molecules found in bacteria and some eukaryotes.
Features: Can carry antibiotic resistance genes and can be transferred between cells through processes like conjugation.
Significance: Important in biotechnology for gene cloning and genetic engineering.
Slide 6: Mechanisms of Extrachromosomal Inheritance
Non-Mendelian Patterns: Do not follow Mendel’s laws of inheritance.
Cytoplasmic Segregation: During cell division, organelles like mitochondria and chloroplasts are randomly distributed to daughter cells.
Heteroplasmy: Presence of more than one type of organellar genome within a cell, leading to variation in expression.
Slide 7: Examples of Extrachromosomal Inheritance
Four O’clock Plant (Mirabilis jalapa): Shows variegated leaves due to different cpDNA in leaf cells.
Petite Mutants in Yeast: Result from mutations in mitochondrial DNA affecting respiration.
Slide 8: Importance of Extrachromosomal Inheritance
Evolution: Provides insight into the evolution of eukaryotic cells.
Medicine: Understanding mitochondrial inheritance helps in diagnosing and treating mitochondrial diseases.
Agriculture: Chloroplast inheritance can be used in plant breeding and genetic modification.
Slide 9: Recent Research and Advances
Gene Editing: Techniques like CRISPR-Cas9 are being used to edit mitochondrial and chloroplast DNA.
Therapies: Development of mitochondrial replacement therapy (MRT) for preventing mitochondrial diseases.
Slide 10: Conclusion
Summary: Extrachromosomal inheritance involves the transmission of genetic material outside the nucleus and plays a crucial role in genetics, medicine, and biotechnology.
Future Directions: Continued research and technological advancements hold promise for new treatments and applications.
Slide 11: Questions and Discussion
Invite Audience: Open the floor for any questions or further discussion on the topic.
THE IMPORTANCE OF MARTIAN ATMOSPHERE SAMPLE RETURN.Sérgio Sacani
The return of a sample of near-surface atmosphere from Mars would facilitate answers to several first-order science questions surrounding the formation and evolution of the planet. One of the important aspects of terrestrial planet formation in general is the role that primary atmospheres played in influencing the chemistry and structure of the planets and their antecedents. Studies of the martian atmosphere can be used to investigate the role of a primary atmosphere in its history. Atmosphere samples would also inform our understanding of the near-surface chemistry of the planet, and ultimately the prospects for life. High-precision isotopic analyses of constituent gases are needed to address these questions, requiring that the analyses are made on returned samples rather than in situ.
Richard's entangled aventures in wonderlandRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
Nutraceutical market, scope and growth: Herbal drug technologyLokesh Patil
As consumer awareness of health and wellness rises, the nutraceutical market—which includes goods like functional meals, drinks, and dietary supplements that provide health advantages beyond basic nutrition—is growing significantly. As healthcare expenses rise, the population ages, and people want natural and preventative health solutions more and more, this industry is increasing quickly. Further driving market expansion are product formulation innovations and the use of cutting-edge technology for customized nutrition. With its worldwide reach, the nutraceutical industry is expected to keep growing and provide significant chances for research and investment in a number of categories, including vitamins, minerals, probiotics, and herbal supplements.
2. 08/12/15 2
Performance of an Aircraft with Parabolic PolarSOLO
Table of Content
Flat Earth Three Degrees of Freedom Aircraft Equations
Performance of an Aircraft with Parabolic Polar
Aircraft Drag
Energy per unit mass E
Load Factor n
Aircraft Trajectories
Summary
Constraints
Horizontal Plan Trajectory ( )0,0 == γγ
Horizontal Turn Rate as Function of ps, n
Horizontal Turn Rate as Function of nV,
References
3. 08/12/15 3
SOLO
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)
•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
Flat Earth Three Degrees of Freedom Aircraft Equations
4. 4
SOLO
• Rotation Matrix from Earth to Wind Coordinates
[ ] [ ] [ ]321 χγσ=W
EC
where
σ – Roll Angle
γ – Elevation Angle of the Trajectory
χ – Azimuth Angle of the Trajectory
Force Equation:
amgmTFA
=++
where:
• Aerodynamic Forces (Lift L and Drag D)
( )
−
−
=
L
D
F
W
A 0
• Thrust T ( )
=
α
α
sin
0
cos
T
T
T W
• Gravitation acceleration
( ) ( )
−
−
−
==
g
cs
sc
cs
sc
cs
scgCg EW
E
W
0
0
100
0
0
0
010
0
0
0
001
χχ
χχ
γγ
γγ
σσ
σσ
( )
g
cc
cs
s
g W
−
=
γσ
γσ
γ
α
T
V
L
D
Bx
Wx
Bz
Wz
Wy
By
Flat Earth Three Degrees of Freedom Aircraft Equations
5. 5
SOLO
α
T
V
L
D
Bx
Wx
Bz
Wz
Wy
By
• Aircraft Acceleration
( )
( )
( ) ( )WW
W
W
VVa
×+=
→
ω
where:
( )
=
0
0
V
V W
and
( )
=
→
0
0
V
V
W
( )
−+
−
+
−
=
=
χ
χχ
χχ
γ
γγ
γγσ
σσ
σσω
0
0
100
0
0
0
0
0
010
0
0
0
0
0
001
cs
sc
cs
sc
cs
sc
r
q
p
W
W
W
W
or ( )
+−
+
−
=
=
γσχσγ
γσχσγ
γχσ
ω
ccs
csc
s
r
q
p
W
W
W
W
therefore
( )
( )
( ) ( )
( )
( )
+−
+−=
−
=×+=
→
γσχσγ
γσχσγω
cscV
ccsV
V
qV
rV
V
VVa
W
W
WW
W
W
Flat Earth Three Degrees of Freedom Aircraft Equations
6. 6
SOLO
α
T
V
L
D
Bx
Wx
Bz
Wz
Wy
By
• Aircraft Acceleration
Flat Earth Three Degrees of Freedom Aircraft Equations
From the Force equation we obtain:
( )
( )
( ) ( ) ( ) ( )
( ) ( )WWW
A
WW
W
W
gTF
m
VVa
++=×+=
→
1
ω
or
( )
( ) ( ) σ
σ
σ
σ
γσαγσχσγ
γσγσχσγ
γα
s
c
c
s
ccgmLTcscVqV
csgccsVrV
sgmDTV
W
W
−−
−
++−=+−=−
=+−=
−−=
/sin
/)cos(
from which we obtain:
( )
( )
+=
−+=
−−=
msLTcV
cgmcLTV
sgmDTV
/sin
/sin
/)cos(
σαγχ
γσαγ
γα
Define the Load Factor
gm
LT
n
+
=
αsin
:
7. 7
SOLO
α
T
V
L
D
Bx
Wx
Bz
Wz
Wy
By
• Velocity Equation
Flat Earth Three Degrees of Freedom Aircraft Equations
( ) ( )
==
=
0
0
V
CVC
h
y
x
V E
W
WE
W
E
−
−
−
=
0
0
0
0
001
0
010
0
100
0
0 V
cs
sc
cs
sc
cs
sc
h
y
x
σσ
σσ
γγ
γγ
χχ
χχ
=
=
=
γ
χγ
χγ
sVh
scVy
ccVx
or
• Energy per unit mass E
g
V
hE
2
:
2
+=
Let differentiate this equation:
( )
W
VDT
W
DT
g
g
V
V
g
VV
hEps
−
=
−
−
+=+==
α
γ
α
γ
cos
sin
cos
sin:
Return to Table of Content
8. 8
SOLO Flat Earth Three Degrees of Freedom Aircraft Equations
Summary
γ
χγ
χγ
sin
sincos
coscos
Vh
Vy
Vx
=
=
=
( )
( )
σ
γ
σ
γ
α
χ
γσγσ
α
γ
α
γ
α
sin
cos
sin
cos
sin
coscoscoscos
sin
cos
sin
cos
n
V
g
W
LT
V
g
n
V
g
W
LT
V
g
W
VDT
Eor
W
DT
gV
=
+
=
−=
−
+
=
−
=−
−
=
where
mgW =
Aircraft Thrust( ) 10, ≤≤= ηη VhTT MAX
( ) ( )MSCVhL L ,
2
1 2
αρ=
( ) ( )LD CMSCVhD ,
2
1 2
ρ=
( ) ( ) ( ) 2
0, LDLD CMKMCCMC +=
( ) 0/
0
hh
eh −
= ρρ
( ) ( ) soundofspeedhaNumberMachMhaVM === &/
Aircraft Weight
Aircraft Lift
Aircraft Drag
Parabolic Drag Polar
Return to Table of Content
9. 9
SOLO Flat Earth Three Degrees of Freedom Aircraft Equations
Constraints:
State Constraints
• Minimum Altitude Limit minhh ≥
• Maximum dynamic pressure limit ( ) ( )hVVorqVhq MAXMAX ≤≤= 2
2
1
ρ
• Maximum Mach Number limit
( ) MAXM
ha
V
≤
Aerodynamic or heat limitation
Control Constraints
• Maximum Lift Coefficient or Maximum Angle of Attack
( ) ( ) ( )VhorMCMC STALLMAXLL ,, _ ααα ≤≤
• Minimum Load Factor
( )
MAXn
W
VhL
n ≤=
,
• Maximum Thrust
( )VhTT MAX ,≤
10. 10
SOLO Flat Earth Three Degrees of Freedom Aircraft Equations
MAXα MAXα
αα
LCDC
MAXLC
0DC
( ) ( )αα
2
0 LDD kCCC +=
Drag and Lift Coefficients as functions of Angle of Attack
11. 11
SOLO Flat Earth Three Degrees of Freedom Aircraft Equations
( ) ( )
( )
Limit
Vhor
MCMC
STALL
MAXLL
,
, _
αα
α
=
=
( )
( )
Limit
hVVor
qVhq
MAX
MAX
=
== 2
2
1
ρ
minhh =
MAXMM =
Mach
Altitude
Flight Envelope of the Aircraft
Return to Table of Content
12. 12
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 assumed 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
13. 13
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
0
*
2
1
:*
*
:
2
*
2
1
:*
Vq
V
V
u
CS
kW
CS
W
V
D
L
ρ
ρρ
2
0
:
LD
L
D
L
CkC
C
CSq
CSq
D
L
e
+
===
Climbing Aircraft Performance
14. 14
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
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
=
=
=
2
0
*
2
1
:*
*
:
2
:*
Vq
V
V
u
CS
kW
V
D
ρ
ρ
15. 15
Performance of an Aircraft with Parabolic PolarSOLO
We obtained
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
u 0
- - - - 0 + + + + +
D ↓ min ↑
n
u
D
∂
∂
+= 2
2
2
*2 u
n
u
e
W
D
16. 16
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
Return to Table of Content
Climbing Aircraft Performance
Maximum Load Factor
=
=
=
2
0
*
2
1
:*
*
:
2
:*
Vq
V
V
u
CS
kW
V
D
ρ
ρ
17. 17
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
* −+−
=
nz
nzzu
nzzu
nuzup constns >
−+=
−−=
→=+−→==
22
2
22
1224
020
( ) ( )
2
224
2
2243
23
*
*244
*
*
u
nuzu
e
V
u
nuzuuuzu
e
V
u
p
constn
s ++−
=
−+−−+−
=
∂
∂
=
3
3
0
22
nzz
u
u
p
MAX
constn
s
++
=→=
∂
∂
=
2
21
2
uu
uu
MAX <<
+
nz >
Climbing Aircraft Performance
18. 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 ==
19. 19
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
nuzu
e
V
ps
224
2
*
* −+−
=
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
=→=+−=
∂
∂
=
Return to Table of Content
Climbing Aircraft Performance
=
=
=
2
0
*
2
1
:*
*
:
2
:*
Vq
V
V
u
CS
kW
V
D
ρ
ρ
20. 20
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
u
V
pe
uzun s
*
*2
2 24
−+−=
( )
3
0412 2
2
22
z
uzu
u
n
constps
=→=+−=
∂
∂
=
( )
*
*2
44 3
2
V
pe
uzu
u
n s
constps
−+−=
∂
∂
=
Climbing Aircraft Performance
=
=
=
2
0
*
2
1
:*
*
:
2
:*
Vq
V
V
u
CS
kW
V
D
ρ
ρ
21. 21
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
Return to Table of Content
Climbing Aircraft Performance
=
=
=
2
0
*
2
1
:*
*
:
2
:*
Vq
V
V
u
CS
kW
V
D
ρ
ρ
22. 22
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 Trajectories
We found
Aircraft Turn Performance
23. 23
Performance of an Aircraft with Parabolic PolarSOLO
( ) ( )
( )
γ
σ
φ
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
25. 25
Performance of an Aircraft with Parabolic PolarSOLO
2. Horizontal Plan Trajectory ( )0,0 == γγ
We can see that for n > 1
1
1
1'
1
11'
2
2
2
2
22
−
≈
−
=
−≈−=
ng
V
ng
V
R
n
V
g
n
V
g
χ
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
26. 26
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
27. 27
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
a 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
28. 28
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
29. 29
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
30. 30
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
31. 31
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
s
χ
Aircraft Turn Performance
32. 32
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
33. 33
Performance of an Aircraft with Parabolic PolarSOLO
Horizontal Turn Rate as Function of ps, n
u
0<sp
0=sp
0>sp
χ
( )12
*
−z
V
g
1=u1u
2u
2
24
1
*
*2
2
* u
up
V
e
uzu
V
g s −−+−
=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
nMAX
LIMIT
C MAXL_
MAX
MAX
L
MAXL
n
n
C
C
V
g 1
**
2
_ −
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
34. 34
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
35. 35
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
36. 36
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
The minimum of 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
37. 37
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
38. 38
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
39. 39
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
40. 08/12/15 40
Performance of an Aircraft with Parabolic PolarSOLO
References
Angelo Miele, “Flight Mechanics, Volume I, Theory of Flight Paths”,
Addison-Wesley, 1962
Return to Table of Content
S. Hermelin, “3 DOF Model of Aircraft in Earth Atmosphere”
41. 41
SOLO
Technion
Israeli Institute of Technology
1964 – 1968 BSc EE
1968 – 1971 MSc EE
Israeli Air Force
1970 – 1974
RAFAEL
Israeli Armament Development Authority
1974 – 2013
Stanford University
1983 – 1986 PhD AA
Editor's Notes
Prof. Earll Murman, “Introduction to Aircraft Performance and Static Stability”, September 18, 2003