Direct Torque Control for Doubly Fed Induction Machine-Based Wind Turbines un...
TVC main report
1. 1
CHAPTER 1
INTRODUCTION
.
Thrust vector control system or thrust vectoring is a technology that deflects the mean
flow of an engine jet from the centerline to transfer some force to the aimed axis. By that
imbalance, a momentum is created and used to control the change of attitude of the
aircraft. Among other things, thrust vector greatly improves maneuverability, even at
high angle of attack or low speeds where conventional aerodynamic control surface loses
are effectiveness. A propulsive system not only provides thrust but also a means in
controlling its flight path by redirecting its thrust vector to provide directional control.
This is known as THRUST VECTOR CONTROL (TVC).
Thrust vectoring, also Thrust vector control or TVC, is the ability of an aircraft or rocket,
or other vehicle to manipulate the direction of the thrust from its engine or motor in order
to control the attitude or angular velocity of the vehicle
There are several methods of thrust vectoring system available but Thrust Vectoring by
JET DEFLECTION method is chosen as it satisfies the requirement of our system.
2. 2
1.1 AIM:
To analyze the thrust vectoring control system using JET DEFLCTION method.
1.2 SCOPE:
Jet deflection thrust vector controlling method is with light weight and has high response.
This can be controlled using manual gears or RC. This method can be applied to small
scale and low thrust available engines.
1.3 SOFTWARES USED:
SOLIDWORKS
ANSYS
FLUENT
1.4 TYPES OF THRUST VECTORING
GIMBALED ENGINE:
In this case, the engine has a hinge or a gimbal ( a universal joint) that allows
rotation about its axis – that is the whole engine is pivoted on a bearing.
Flexible Laminated Bearing
The swiveled nozzle changes the direction of throat and nozzle. It is similar to
gimbal engine. The main drawback in using this method is the difficulty in
fabricating the seal joint of the swivel since the swivel is exposed to extreme high
pressure and temperature.
3. 3
JET VANES
JET VANES are small airfoils located at the nozzle exit plane, and behave like
aileron or elevator on an aircraft, and cause the vehicle to change the direction.
This control system causes a loss of thrust (2 to 3%), and erosion of vanes.
JETAVATORS
This system has rotating airfoil shaped collar, and gives an unsymmetrical
distribution of gas flow. This provides a side force thereby changing the direction
of the flight.
.
JET TABS
This system has tabs rotated by hydraulic actuators. Power is supplied from
compressed nitrogen. Usually, this type of TVC methods is used in military
missiles.
1.5 BENEFITS OF THRUST VECTORING
Enhanced performance in conventional flight.
Extended flight envelope.
Increased safety
Reduction of aero controls
4. 4
CHAPTER 2
2. LITERATURE STUDY
Our aim is to design lightweight with fast response RC type thrust vectoring system.
2.1 SELECTION OF THRUST VECTORING METHOD
Secondary fluidic injection TVC system requires modifying the engine nozzle to
create a cavity for flow manipulation as well as injection ports, the lack of readily
available source of engine bleed air necessitates the use of storage tank which
would have a large weight and occupies large area and also plumbing work
required for this system increases weight and increases the design difficulties.
Gimbal engine is relatively higher in weight and complex in design makes it
difficult for this application.
Jetavators and paddles posses high thermal stress and bulky which results in high
weight hence it is not suitable for the design.
Adjustable internal nozzle contouring does not produce satisfactory degrees of
vectoring, this method also greatly reduces the overall thrust level hence it is not
valuable.
5. 5
Post nozzle exit exhaust flow manipulation leaves the original engine design intact
and has small weight penalties. Thrust loss associated with supersonic flow
turning will not be an issue since our system is designed for subsonic flow turning.
Hence vectoring method using airfoils mounted directly on exhaust flow is
selected for TVC system. This method reduces the available thrust but care has
been taken to ensure that sufficient thrust is produced.
2.2 AIRFOIL DESIGN:
S.No. Type Cause Result Decision
1. Circular leading edge
which directly extends
into flat section before
tapering to a sharp trailing
edge
In pressure
distribution there are
two rise in pressure.
One at leading edge
and at transition region
of taper region
Flow separation
starts at low angle
of attack
No
2. Circular leading region
without flat region but
start tapering
Rise in pressure
distribution is very
high at leading edge
Flow separation
starts at low angle
of attack
No
3. Cambered airfoil No uniform flow after
passing the airfoil
No
4. Symmetrical airfoil Rise in pressure
distribution
is not so high
No flow
separation at low
angle of attack
Yes
Hence symmetrical airfoil is chosen.
6. 6
2.2.1 NACA 0012 AIRFOIL
From classical thin airfoil theory 0f symmetrical airfoil
Coefficient of lift (Cl) = 2πα
Lift slope = 2π
The center of pressure and the aerodynamic center are both located at the
quarter – chord point.
Hence,
From thin airfoil theory – quarter chord is the point where center of
pressure and aerodynamic center are present.
For the required geometry fitting should be at the quarter chord.
Thickness should be around 12% of the airfoil
NACA 0012 satisfies all the above requirements.
Since thickness above 12% does not satisfy thin airfoil theory, our selection
is NACA 0012 airfoil.
7. 7
CHAPTER 3
3. THEORY
3.1 CFD (Computational Fluid Dynamics)
CFD is a technology that enables us to study the dynamics of things that
flow. Using CFD we can develop a computational model of the system or device that we
want to study. Fluid flow physics is applied along with some chemistry and the software
will output the fluid dynamics and the related physical phenomena. With the advent of
CFD, one has the power to simulate the flow of gases and liquids, heat and mass transfer
moving bodies, multiphase physics, chemical reaction, fluid structure interaction and
acoustics through computer modeling. A virtual prototype of the system can be built
using CFD (Fluent Inc.). Thus CFD can be applied for predicting the fluid flow
associated with the complications of simultaneous flow of heat, mass transfer, phase
change, chemical reaction etc. using computers. CFD has now become an integral part of
the engineering design and analysis. Engineers can make optimal use of the CFD tools to
simulate fluid flow and heat transfer phenomena in a system model and can even predict
the system performance before actually manufacturing it.
3.2 Navier-Stokes Equation:
It is named after Claude-Louis Navier and George Stokes
and describes the motion of fluid substance. It’s also a fundamental equation being used
by ANSYS and even in the present project work being carried out. These equations arise
from applying second law of Newton to fluid motion, together with the assumptions that
the fluid stress is the sum of a diffusing viscous term (proportional to the gradient of
velocity), plus a pressure term.
8. 8
The derivation of the Navier-Stokes equation begins with an application of second law of
Newton i.e. conservation of momentum (often alongside mass and energy conservation)
being written for an arbitrary portion of the fluid. In an inertial frame of reference, the
general form of the equations of fluid motion is
Where
V is the flow velocity,
is the fluid density,
P is the pressure,
T is the (deviatoric) stress tensor
F represents body forces (per unit volume) acting on the fluid.
stands for del operator.
This equation often written using the material derivative, denoted as Dv/Dt, making it
more apparent that is a statement of second law of Newton:-
The left side of the equation describes acceleration, and may be composed of time
dependent or convective effects (also the effects of non-inertial coordinates if present).
The right side of the equation is in effect a summation of body forces (such as gravity)
and divergence of stress (pressure and shear stress).
9. 9
3.3 Benefits of CFD
Insight- if there is a design or system design which is difficult to analyze or test
through experimentation, CFD analysis enables us to virtually sneak inside the
design and see how it performs. CFD gives a deep perception into the designs.
There are many occurrences that we can witness through CFD which wouldn’t be
visible through any other means.
Foresight- under a given set of circumstances, we can envisage through the CFD
software what will happen. In short time we can predict how the design will
perform and test many variants until we arrive at an ideal result.
Efficiency- the foresight we gain help us to design better to achieve good results.
CFD is a device for compressing the design and development cycle allowing for
rapid prototyping
.
3.4 Advantages of CFD can be summarized as
The effect of various parameters and variables on the behavior of the system can
be studied instantaneously since the speed of computing is very high. To study the
same in an experimental setup is not only difficult and tedious but also sometimes
may be impossible.
In terms of cost factor, CFD analysis will be much cheaper than setting up
experiments or building sample model of physical systems.
Numerical modeling is flexible in nature. Problems with different level of
complexity can be simulated.
It allows models and physical understanding of the problem to be improved, very
much similar to conducting experiments.
In some cases it may be the only practicable ancillary for experiments.
10. 10
3.5 CFD Process
The steps underlying the CFD process are as follows:
Geometry of the problem is defined.
Volume occupied by fluid is divided into discrete cells.
Physical modeling is well-defined.
Boundary conditions are defined which involves specifying the fluid behavior and
properties at the boundaries.
Equations are solved iteratively as steady state or transient state.
Analysis and visualization of resulting solution is carried out.
3.6Limitations of CFD
Even if there are many advantages of CFD, there are few shortcomings of it as follows :-
CFD solutions rely upon physical models of real world processes. Solving
equations on a computer invariably introduces numerical errors.
Truncation errors due to approximation in the numerical models.
Round-off errors due to finite word size available on the computer.
The accuracy of the CFD solution depends heavily upon the initial or
boundary conditions provided to numerical model.
3.7 Comparative Study of Experimental, Analytical and Numerical Methods
Experimental Method - Experimental methods are used to obtain consistent
information about physical processes which are not clearly understood. It is the
most realistic approach for problem solving. It may involve full scale, small scale
or blown up scale model. However disadvantages are high cost, measurement
difficulties and probe errors.
11. 11
Analytical Method - These methods are used to obtain solution of mathematical
model which consists of a set of differential equations that represent a physical
process within the limit of conventions made. The systematic solution often
contain infinite series, special functions etc. and hence their numerical evaluation
becomes difficult to handle.
Numerical method – Numerical prediction works on the results of the
mathematical model. The solution is obtained for variables at distinct grid points
Within the computational field. It provides for greater handling of complex
geometry and nonlinearity in governing equations or boundary conditions. The
kind of ease provided by numerical methods makes it the powerful and widely
applicable. The above said discussion is represented in tabular form in table 3.1.
3.8 Table Comparison of Experimental, Analytical and Numerical Methods of Solution
Name of the Method Advantages Disadvantages
1.Experimental Capable of being most
realistic Equipment required
Scaling problem
Measurement
difficulties
Probe errors
High operating costs
2.Analytical Clean, general information
which is usually in
formula form
Restricted to simple
geometry and physics
Usually restricted to
linear problems
Cumbersome results-
difficult to compute
12. 12
3.Numerical No restriction to linearity.
Ability to handle irregular
geometry and complicate
physics.
Low cost and high speed
of computation
Truncation and round-
off errors
Boundary condition
problems
An assessment of advantages and disadvantages of numerical methods vis-à-vis
analytical and experimental method shows that even though the Numerical Method has
few shortcomings but it has many advantages associated with it and is hence suited.
13. 13
CHAPTER 4
4. EXPERIMENTAL THEORY OF COMPUTATIONAL FLUID DYNAMICS
4.1 SOLVER
4.1.1 DENSITY BASED SOLVER:
This method was developed for low speed incompressible flows, while density –
based is used for high speed compressible flows.
In both methods the velocity field is obtained from the momentum equations.
In the density – based approach, the continuity equation is used to obtain the
density field while the pressure field is determined from the equation of state.
In pressure based approach, the pressure field is extracted by solving a pressure or
pressure correction equation which is obtained by manipulating continuity and
momentum equations.
Hence the pressure based solver has been mainly used for incompressible and
mildly compressible flows.
4.1 SOLVER
4.1.2K- EPSILON TURBULENT MODEL:
K-epsilon (k-ε) turbulence model is the most common model used in Computational
Fluid Dynamics (CFD) to simulate mean flow characteristics for turbulent flow
conditions. It is a two equation model which gives a general description of turbulence by
means of two transport equations (PDEs). The original impetus for the K-epsilon model
was to improve the mixing-length model, as well as to find an alternative to algebraically
prescribing turbulent length scales in moderate to high complexity flows.
14. 14
The first transported variable determines the energy in the turbulence and is
called turbulent kinetic energy (k).
The second transported variable is the turbulent dissipation ( ) which determines the
rate of dissipation of the turbulent kinetic energy.
The k-ε model has been tailored specifically for planar shear layers and recirculation
flows. This model is the most widely used and validated turbulence model with
applications ranging from industrial to environmental flows, which explains its
popularity. It is usually useful for free-shear layer flows with relatively small
pressure gradients as well as in confined flows where the Reynolds shear stresses are
most important. It can also be stated as the simplest turbulence model for which only
initial and or boundary conditions needs to be supplied.
However it is more expensive in terms of memory than the mixing length model as it
requires two extra PDEs. This model would be an inappropriate choice for problems such
as inlets and compressors as accuracy has been shown experimentally to be reduced for
flows containing large adverse pressure gradients. The k-ε model also performs poorly in
a variety of important cases such as unconfined flows, curved boundary layers, rotating
flows and flows in non-circular ducts.
4.3 BOUNDRY CONDITIONS:
General: Pressure inlet, Pressure outlet.
Incompressible: Velocity inlet, Outflow.
Compressible: Mass flow inlet, Pressure far-field, Mass flow outlet.
Other: Wall, Symmetry, Axis.
Special: Inlet vent, Outlet vent, Intake fan, Exhaust fan.
15. 15
4.3.1 Velocity Inlet:
1. Define velocity and other properties at the inlet.
2. The flow is incompressible.
3. The velocity is applied at the inlet.
4.3.2 Pressure inlet:
1. Define total pressure and other properties at the inlet.
2. The flow direction is also defined.
3. If the inlet is Supersonic, then the static pressure is also to be specified.
4.3.3 Pressure outlet:
1. The static pressure is defined at the inlet.
2. This process works well in case there if is a back flow.
3. This condition is used the model is set up with pressure inlet.
4. This condition is suitable for compressible and incompressible flow.
4.3.4 Pressure far-field:
1. This condition is used in free stream compressible flow at infinity, with free
stream Mach number and static conditions are specified.
2. This condition is applied for compressible flow when density is calculated by
ideal gas.
16. 16
4.3.5 Outflow:
1. It is used when flow velocity and pressure are not known before the flow.
2. This condition cannot be used for compressible flow with pressure inlet.
3. This condition cannot be used for unsteady flow with variable density.
4.3.6 Mass flow inlet:
1. This condition is used in compressible flow to find mass flow rate.
2. Not mostly used in incompressible flow since velocity will fix the mass flow
rate.
4.4 MATERIAL PROPERTIES:
4.4.1 STAINLESS-STEEL (SOLID MEDIUM)
Density ( = 7480-8000 / 3
Specific heat ( = 510 /
Thermal conductivity (k) = 15 /
4.4.2 AIR (FLUID MEDIUM)
Density ( = 1.225 / 3
Specific heat ( = 1006.43 /
Thermal conductivity (k) = 0.0242 /
17. 17
4.5 SOLUTION METHODS
4.5.1 EXPLICIT AND IMPLICID METHODS:
These are approaches used in numerical analysis for obtaining numerical solutions of
time dependent ordinary and partial differential equations, as is required in computer
simulation of physical processes.
Y (t+ = F(Y (t))
While for an implicit method one soles an equation
G (Y (t), Y (t+ )) = 0 (1)
To find Y (t+ .
It is clear that implicit methods require an extra computation (solving the above
equation), and they can be much harder to implement. Implicit methods are used because
many problems arising in practice are stiff, for which the use of an explicit method
requires impractically small time steps t to keep the error in the result bounded. For
such problems to achieve given accuracy, it takes much less computational time to use an
implicit method with larger time steps even taking into account that one needs to solve an
equation of the form( 1) at each time step . These explain, whether one should use an
explicit or implicit method depend upon the problem to be solved.
4.5.2 UPWIND SCHEME
In computational physics, upwind schemes denote a class of
numerical discretization methods for solving hyperbolic partial differential equations.
Upwind schemes use an adaptive or solution-sensitive finite difference stencil to
numerically simulate the direction of propagation of information in a flow field. The
upwind schemes attempt to discretize hyperbolic partial differential equations by using
18. 18
differencing biased in the direction determined by the sign of the characteristic speeds.
Historically, the origin of upwind methods can be traced back to the work of Courant,
Isaacson, and Rees who proposed the CIR method.
4.5.2.1MODEL EQUATION
To illustrate the method, consider the following one-dimensional linear advection
equation
which describes a wave propagating along the -axis with a velocity . This equation is
also a mathematical model for one-dimensional linear advection. Consider a typical grid
point in the domain. In a one-dimensional domain, there are only two directions
associated with point – left and right. If is positive the left side is called upwind side
and right side is the downwind side. Similarly, if is negative the left side is
called downwind side and right side is the upwind side. If the finite difference scheme for
the spatial derivative, contains more points in the upwind side, the scheme is
called an upwind-biased or simply an upwind scheme.
4.5.2.2 FIRST ORDER UPWIND
The simplest upwind scheme possible is the first-order upwind scheme. It is given by[2]
19. 19
4.5.2.2.1Compact form
Defining
and
The two conditional equations (1) and (2) can be combined and written in a compact
form as
Equation (3) is a general way of writing any upwind-type schemes.
4.5.2.2.2Stability
The upwind scheme is stable if the following Courant–Friedrichs–Lewy condition (CFL)
condition is satisfied.
A Taylor series analysis of the upwind scheme discussed above will show that it is first-
order accurate in space and time. The first-order upwind scheme introduces
severenumerical diffusion in the solution where large gradients exist
4.5.3 SECOND ORDER UPWIND SCHEME
The spatial accuracy of the first-order upwind scheme can be improved by including 3
data points instead of just 2, which offers a more accurate finite difference stencil for the
approximation of spatial derivative. For the second-order upwind scheme, becomes
the 3-point backward difference in equation (3) and is defined as
20. 20
and is the 3-point forward difference, defined as
This scheme is less diffusive compared to the first-order accurate scheme and is called
linear upwind differencing (LUD) scheme.
21. 21
CHAPTER 5
5. DESIGN AND MODELLING
INTRODUCTION TO SOLIDWORKS
The geometry is modeled as per the required dimensions.
5.1 DIMENSIONS
5.1.1 Cylinder
Main
Length = 880 mm
Diameter = 117 mm
CONNECTING RODS:
Length = 130 mm
Diameter = 3.3 mm
5.1.2 Jet vane:
Chord length = 44 mm
Span = 33 mm
Thickness = 5.16 mm
Separation distance = 16 mm
22. 22
5.1.3 FLAT PLATE
Length = 43 mm
Breadth = 33 mm
Width = 5.16 mm
Separation distance = 16 mm
5.2 THRUST VECTORING USING FLATE PLATE:
23. 23
5.3 THRUST VECTORING USING SYMMETRIC AEROFOIL:
5.4 MESH MODELING
In order to analyze the flow after the exit there is a need of an ambient cylinder
ABIENT CYLINDER
Diameter - 2500 mm
Length - 2500 mm
24. 24
CHAPTER 6
6. COMPUTATIONAL SET-UP
6.1 INTRODUCTION TO ANSYS:
The modeled geometry is imported to ANSYS for further analysis.
6.2PRE-PROCESSING
Step 1: Creating the domain
The geometry is designed in Solid Works for the required THRUST VETORING
SYSTEM (JET DEFLECTION TYPE) and it is exported to ANSYS.
6.2.1GRID GENERATION
Step 2: Meshing the edges
The geometry is meshed.
25. 25
6.3PROCESSING
Step 1: Creating the domain
The geometry is designed in Solid Works for the required THRUST VETORING
SYSTEM (JET DEFLECTION TYPE) and it is exported to ANSYS.
6.3.1 SOLVER
1. Enable energy equation
2. Reynolds number calculation
Re =
= density of the material
v = velocity
= diameter of the tube
= dynamic viscosity
for air flowing through a duct
= 1.225
v = 265.5 m/s
= 117
= 1.846 * Kg/m s
Hence our case is a turbulent case
3. Turbulence selection - K EPSILON MODEL
.
6.3.2 BOUNDARY CONDITIONS
Step 3 : Set Boundary Types
General : Pressure inlet, Pressure outlet.
Incompressible : Velocity inlet, Outflow.
Compressible : Mass flow inlet, Pressure far-field, Mass flow outlet.
26. 26
Other : Wall, Symmetry, Axis.
Special : Inlet vent, Outlet vent, Intake fan, Exhaust fan.
6.3.3 OPERATING CONDITIONS
Step 4: Operating condition
Operating pressure = 0 Pa
Give the value for x, y, z based on flow direction
6.3.4 MATERIAL PROPERTIES
Step 5: MATERIAL PROPERTIES
1. Stainless Steel (Solid Medium)
Density = 7480-8000 / 3
Specific heat = 510 /
Thermal conductivity = 15 /
2. Air (Fluid Medium)
Density = 1.225 / 3
Specific heat = 1006.43 /
Thermal conductivity = 0.0242 /m
6.4 POST PROCESSING
1. Display contour.
2. Display vectors.
3. Display path lines
27. 27
4. Take input summary and results
5. File - write - case and data.
6. File – exit.
The step by step solution consists of 7 major steps discussed as follows:-
Step 1 - Pre-analysis and start-up
1. Start workbench.
2. Drag and drop “Fluid Flow (FLUENT)” on “Project Schematic”.
3. Select “3D”.
Step 2 - Geometry
The required geometry is imported from SlidWorks.
Now for creating Named Selection
1. Select Face Selection– Select th inlet – Right click – Create name selection –
Define as INLET.
2. Select Face Selection – Select the first cylinder – Right click – Create name
selection – Define as MAIN CYLINDER.
3. Select Face Selection – Select the Second Cylinder’s inner face – Right click –
Create name selection – Define as PRESSURE OUTLET.
4. Select Face Selection – Select the Second Cylinder’s outer face – Right click –
Create name selection – Define as PRESSURE OUTLET.
5. Select Face Selection – Select the Second Cylinder’s Body – Right click – Create
name selection – Define as AMBIENT CYLINDER.
6. Close geometry.
Step 3 – Mesh
1. Double click on Mesh
28. 28
2. Sizing – Relevance center – Fine
3. Generate mesh
4. Save – Exit – Update.
Step 4 - Setup
1. Double click on setup – Select "Double Precision“.
2. Define - Material – fluid – Air
(Here, set Density = 1.225 , Cp=1006.43 J/kgK, k=0.0242 W/m K)
3. Define – Material – Solid – Aluminum.
(Here, Set Density= 7500 , Cp=510J/kg-K, k=16 W/m K
4. Set – Boundary conditions
I. Inlet - Pressure inlet
At mach no =0.8 with total pressure of 1atm
M = mach no
Hence
II. Main Cylinder – Wall
III. Outlet – Pressure Outlet
IV. Ambient Cylinder – Pressure Outlet
V. Outer Cylinder – Pressure Outlet
VI. Object 1 – Wall
VII. Object 2 – Wall
VIII. Object – Wall
IX. Object 4 – Wall
29. 29
Step 5 – Solutions
1. Solution Methods – Momentum – Second order upwind method
2. Solution Initialization – Initialize
3. Run Calculations – Number of iterations – Calculate
4. Save project
The remaining two steps in the simulation procedure consist of
i. Results
ii. Verification and Validation
These steps are further discussed in detail and the conclusion is drawn from them
30. 30
CHAPTER 7
RESULT AND DISCUSSION
7.1 RESULT
7.1.1 NO PLATE RESULTS
7.1.1.1 Velocity magnitude sectional view
7.1.1.2 Velocity vector sectional view
42. 42
7.2 DISCUSSION
VELOCITY VECTOR AT ZERO DEGREE ANGLE OF ATTACK FOR FLAT PLATE
Deflecting Angle = 0.502 degree
VELOCITY PATH LINE AT FIVE DEGREE ANGLE OF ATTACK FOR FLAT
PLATE
43. 43
Deflecting Angle = 5.73 degree
VELOCITY PATH LINE AT ZERO DEGREE ANGLE OF ATTACK FOR AN
AIRFOIL
Deflecting Angle = 0.85 degree
VELOCITY PATH LINE AT FIVE DEGREE ANGLE OF ATTACK FOR AN
AIRFOIL
45. 45
CHAPTER 8
CONCLUSION
From the above tabulation and calculations, parameters such as the velocity magnitudes,
flow deflection angle and thrust for the THRUST VECTORING SYSTEM using the Jet
Vane method has been analyzed and it is concluded that a system with THRUST
VECTORING METHOD using a symmetrical airfoil has better results for mach 0.8 at
pressure of 6.64 atm as flow deflected by airfoil is comparatively higher than flat
plate of same dimension and condition.
46. 46
Reference
1.M.Ryan Schafermeyer “Aerodynamic Thrust Vectoring For Attitude Control Of A
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4. Subanesh Shyam Rajendran, Aravind Kumar T.R “Studies on thrust vector control
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5. F.J.Enright “Thrust vector control algorithm design for the cassini spacecraft”
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6. John D Anderson Jr. “Fundamentals of Aerodynamics” 5th
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and chapter10.
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