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Preface
It is fitting in this beginning of the second century of powered flight to be writing
a book on flight dynamics—in particular, the modeling and simulation of
flight dynamics. No one had heard of flight dynamics, or indeed, modeling and
simulation, in the early days of aviation. Pioneers such as the Wright brothers,
Langley, Curtiss, and Bleriot preferred experimentation with flying models to
working with the equations of motion. Until the 1940s, flight dynamics was
more of an art than an engineering discipline, with books such as Stick and
Rudder: An Explanation of the Art of Flying by Langewiesche (McGraw-Hill,
1944) imparting a fundamental, but purely qualitative understanding of airplane
flight, which was useful to both budding aeronautical engineers and
pilots. After the Second World War, large strides made in mathematical modeling
of airplane flight dynamics were first documented in classical texts such
as Airplane Performance, Stability and Control by Perkins and Hage (Wiley,
1949), which covered linearized, time-invariant flight dynamics, and inspired
a multitude of textbooks in the field, such as those by Etkin, McCormick,
Miele, etc. With the advent of supersonic, hypersonic, and space flight in
the 1950s and 1960s, there was a great need for modeling and simulation of
high-performance flight dynamics, which essentially involves a set of coupled,
time-varying, nonlinear, ordinary differential equations. While a sprinkling of
mathematical models for high-speed flight was available in the introductory
texts of the 1960s and 1970s, a thorough analytical treatment of these topics
was confined to specialized technical reports. Yet, there was little mention of
numerical modeling and simulation. Research articles first began to appear in
the 1960s, which presented numerical calculations for special examples with
analog—and later—digital computers, concerning high-performance flight dynamics.
As numerical science evolved and computing power grew, the sophistication
of flight dynamic modeling and simulation progressed to missiles,
launch vehicles, re-entry vehicles, and spacecraft. Guidance and navigation
for the manned lunar missions (1968–72) essentially utilized such numerical
capability. Numerical modeling and simulation of flight dynamics has now
emerged as a major discipline. The chief motivation for writing this book is to present a unified approach
to both aircraft and spacecraft flight dynamics. Modern aerospace vehicles,
such as the space shuttle, other launch vehicles, and long-range ballistic missiles,
do not discriminate between atmospheric and space flight. Unfortunately,
nearly all textbooks on flight dynamics do so, and seldom do we find aircraft
and spacecraft co-existing within the covers of the same book. Many excellent
textbooks are available on modern aircraft dynamics (such as those by
Zipfel, Pamadi, Stengel, Etkin, and Schmidt), but they stop short of hypersonic
aircraft and sub-orbital trajectories. Similarly, the available textbooks
on space dynamics (Hale, Brown, Curtis, etc.) do not go below hypersonic
speeds of re-entry vehicles. While it is easy to understand the separate evolution
of aircraft and spacecraft in the past, the future of flight lies in integrating
the two vehicles into a single unit. The single-stage-to-orbit (SSTO) reusable
launch vehicle, which takes off and lands like an aircraft and delivers payload
to an orbiting space station, exemplifies the vision of aerospace engineering
for the future. Therefore, it is imperative that this new generation of engineers
is taught to remove the artificial distinction between atmospheric and
space flight. Many aerospace engineering departments realize this need, offering
courses that integrate atmospheric and space flight. Examples of such
courses are AE-520 Flight Vehicle Dynamics, AE-580 Analytical Methods in
Aeronautical and Astronautical Engineering, and AE-621 Aircraft and Spacecraft
Automatic Control Systems, offered by the Department of Aerospace
Engineering at Ohio State University.
This book is an attempt to bridge the gap between aircraft and spacecraft
dynamics, by demonstrating that the two evolve logically from the same set of
physical principles. The breadth of topics covered is unparalleled by any other
book on the subject. Beginning with kinematics and translational dynamics
over a rotating planet, nonspherical gravity model, leading to two-body orbits,
orbital maneuvers, rendezvous in space, and lunar and interplanetary travel,
atmospheric flight follows logically after chapters on atmospheric modeling,
aerodynamics, and propulsion. The attitude dynamics and control of aircraft
and spacecraft are presented in an integrated and continuous fashion. Modeling
of nonlinear flight dynamics is covered in numerous examples, such as
simulation of long-range airplane flight, supermaneuvers, rocket ascent, suborbital
flight and atmospheric entry, multi-axis rotations of spacecraft, and inertia
coupled, open- and closed-loop airplane dynamics, which are not found in
other textbooks on flight dynamics. The book culminates with a final chapter
covering advanced concepts with six-degree-of-freedom simulation examples,
and modeling of structural dynamics, unsteady aerodynamics, aeroelasticity,
and propellant slosh dynamics. From the solved examples, the reader can
easily build his/her own simulations as independent semester projects. The
choice of gravitational models, coordinate frames, attitude control systems,
propulsion systems, and flow models to use is left up to the reader, in order
to provide an almost unlimited capability to build various simulations. This book is primarily designed as a textbook for junior and senior undergraduates,
as well as graduate students in mechanical, aerospace engineering/
aeronautics, and astronautics departments. The book may also be used
as a reference for practicing engineers and researchers in aerospace engineering,
aeronautics, and astronautics, whose primary interest lies in modeling
and simulation of flight dynamics. The contents have evolved from the lecture
notes of several 3rd–4th year undergraduate, and graduate-level courses
I have taught over the past 14 years. The material in the book has been especially
selected to be useful in a modern course on flight dynamics, where the
artificial distinction between atmospheric and space flight is removed. At the
same time, the material offers the choice of being adopted in separate traditional
courses on space dynamics and atmospheric flight mechanics. In this
respect, the text is quite flexible and can be utilized by even those instructors
who do not necessarily agree with the comprehensive approach adopted in the
book. It is, however, suggested that the mix of atmospheric and space dynamics
be retained in each course. A detailed discussion of the usage of material
by course instructors is given below. The chapters are designed to follow in
a sequence such that their concepts evolve logically and fit into each other
like a glove. The concepts are introduced in an easy-to-read manner, while
retaining mathematical rigor. The theory behind flight dynamic modeling is
highlighted and fundamental results are derived analytically. Examples and
problems have been carefully chosen to emphasize the understanding of underlying
physical principles. Each chapter begins with a list of clearly defined
aims and objectives. At the end of each chapter, short summaries and a number
of exercises are provided in order to help readers consolidate their grasp of
the material presented. Answers to selected problems are included at the back
of the book so that a reader can verify his/her own solutions. Full step-by-step
solutions to all of the exercises will be available upon request to the publisher
in a separate solutions manual designed for course instructors to use with
their students. The manual may also be made available to researchers and
professionals (nonstudents) who are using the book for self-study purposes.
Requests for the solutions manual should be sent to the publisher on an official
letterhead with full particulars, including the course name and number
for which the book is being adopted.
Perhaps the greatest distinguishing feature of the book is the ready and
extensive use of MATLAB and Simulink, as practical computational
tools to solve problems across the spectrum of modern flight dynamics. The
MATLAB/Simulink codes are integrated within the text in order to readily illustrate
modeling and simulation of aerospace dynamics. MATLAB/Simulink
is standard, easy-to-use software that most engineering students learn in the
first year of their curriculum. Without such a software package, the numerical
examples and problems in a text of this kind are difficult to understand and solve. In giving the reader a hands-on experience with MATLAB/Simulink
as applied to practical problems, the book is useful for a practicing engineer,
apart from being an introductory text for the beginner. The book uses
the software only as an instructional tool, discouraging the “black-box” approach
found in many textbooks that carry “canned” software. The reader is
required to write his/her own codes for solving many of the problems contained
as exercises. An appendix contains a brief review of some important
methods of numerically integrating ordinary differential equations that are
commonly encountered in flight dynamics. In summary, the primary features
of this book are a unified approach to aircraft and spacecraft flight, a wide
range of topics, nontrivial simulations, logical and seamless presentation of
material, rigorous analytical treatment that is also easy to follow, and a
ready use of MATLAB/Simulink software as an instructional tool. All the
codes used in the book are available for downloading at the following website:
http://home.iitk.ac.in/∼ashtew/page10.html.
The text focuses on the modeling and simulation aspects of flight mechanics
in a wide range of aerospace applications. This treatment is more general
than that found in many textbooks on atmospheric flight dynamics, which
only cover the approximate equations of motion offering analytical closedform
solutions (considered trivial from a modeling and simulation viewpoint).
However, it is recognized that the analytical solutions impart an insight into
the science of flight dynamics, especially in a junior-level course. For this
reason, the discussion of approximate, analytical solutions to special flight
situations is offered in the form of exercises at the end of the chapters. The
reader is referred to traditional flight mechanics texts for details on approximate,
analytical treatment wherever necessary. A course instructor has the
freedom to begin with the general derivations of the equations of motion presented
in the book, proceeding to the special approximate flight situations
(planar, quasi-steady, constant mass, flat nonrotating earth, etc.) for which
the traditional, analytical solution is available.
A reader is assumed to have taken basic undergraduate courses in mathematics
and physics—particularly calculus, linear algebra, and dynamics—and
is encouraged to review these fundamental concepts at several places in the
text. I will now briefly discuss the organization and highlights of the topics
covered in each chapter in order to provide a ready guide to the reader and
the classroom instructor. This will help readers and instructors select what
parts of the book will be relevant either in a particular course, or for specific
professional study and reference.
It is sometimes felt that a “logical” sequence of topics should begin with
atmospheric flight and end with space flight. While such an “earth-to-space”
arrangement may appear natural in a documentary on flight, it is not suitable
for a textbook. As pointed out above, the material in the text has been ordered
such that the physical and mathematical concepts evolve logically and
sequentially. Chapters 2–4, which cover kinematics and analytical dynamics,
are equally relevant to both space and atmospheric flight. The next three chapters (5–7) on orbital mechanics logically follow from this foundation, as they
do not require a model of the atmosphere and aerodynamics. Chapter 8, on
rocket propulsion, follows for the same reason. However, before beginning the
treatment on atmospheric flight, it is necessary to introduce an atmospheric
model, aerodynamic concepts, and air-breathing propulsion, which are carried
out in Chapters 9–11. I would add that for an undergraduate student
in flight dynamics, the introductory chapters on aerodynamics and propulsion
(Chapters 8–11) are especially relevant. Chapters 12–14—the “meat” of
the book—put together the concepts of the foregoing chapters in order to
present a comprehensive modeling, simulation, control, and analysis of atmospheric,
trans-atmospheric, and space trajectories. Chapter 15 culminates
the treatment with advanced modeling and simulation concepts applicable to
aerospace vehicles. Hence, the first four and the last four chapters pertain to
both atmospheric and space flight, whereas the intervening chapters present
specialized treatment of either of the two aspects of flight. The following is a
detailed overview of each chapter:
Chapter 1 offers a basic introduction and motivation for studying flight
dynamics in a comprehensive manner and includes the classification of flight
vehicles, as well as the important assumptions made in their modeling and
simulation.
Chapter 2 presents the kinematic modeling and coordinate transformations
useful in all aspects of flight dynamic derivations. The rigorous vector
analysis of rotational kinematics is presented with many numerical examples.
Basic identities—such as the time derivative of a vector, its rotation, and representation
in various reference frames—are derived in a manner that can be
easily utilized for derivation of both translational and rotational equations of
motion in subsequent chapters. Several alternative kinematic representations
[Euler angles, Euler-axis/principal rotation, rotation matrix, Euler symmetric
parameters (quaternion), Rodrigues and modified Rodrigues parameters] are
introduced and their time-evolution derived. A reader can cover the first two
sections in a first reading, proceed to Chapters 3–12, and then return to the
other sections of Chapter 2 before beginning Chapter 13.
Chapter 3 discusses planetary shape and gravity. While a spherical gravity
model serves most atmospheric flight applications reasonably well, it is
necessary to model the spherical harmonics of a nonspherical mass distribution
(Sections 3.2–3.4) for accurate space-flight, rocket-ascent, and entry-flight
trajectories.
Chapter 4 is an introduction to analytical dynamics. While presenting
the analytical tools for deriving a general model for translational motion, the
chapter also discusses the relationship between translational and rotational
dynamics of a flight vehicle. This chapter is essentially the starting point for
deriving the basic kinetic equations for aerospace flight, and includes dynamics
in moving frames, variable mass bodies, the N-body gravitational problem in
space dynamics, and its specialization to two-body trajectories with analytical
and numerical solutions. The problems at the end of the chapter test the reader’s understanding of the important concepts in analytical (Newtonian)
dynamics. After reading Chapter 4, a reader can either proceed to space flight
dynamics (Chapters 5–8) or go to Chapters 9–12 on atmospheric flight.
Chapter 5 covers orbital mechanics concepts, orbital maneuvers, relative
motion in orbit, and orbit determination for three-dimensional guidance, with
examples. A basic course can take advantage of the special coordinate frames
(celestial, local horizon, planetary) discussed in Sections 5.1–5.3, which are
useful not only in space flight, but also in long-range atmospheric trajectories.
The later sections of the chapter are useful in designing trajectories for
interplanetary missions, where the associated Lambert’s problem is solved numerically.
An interesting example of Lambert’s problem in the chapter is a
nonplanar orbital rendezvous between two spacecraft, which is not found in
other textbooks on orbital mechanics.
Chapter 6 discusses orbital perturbations caused by gravitational asymmetry
(oblateness, and presence of a third body) as well as atmospheric drag.
Oblateness effects lead to sun-synchronous and Molniya orbits, whereas a simple
atmospheric model is used to predict the life of a satellite in low orbits.
Third-body perturbations result in the sphere of influence and the patched
conic approach for the design and analysis of lunar and interplanetary missions.
Chapter 7 is devoted to the restricted three-body problem, its solvability,
equilibrium points, and numerical solutions, with examples of the earth-moonspacecraft
trajectories. This chapter can be skipped in a basic-level course on
flight dynamics.
Chapter 8 introduces the elements of rocket propulsion. The first two sections
of this chapter are strongly recommended in all basic courses on flight
dynamics. The design of optimal multistage rockets—a crucial problem in
sub-orbital and space flight—is covered in Section 8.3 with examples of twoand
three-stage rockets.
Chapter 9 begins the modeling of atmospheric flight with a detailed standard
atmosphere model, including nondimensional aerodynamic parameters.
For example, a 21 layer U.S. Standard Atmosphere is considered, ranging
from sea level to a geometric altitude of 700 km. This model is utilized in all
simulations of atmospheric and trans-atmospheric trajectories in the book.
Chapter 10 introduces aerodynamics, ranging from elementary concepts
to models of viscous hypersonic and rarefied flows using computational fluid
dynamics. The discussion is aimed at building an appropriate model of aerodynamic
force and moment vectors for each flow regime for the purpose of
flight dynamic calculations. This chapter is a must for a beginning course on
flight dynamics. Those interested in the details of aerodynamic and thermodynamic
models may consider the multitude of specialized texts cited in the
chapter.
Chapter 11 covers the elements of air-breathing propulsion from the point
of view of flight dynamic modeling of the thrust vector and the rate of fuel
consumption. The discussion of characteristics and operational limitations of piston–propeller, turbine, and ramjet engines is comprehensive and selfcontained.
A numerical model of the thrust and specific fuel consumption
with altitude and Mach number of a low-bypass turbofan engine is presented
and utilized in Chapter 12 for the simulation of fighter airplane trajectories.
This chapter also must be a part of the basic undergraduate course in flight
dynamics.
Chapter 12 is the heart of the book with its three-degree-of-freedom
flight models, including planetary form, rotation, aerodynamics, and propulsion.
General equations of motion in the planet-fixed frame are derived from
first principles in a systematic fashion. These equations govern the translational
flight of all aerospace vehicles (airplanes, rockets, spacecraft, entry
vehicles). The chapter contains nontrivial examples of atmospheric and transatmospheric
trajectories and provides detailed analytical insight into airplaneflight,
rocket-ascent, and entry trajectories.
Chapter 13 presents the universal rotational dynamics model applicable
to all aerospace vehicles, emphasizing the commonality between the stability
and control characteristics of aircraft and spacecraft. The chapter derives
several attitude dynamics models based on various useful kinematic parameters
introduced in Chapter 2. Single-axis, open-loop, time-optimal impulsive
maneuvers are an important part of this chapter. After exhaustively covering
spacecraft dynamics with many examples, the chapter culminates with
a rigorous derivation, modeling, and simulation of attitude motion in the
atmosphere. Examples in the chapter range from spin-stabilized, rotor and
thruster controlled spacecraft, to gravity-gradient satellites, thrust-vectored
rockets, and six-degree-of-freedom, inertia-coupled, fighter airplanes. The rotational
dynamic models can be easily added to the three-degree-of-freedom
translational models of Chapter 12 in order to simulate the complete sixdegree-
of-freedom motion of rigid craft.
Chapter 14 offers the modeling and simulation of closed-loop control systems
for a large variety of aerospace applications based upon modern control
concepts. The first part of the chapter presents an introduction of linear
systems theory, while the later sections cover multivariable control systems
applied to aircraft, spacecraft, and rockets with a multitude of interesting
examples.
Chapter 15 introduces advanced concepts, such as six-degree-of-freedom
and nonlinear modeling and simulation, flexible vehicle dynamics, unsteady
aerodynamics and aeroelasticity, and propellant slosh dynamics. The importance
of these topics to flight dynamics, and their inclusion in advanced models,
are discussed along with several important references for further study.
The following is a suggested coverage of material by course instructors. It
is not envisaged that the entire contents can be followed in a single course.
The basic undergraduate curriculum traditionally includes flight dynamics
as a pair of courses: one on translational flight (airplane performance/space
flight dynamics), and the other on attitude motion (airplane stability and control/
spacecraft dynamics and control). I suggest that the first course (called Flight Dynamics-I )—typically offered at the third-year level—should address
the translational aspects of both atmospheric and space flight dynamics. Ideally,
this course would cover Chapter 1, Sections 2.1–2.4, 3.1, Chapter 4, Sections
5.1–5.2, 8.1–8.2, 9.1–9.3, 10.1, 10.2.1–10.2.4, and 10.3, Chapter 11, and
Sections 12.1 and 12.2. The second undergraduate course, Flight Dynamics-II,
(taken in the fourth year) would focus on the attitude motion of aerospace
vehicles, and would cover Sections 2.5–2.8, 13.1–13.6, 13.7.1–13.7.6, 14.1–
14.5, and 15.1–15.3. An advanced elective undergraduate- and graduate-level
course can be designed to cover rocket and entry trajectories, such as AE-644
Hypersonic and Trans-atmospheric Flight offered at the Indian Institute of
Technology, Kanpur, having the two basic undergraduate courses discussed
above as its prerequisites. Such a course may cover Sections 3.2–3.4, 5.3–5.5,
6.1, 6.2, 8.3, 9.4, 10.2.5, 12.3, 12.4, 13.7.7, 13.7.8, 14.6, and 15.4. Another
advanced senior undergraduate and graduate course on interplanetary flight
would consist of material covered in Sections 3.2–3.4, 5.3–5.7, 6.1–6.4, Chapter
7, Sections 8.3 and 12.4, having Flight Dynamics-I as its prerequisite. Such a
course would rely heavily on modeling and simulation, with trajectory design
semester projects based upon numerical solutions to Lambert’s problem. The
following flowchart highlights the suggested coverage of material:
My motivation to study flight dynamics began early in life. I recall, as a
five-year-old, being inspired by my late father—a doctor and medical officer—
to listen at late hours to a scratchy, live commentary of the Apollo-11 mission
over a radio set in the remote town of Chunar in India. My incessant curiosity
about flight in the growing years was sought to be satisfied by him through numerous
illustrated books, magazines, and newspaper articles (of which I kept
a careful catalog in an old scrapbook). His encouragement was partly responsible
for my taking up aeronautical engineering in college at IIT–Kanpur, and
later on when he was no longer with us, completing my doctorate in aerospace
engineering at the University of Missouri–Rolla, as well as obtaining a privatepilot
certificate. I still remember his ideas about hydrogen-fueled airplanes,
and his suggestion that I also study astronomy in graduate school. My family,
friends, students, and colleagues have continually supported my enthusiasm
for all forms of flight to the present day.
I would like to thank the editorial and production staff of Birkh¨auser
Boston, especially Tom Grasso, for their constructive suggestions and valuable
insights during the preparation of the manuscript. I am also grateful to The
MathWorks, Inc., for providing the latest MATLAB/Simulink version, utilized
in the examples throughout the book.
Ashish Tewari
May 2006
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Aims and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Atmospheric and Space Flight. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Modeling and Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Attitude and Kinematics of Coordinate Frames . . . . . . . . . . . . 9
2.1 Aims and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Basic Definitions and Vector Operations . . . . . . . . . . . . . . . . . . . . 9
2.3 Coordinate Systems and Rotation Matrix. . . . . . . . . . . . . . . . . . . 13
2.4 Euler Axis and Principal Angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.5 Euler Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.6 Euler Symmetric Parameters (Quaternion). . . . . . . . . . . . . . . . . . 23
2.7 Rodrigues Parameters (Gibbs Vector) . . . . . . . . . . . . . . . . . . . . . . 27
2.8 Modified Rodrigues Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.9 Attitude Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.10 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3 Planetary Form and Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1 Aims and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2 Newton’s Law of Gravitation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3 Gravity of an Axisymmetric Planet . . . . . . . . . . . . . . . . . . . . . . . . 46
3.4 Radius of a Nonspherical Planet . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.5 Gravitational Anomalies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.6 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4 Translational Motion of Aerospace Vehicles . . . . . . . . . . . . . . . . 59
4.1 Aims and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2 Particle and Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2.1 Particle Kinematics in a Moving Frame . . . . . . . . . . . . . . 61
4.3 Newton’s Laws of Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.3.1 Variable Mass Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.3.2 Rotation and Translation of a Body . . . . . . . . . . . . . . . . . 76
4.4 Energy and Angular Momentum. . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.4.1 The N-Body Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.5 The Two-Body Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.5.1 Geometry of Two-Body Trajectories . . . . . . . . . . . . . . . . . 94
4.5.2 Lagrange’s Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.5.3 Kepler’s Equation for Elliptical Orbit . . . . . . . . . . . . . . . . 101
4.5.4 Position and Velocity in a Hyperbolic Trajectory . . . . . . 108
4.5.5 Parabolic Escape Trajectory . . . . . . . . . . . . . . . . . . . . . . . . 111
4.6 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5 Orbital Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.1 Aims and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.2 Celestial Frame and Orbital Elements . . . . . . . . . . . . . . . . . . . . . . 117
5.2.1 Orbit Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.3 Spherical Celestial Coordinates and Local Horizon . . . . . . . . . . . 123
5.4 Planet Fixed Frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.5 Single Impulse Orbital Maneuvers . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.6 Multi-Impulse Orbital Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.7 Relative Motion in Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.8 Lambert’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.9 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6 Perturbed Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6.1 Aims and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6.2 Perturbing Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6.3 Effects of Planetary Oblateness. . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.3.1 Sun Synchronous Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.3.2 Molniya Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.4 Effects of Atmospheric Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.5 Third-Body Perturbation and Interplanetary Flight . . . . . . . . . . 159
6.5.1 Sphere of Influence and Patched Conics . . . . . . . . . . . . . . 161
6.6 Numerical Solution to the Perturbed Problem. . . . . . . . . . . . . . . 164
6.7 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .170
7 The Three-Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
7.1 Aims and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
7.2 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
7.3 Lagrange’s Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
7.4 Restricted Three-Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
7.4.1 Lagrangian Points and Their Stability . . . . . . . . . . . . . . . 180
7.4.2 Jacobi’s Integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
7.4.3 Numerical Solution of the Restricted Problem . . . . . . . . . 185
7.5 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
8 Rocket Propulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
8.1 Aims and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
8.2 The Rocket Engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
8.3 The Rocket Equation and Staging . . . . . . . . . . . . . . . . . . . . . . . . . 197
8.3.1 The Single-Stage Rocket . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
8.3.2 The Multi-Stage Rocket . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
8.3.3 Parallel Staging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
8.3.4 Mission Trade-Off . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
8.4 Optimal Rockets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
8.4.1 Optimal Two-Stage Rocket . . . . . . . . . . . . . . . . . . . . . . . . . 211
8.4.2 Optimal Three-Stage Rocket . . . . . . . . . . . . . . . . . . . . . . . . 213
8.5 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
9 Planetary Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
9.1 Aims and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
9.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
9.3 Hydrostatic Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
9.4 Standard Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
9.5 Exponential Model for Planetary Atmospheres . . . . . . . . . . . . . . 230
9.6 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
10 Elements of Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
10.1 Aims and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
10.2 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
10.2.1 Aerodynamic Force and Moment . . . . . . . . . . . . . . . . . . . . 233
10.3 Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
10.3.1 Flow Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
10.3.2 Continuum Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
10.3.3 Continuum Viscous Flow and the Boundary Layer . . . . . 242
10.3.4 Continuum Compressible Flow . . . . . . . . . . . . . . . . . . . . . . 247
10.3.5 Rarefied Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
10.4 Force and Moment Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
10.5 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
11 Airbreathing Propulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
11.1 Aims and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
11.2 Ideal Momentum Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
11.3 Propeller Engines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
11.4 Jet Engines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
11.4.1 Ramjet Engines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
11.4.2 Turbojet and Turbofan Engines . . . . . . . . . . . . . . . . . . . . . 275
11.5 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
12 Atmospheric and Transatmospheric Trajectories . . . . . . . . . . . 283
12.1 Aims and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
12.2 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
12.3 Airplane Flight Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
12.3.1 Long-Range Cruising Flight . . . . . . . . . . . . . . . . . . . . . . . . 297
12.3.2 Effect of a Steady Wind on an Airplane Flight . . . . . . . . 304
12.3.3 Take-Off Maneuver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
12.3.4 Accelerated Climb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
12.3.5 Maneuvers and Supermaneuvers . . . . . . . . . . . . . . . . . . . . . 325
12.4 Entry Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
12.4.1 Ballistic Entry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
12.4.2 Maneuvering Entry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
12.5 Rocket Ascent Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
12.6 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
13 Attitude Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
13.1 Aims and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
13.2 Euler Equations of Rotational Motion . . . . . . . . . . . . . . . . . . . . . . 369
13.3 Rotational Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
13.4 Principal Body Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374
13.5 Torque-Free Rotation of Spacecraft . . . . . . . . . . . . . . . . . . . . . . . . 376
13.5.1 Axisymmetric Spacecraft . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
13.5.2 Asymmetric Spacecraft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
13.6 Spacecraft with Attitude Thrusters . . . . . . . . . . . . . . . . . . . . . . . . 384
13.6.1 Single-Axis Impulsive Rotation . . . . . . . . . . . . . . . . . . . . . . 386
13.6.2 Attitude Maneuvers of Spin-Stabilized Spacecraft . . . . . . 387
13.6.3 Asymmetric Spacecraft Maneuvers by Attitude
Thrusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394
13.7 Spacecraft with Rotors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
13.7.1 Variable-Speed Control Moment Gyroscope . . . . . . . . . . . 400
13.7.2 Dual-Spin Spacecraft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404
13.7.3 Gravity Gradient Spacecraft . . . . . . . . . . . . . . . . . . . . . . . . 409
13.8 Attitude Motion in Atmospheric Flight . . . . . . . . . . . . . . . . . . . . . 414
13.8.1 Equations of Motion with Small Disturbance . . . . . . . . . . 416
13.8.2 Stability Derivatives and De-coupled Dynamics. . . . . . . . 425
13.8.3 Longitudinal Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
13.8.4 Airplane Longitudinal Modes . . . . . . . . . . . . . . . . . . . . . . . 431
13.8.5 Lateral Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
13.8.6 Airplane Lateral Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
13.8.7 Rotational Motion of a Launch Vehicle . . . . . . . . . . . . . . . 442
13.8.8 Inertia Coupled Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 446
13.9 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452
14 Attitude Control Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
14.1 Aims and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
14.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
14.3 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461
14.3.1 Time-invariant, Linear Systems . . . . . . . . . . . . . . . . . . . . . 462
14.3.2 Linear Stability Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
14.3.3 Transfer Matrix and Second-Order Systems . . . . . . . . . . . 465
14.4 Basic Closed-Loop Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468
14.5 Implementation of Control System Elements . . . . . . . . . . . . . . . . 472
14.5.1 Gyroscopic Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474
14.6 Single-Axis, Closed-Loop Attitude Control . . . . . . . . . . . . . . . . . . 479
14.6.1 Control of Single-Axis Spacecraft Maneuvers . . . . . . . . . . 479
14.6.2 Roll Control of Aircraft and Missiles . . . . . . . . . . . . . . . . . 484
14.7 Multi-Axis Closed-Loop Attitude Control . . . . . . . . . . . . . . . . . . . 486
14.7.1 Attitude Stabilization of a Launch Vehicle . . . . . . . . . . . . 486
14.7.2 Reaction Wheel and Magnetic Denutation of Gravity
Gradient Spacecraft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491
14.7.3 Control of Aircraft and Missiles with Inertia Coupling. . 499
14.8 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504
15 Advanced Modeling and Simulation Concepts . . . . . . . . . . . . . 507
15.1 Aims and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507
15.2 Six-Degree-of-Freedom Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 507
15.2.1 Wing-Rock Motion of a Fighter Airplane . . . . . . . . . . . . . 510
15.2.2 Trajectory and Attitude of a Ballistic Entry Vehicle . . . 512
15.3 Structural Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518
15.4 Unsteady Aerodynamics and Aeroelasticity . . . . . . . . . . . . . . . . . 521
15.5 Propellant Slosh Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527
15.6 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529
A Numerical Integration of Ordinary Differential Equations . . 531
A.1 Fixed-Step Runge–Kutta Algorithms . . . . . . . . . . . . . . . . . . . . . . . 531
A.2 Variable-Step Runge–Kutta Algorithms . . . . . . . . . . . . . . . . . . . . 532
A.3 Runge–Kutta–Nystr¨om Algorithms . . . . . . . . . . . . . . . . . . . . . . . . 534
Answers to Selected Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547
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