Table Of ContentAdvanced Dynamics
Shuh-Jing (Benjamin) Ying
University of South adirolF
,apmaT adirolF
¢/U L
EDUCATION SERIES
J. S. Przemieniecki
Series Editor-in-Chief
Air Force Institute of Technology
Wright-Patterson Air Force Base, Ohio
Published by
American Institute of Aeronautics dna Astronautics, Inc.
1081 Alexander Bell Drive, Reston, AV 19102
American Institute of Aeronautics and Astronautics, Inc., Reston, Virginia
Library of Congress Cataloging-in-Publication Data
Ying, Shuh-Jing
Advanced Dynamics / Shuh-Jing (Benjamin) Ying.
p. cm. -- (AIAA education series)
Includes bibliographical references and index.
.1 Dynamics. 2. Mechanics, Applied. I. Title. II. Series.
TA352.Y56 1997 620.1'04--DC21 97-22864
ISBN 1-56347-224-4
Copyright @ 1997 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
Printed in the United States of America. No part of this publication may be reproduced, distributed, or
transmitted, in any form or by any means, or stored in a database or retrieval system, without the prior
written permission of the publisher.
Data and information appearing in this book are for informational purposes only. AIAA is not re-
sponsible for any injury or damage resulting from use or reliance, nor does AIAA warrant that use or
reliance will be free from privately owned rights.
Foreword
Advanced Dynamics by Shuh-Jing (Benjamin) Ying provides a comprehensive
introduction to this important topic for aeronautical or mechanical engineering
students. It is written with the student in mind by explaining in great detail the
fundamental principles and applications of advanced dynamics. The applications
are first illustrated on simple problems, such as the collision of two bodies, and then
demonstrated on much more complex problems, such as a two-impulse trajectory
for space probes. Dr. Ying is a Professor at the University of South Florida in
the Department of Mechanical Engineering, and his research interests include
dynamics, vibrations, mechanical design, and heat transfer. Also, in addition to
his extensive research activity and numerous publications, Dr. Ying has taught 34
different courses in mechanical engineering.
The text covers all the essential mathematical tools needed to analyze the dynam-
ics of systems: vector algebra, conversion of coordinates, calculus of variations,
matrix algebra, Cartesian tensors and dyadics, rotation operators, Fourier series,
Fourier integrals, Fourier transforms, and Laplace transforms (in Chapters ,1 6,
and 8). Chapters 1 through 3 start with a review of elementary statics and dynam-
ics, followed by a discussion of Newton's laws of motion, D'Alembert's principle,
virtual work, and kinematics and dynamics of a single particle or system of parti-
cles. Chapter 4 introduces Lagrange's equations and the variational principle used
in dynamics. Chapter 5 is devoted to the dynamics of rockets and space vehicles,
while Chapters 7, 8, and 9 discuss the dynamics of a rigid body and vibrations of
continuous systems as well as lumped parameter systems with a single degree or
multiple degrees of freedom. Nonlinear vibrations are also included. Chapter 10
discusses the Special Theory of Relativity and its consequences in kinematics and
dynamics.
The Education Series of textbooks and monographs published by the American
Institute of Aeronautics and Astronautics embraces a broad spectrum of theory
and application of different disciplines in aeronautics and astronautics, including
aerospace design practice. The series also includes texts on defense science, en-
gineering, and management. Over 50 titles are now included in the series, and the
books serve as both teaching texts for students and reference materials for practic-
ing engineers, scientists, and managers. This recent addition to the series will be
a valuable text for courses in engineering dynamics in aeronautical or mechanical
engineering programs.
J. S. Przemieniecki
Editor-in-Chief
AIAA Education Series
Preface
Dynamics is the foundation of physical science and is an important subject of
study for all engineering students. Although the fundamental laws of dynamics
have remained unchanged, their applications are constantly changing. One hun-
dred years ago, there were no automobiles, no airplanes, and no space vehicles.
Advances in science and technology provide us with many new dynamic devices.
For example, the gyroscopic effect of the rotating propeller in airplanes creates
diving during yawing. When a satellite travels in a circular orbit, the motions of
rolling and yawing also can produce pitching. During times of war, shooting a
missile flying in its orbit is another subject with real and important implications.
Is it possible to shoot a space probe from the surface of Earth to Mars by one im-
pulse? All these scenarios are important and interesting, and understanding them
begins with the study of dynamics.
As I teach advanced dynamics, I feel that there is a need for a textbook that
covers subjects related to recent developments. A book that includes my lecture
notes may fulfill this need, and this is my primary motivation for writing this book.
In addition, this book is intended not only for students in the classroom but also
for practicing engineers who wish to update their knowledge. For this reason, the
book is self-contained with fundamentals in vector algebra, vector analyses, matrix
operations, tensors and dyadics. The details are clearly and explicitly presented. I
have been teaching advanced dynamics for more than l0 years, and I often tell my
students that I have nothing to hide. This is the spirit of this book. Anyone who
reads the book should not only understand current developments in dynamics but
also can learn some of the foundations of mechanical engineering necessary to
understand papers published in recent journals.
Further, I hope this book will show the reader that dynamics is an exciting
field with many new problems to be solved. For example, there are challenging
problems concerning the motion of a space vehicle traveling in a general orbit, and
also in the design of robots and complex automatic machines. Lastly, a chapter
on the special relativity theory is included. This is intended to show that space
and time are related. Just a few days in one system can be many years in the
other system. Past events in the stationary system can be observed at present in
another system traveling near the speed of light. All these are not fairy tales, but
are scientifically true. The purpose of this part of the book is to broaden readers'
minds. Anything is possible.
The contents of this book are briefly described as follows: In Chapter ,1 funda-
mental principles and vector algebra are reviewed. This chapter may be skipped
by well-prepared students. Chapter 2 deals with kinematics and dynamics of a par-
ticle. First, the kinematics of a particle in various coordinate systems is discussed.
Next, examples concerning trajectories of missiles and reentry of space vehicles
xi
xii
are presented. Lastly, fundamental concepts such as work, conservative force, and
potential energy are reviewed. Chapter 3 is devoted to the dynamics of a system of
particles. Besides items commonly introduced in this chapter, the mid-air collision
of missiles is given in detail including a computer program that determines the
trajectory of the second missile. Collisions of solid spheres are also introduced
in this chapter. This can be considered as the first approximation for automobile
collisions. To balance theoretical aspects and practical applications, gravitational
force and potential energy also are studied in this chapter.
Chapter 4 is a major chapter in this book. Many important topics are included.
Many engineering students have difficulty formulating equations for motion for
a particle or a body. Lagrange's equation is intended to help students find the
equation of motion. Students only need to have the knowledge of kinetic and po-
tential energies of the mass for formulating the equations. Hamilton's principle is
a parallel approach to Lagrange's equations. With the study of Hamilton's princi-
ple, students will better understand the equations of motion. Lagrange's equations
with constraints also are introduced. Constraint forces and Lagrange multipliers
are derived. Many examples are given for Lagrange's equations. Students should
be familiar with this subject if a proper effort is devoted to study. The variational
principle is included in this chapter. Through this approach, Lagrange's equation
for a conservative system also can be reached. The purpose of the variational
principle is for optimization. A case of optimization with a constraint condition
is studied also. Many examples are given to demonstrate the application of the
variational principle.
Chapter 5 is devoted to the dynamics of rockets and space vehicles. This is
another demonstration of the balance of theory and practice in this book. Essential
characteristics of rockets are studied in a single-stage rocket. The advantage of
multistage rocket and use of the Lagrangian multiplier for maximizing the burnout
velocity are included. A space vehicle traveling in a gravitational field is treated
extensively in Section 5.3. Different trajectories are discussed. Special attention
is devoted to the elliptical orbit. The trajectory for an electrical-propulsion rocket
is given in Section 5.4. The equations involved in electrical propulsion typically
belong to a small perturbation theory. Equations of motion are solved analytically
in the chapter. Interplanetary trajectories are discussed in Section 5.5. The journey
from Earth to Mars' surface is used to demonstrate the procedure for calculating
the impulses required for the whole trajectory. After a review of previous work,
the use of two impulses for sending a space probe from Earth to Mars' orbit and
spiraling down to the surface of Mars is discussed in detail. In this way the long
and detailed observations can be made by the space probe.
Chapter 6 is for matrices, tensors, dyadics, and rotation operators. This chapter
is entirely mathematical, so that engineering students are exposed to more applied
mathematics. Some applications are included with each subject to make them eas-
ily understandable and more interesting. For example, through rotation operators
it is proved that two successive rotations can be combined into a single rotation.
This can actually reduce the time for rotational motions. Engineers wishing to
xiii
extend their knowledge through journal papers should pay special attention to this
chapter.
The dynamics of a rigid body are studied in Chapter 7. Because many objects
may be modeled as rigid bodies, the analyses presented in this chapter play an
important role in this book. The first three sections present fundamental principles.
Some additional sections are included here describing the gyroscope and the
orbiting space vehicle. The gyroscopic effect of a rotating propeller in an airplane
causing the plane to dive during yawing is studied here in detail. The major
application of the angular momentum of a rigid body is the gyro-compass. Two
examples are particularly aimed in that direction. Furthermore, the motion of a
heavy symmetrical top and induced torques because of flight operations on a
satellite in circular orbit also are treated in detail in this chapter.
Chapters 8 and 9 are devoted to the study of vibrations. In Chapter 8, math-
ematical topics that are necessary for analyzing vibration problems are first pre-
sented. These topics are Fourier series, Fourier integral, and Fourier and Laplace
transforms. The Laplace transform is treated as a special case in Fourier transfor-
mation. Applications include one-dimensional damped oscillations and transient
vibrations. Advanced topics in vibration are treated in Chapter 9. Starting from
a two-degree-of-freedom system, some examples in a lumped parameter system,
a continuous system, and nonlinear vibrations are studied. Stability analysis of
vibrations in a phase plane is also discussed.
Chapter 01 covers the Special Relativity Theory. This is arranged here to broaden
readers' minds. The time and space coordinates are related such that for one person
traveling near the speed of light, just a few days for this person can be many years
to a person in a stationary system. This is proved to be true scientifically. Moreover
one also can prove that an event in the past could be observed as a present event in
another system. Readers are urged to consider that, just as space and time are now
interrelated through the relativity theory, new developments may one day modify
our thoughts concerning our most basic scientific concepts and principles.
In conclusion, I wish to thank Sue Britten for providing valuable support in the
process of accomplishing this book.
gniJ-huhS )nimajneB( gniY
July 1997
Table of
Contents
Preface ................................................ xi
Chapter 1. Review of Fundamental Principles ................... 1
1.1 Dimensions and Units ................................. 1
1.2 Elements of Vector Analysis ............................ 2
1.3 Statics and Dynamics ................................. 5
1.4 Newton's Laws of Motion .............................. 6
1.5 D'Alembert's Principle ................................ 6
1.6 Virtual Work ....................................... 7
Problems ........................................ 10
Chapter 2. Kinematics and Dynamics of a Particle ............... 13
2.1 Kinematics of a Particle .............................. 13
2.2 Particle Kinetics ................................... 16
2.3 Angular Momentum (Moment of Momentum) of a Particle ...... 19
2.4 Work and Kinetic Energy ............................. 21
2.5 Conservative Forces ................................. 21
Problems ........................................ 23
Chapter 3. Dynamics of a System of Particles .................. 27
3.1 Conversion of Coordinates ............................ 27
3.2 Collision of Particles in Midair .......................... 31
3.3 General Motion of a System of Particles ................... 37
3.4 Gravitational Force and Potential Energy ................... 40
3.5 Collision of Two Spheres on a Plane ...................... 44
Problems ........................................ 50
Chapter 4. Lagrange's Equations and the Variational Principle ..... 53
4.1 Generalized Coordinates, Velocities, and Forces .............. 53
4.2 Lagrangian Equations ................................ 55
4.3 Hamilton's Principle ................................. 66
4.4 Lagrangian Equations with Constraints .................... 70
4.5 Calculus of Variations ................................ 76
Problems ........................................ 83
Chapter 5. Rockets and Space Vehicles ....................... 85
5.1 Single-Stage Rockets ................................ 85
5.2 Multistage Rockets .................................. 90
iiv
iiiv
5.3 Motion of a Particle in Central Force Field .................. 92
5.4 Space Vehicle with Electrical Propulsion (equations solved by
small perturbation method) ........................ 103
5.5 Interplanetary Trajectories ............................ 107
Problems ....................................... 112
Chapter 6. Matrices, Tensors, Dyadics, and Rotation Operators .... 115
6.1 Linear Transformation Matrices ........................ 115
6.2 Application of Linear Transformation to Rotation Matrix ....... 119
6.3 Cartesian Tensors and Dyadics ......................... 121
6.4 Tensor of Inertia .................................. 126
6.5 Principal Stresses and Axes in a Three-Dimensional Solid ...... 129
6.6 Viscous Stress in Newtonian Fluid ...................... 133
6.7 Rotation Operators ................................. 136
Problems ....................................... 147
Chapter 7. Dynamics of a Rigid Body ....................... 151
7.1 Displacements of a Rigid Body ........................ 152
7.2 Relationship Between Derivatives of a Vector for Different
Reference Frames .............................. 152
7.3 Euler's Angular Velocity and Equations of Motion ........... 156
7.4 Gyroscopic Motion ................................ 162
7.5 Motion of a Heavy Symmetrical Top ..................... 168
7.6 Torque on a Satellite in Circular Orbit .................... 172
Problems ....................................... 177
Chapter 8. Fundamentals of Small Oscillations ................ 181
8.1 Fourier Series and Fourier Integral ...................... 182
8.2 Fourier and Laplace Transforms ........................ 195
8.3 Properties of Laplace Transforms ....................... 197
8.4 Forced Harmonic Vibration Systems with Single
Degree of Freedom ............................. 203
8.5 Transient Vibration ................................. 214
8.6 Response Spectrum ................................ 221
8.7 Applications of Fourier Transforms ...................... 224
Problems ....................................... 228
Chapter 9. Vibration of Systems with Multiple
Degrees of Freedom .................................... 233
9.1 Vibration Systems with Two Degrees of Freedom ............ 234
9.2 Matrix Formulation for Systems with Multiple
Degrees of Freedom ............................. 244
9.3 Lumped Parameter Systems with Transfer Matrices ........... 255
9.4 Vibrations of Continuous Systems ...................... 266
9.5 Nonlinear Vibrations ............................... 289
ix
9.6 Stability of Vibrating Systems ......................... 294
Problems ....................................... 299
Chapter 10. SpecialRelativity Theory ....................... 305
10.1 Lorentz Transformation ............................. 306
10.2 Brehme Diagram .................................. 310
10.3 Immediate Consequences in Kinematics and Dynamics ........ 314
Problems ....................................... 317
Appendix A: Runge-Kutta Method .......................... 319
Appendix B: Stoke's Theorem .............................. 321
Appendix C: Planetary Data ............................... 325
Appendix D: Determinants and Matrices ...................... 327
Appendix E: Method of Partial Fractions ..................... 331
Appendix F: Tables of Fourier and Laplace Transforms ........... 335
Appendix G: Contour Integration and Inverse Laplace Transform ... 339
Appendix H: Bessel Functions ............................. 349
Appendix I: Instructions for Computer Programs ............... 363
Appendix J: Further Reading .............................. 365
Subject Index ......................................... 367
1
Review of Fundamental Principles
T HIS chapter reviews the fundamental principles necessary for the study of
advanced dynamics. Although these principles may be familiar to students
who have studied elementary mechanics, they are included here so that this book
is reasonably self-contained.
The concepts of dimensions and units are reviewed in Section 1.1. Familiarity
with these concepts will greatly facilitate formulating equations, checking di-
mensional homogeneity of an equation, and converting units. A brief review of
vector analysis is given in Section 1.2. Formulas frequently used in this book
are presented. Section 3.1 contains the definitions of statics and dynamics and a
discussion of the difference between kinematics and kinetics. Section 1.4 presents
Newton's laws of motion. The second law is written in an expanded form to in-
clude the effect of changing mass, which is essential for analyzing the dynamics
of a rocket or any object with variable mass. D'Alembert's principle is presented
in Section 1.5. Through the use of D'Alembert's principle, dynamic problems are
simplified to static ones. Section 1.6 reviews the principles of virtual displace-
ment and virtual work, which are the foundation for the derivation of Lagrange's
equations discussed in Chapter 4.
1.1 Dimensions and Units
A dimension is the measure by which the magnitude of a physical quantity is
expressed. In dynamics, there are usually four dimensions: mass, length, time,
and force. A unit is a determinate quantity adopted as a standard of measurement.
As shown in Table 1.1, the International System of Units (SI) specifies mass in
kilograms (kg), length in meters (m), time in seconds (s), and force in newtons
(N). In the British Gravitational System (BG), mass is measured in slugs, length in
feet (ft), time in seconds (s), and force in pounds (lbf). It is important to mention
that understanding dimensions and units will prevent errors from occurring when
analyzing problems and converting units. The conversion factors for the two
systems are given in Table 1.1.
Of the four dimensions mentioned in Table 1.1, mass, length, and time are
considered as primary dimensions and force as a secondary dimension. Force can
be expressed in terms of mass, length, and time as follows:
1 N = 1 kg-m/s 2 (1.1)
1 lbf = 1 slug ft/s 2 (1.2)
The following example illustrates the technique used in the conversion of units.
m 1 ft 1 mile 3600 s
1 km/s = 1000
s 0.3048m 5280ft 1 h
= 2236.94 mph
