fundamentals of applied dynamics

Fundamentals of Applied Dynamics


Advanced Texts in Physics
This program of advanced texts covers a broad spectrum of topics that are of
current and emerging interest in physics. Each book provides a comprehensive and
yet accessible introduction to a field at the forefront of modern research. As
such, these texts are intended for senior undergraduate and graduate students at the
M.S. and Ph.D. levels; however, research scientists seeking an introduction to
particular areas of physics will also benefit from the titles in this collection.


Roberto A. Tenenbaum

Fundamentals of
Applied Dynamics
With 568 Figures


Roberto A. Tenenbaum
Departamento de Engenharia Mecaˆ nica, EE
Programa de Engenharia Mecaˆ nica COPPE
Universidade Federal do Rio de Janeiro
Caixa Postal 68503
Rio de Janeiro, 21945-970 RJ
Brasil
Translated by Elvyn Laura Marshall based on the Portuguese edition, (Dinaˆ mica, Editora da
UFRJ, Rio de Janeiro, 1997).

Library of Congress Cataloging-in-Publication Data

Tenenbaum, Roberto A.
Fundamentals of Applied Dynamics / Roberto A. Tenenbaum.
p. cm.
Includes bibliographical references and index.
ISBN 0-387-00887-X (acid-free paper)
1. Dynamics. I. Title.
QA845.T37 2003
531′.11—dc21
2003045454
ISBN 0-387-00887-X

Printed on acid-free paper.

 2004 Springer-Verlag New York, Inc.
All rights reserved. This work may not be translated or copied in whole or in part without the
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I dedicate this book to everyone who makes a
real effort to improve himself or herself, and
especially to Viviane, Isabela, and Miguel.


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Preface

When faced with a new textbook on dynamics, a natural question confronts the reader: What is the textbook contribution, if any, relative to
the many others already available in the field? With regard to fundamental theory, there is clearly no possible difference, for since Newton,
Euler, Lagrange, and D’Alembert there have been no significant developments in the realm of classical mechanics. Nonetheless there has
been a generalized and growing dissatisfaction with available textbooks
on dynamics. The difficulty encountered by engineering students, or
even recent graduates in this area, in correctly analyzing a somewhat
more complex mechanical system can be seen as evidence of this dissatisfaction. In an era when engineers face challenges such as modeling
a system with several degrees of freedom, designing a mechanical arm,
analyzing the stability of an underwater robot, actively controlling the
chassis movement of a motor car, or accurately predicting the trajectory of a satellite, as examples, a thorough understanding of dynamics
is indispensable.
When confronted with the challenge of a nonconventional problem on classical mechanics, the engineer must not and may not lose
himself in a multitude of formulae and methods. In order to safely obtain a solution it is necessary to recognize with accuracy the forces and
torques which act upon the system, to identify the number of degrees
of freedom with absolute certainty, and to choose appropriate reference
coordinates, bases, and axes, describing the motion of the system as a

viii

Preface

function of the chosen coordinates. To correctly describe the system he
or she must master the use of intermediary reference frames. One must
also be able to set up the inertia matrices of the system, write a coherent
set of equations of motion and kinematics constraints equations, and, finally, solve them or extract relevant information from them. To master
all these techniques and consequently be capable of obtaining a reliable
result, it is of the utmost importance to have a thorough knowledge of
the fundamental concepts of dynamics and, at the same time, to have a
solid training in problem-solving methodology.
This book gradually began to take shape as the result of the
experience gained over 30 years of teaching — and learning — dynamics
and related subjects. It originated from the need for a textbook more in
accordance with the methodological unity dictated by the subject and
which could simultaneously fulfill the tasks of teaching and training.
The reader will find that the text almost always introduces general concepts before introducing specific ones. The author’s deliberate choice
to do so only appears to present more difficulties at the very beginning. Didactic experience, however, demonstrates the exact opposite:
The student, exposed to a concept in its most general form, will rapidly
become accustomed to it and will easily master the simplifications which
occur in special cases and, most important of all, will not hesitate when
faced with more complex situations. In this book, each new concept is
introduced along with an illustrative example. Since theory and practice accompany each other, the student is able to implicitly learn useful
problem-solving techniques.
It is precisely the methodological approach used in this book,
the author believes, that characterizes its contribution, modest though
it may be. Although the presentation of concepts is somewhat rigorous,
the purpose of this approach is to avoid ambiguities and to develop in

the reader the habit of thinking a little more abstractly. Aside from
this, several concepts, such as the definition of vector systems, the notion of the angular velocity of a rigid body, or the introduction of the
concept of a particle inertia tensor, among others, are presented in a
manner considered unusual in basic textbooks of mechanics. The textbook presents a unity within the discipline that is evident to any minimally attentive reader and that is supported by the consistent notation


Preface

ix

and the methodology used throughout the book. In this manner particle
dynamics, system dynamics, and rigid body dynamics, notwithstanding
their specificities, are treated uniformly, so that a beginner in the subject
will always recognize the principles which permeate the discipline.
The text presents the so-called Newtonian mechanics. Hamilton’s, Lagrange’s, or Kane’s formulations are therefore not discussed
here. Experience has shown that a solid basis in Newton-Euler mechanics is a prerequisite for readily mastering the methods of analytic mechanics, thus strengthening the intuition of the future engineer. This is
a deliberate choice of the author. This texbook can be seen as a support
for an undergraduate first course in dynamics. However, it is intended
to prepare engineers to solve simple problems in dynamics and, on the
other hand, to create a solid base for a graduate course on analytical
mechanics. In this way, graduate students in physics, engineering, and
correlate areas will find the text useful.
Instructors will find the text to be reasonably complete, including theory, examples, and problems, covering the essential material to
be taught in a two-semester dynamics course, each semester consisting
of around 60 hours. Usually the first four chapters can be covered during the first semester and the last four during the second. The natural
prerequisites are at least one year of undergraduate-level calculus, one
linear algebra course, and a physics course covering the principles of
classical mechanics. It it also desirable, but not essential, for the reader
to have taken a basic mechanics course, usually offered in all engineering
departments, so as to have acquired notions of statics and link analysis.

No textbook, regardless of its excellence, can substitute for the
instructor’s work in the classroom. It is, naturally, the instructor who
must determine the best method to be followed, excluding some topics or
adding others according to his or her personal convenience. For example,
Section 5.8, which deals with fluids, can be omitted without hindering
in any way the understanding of the material that follows. Aside from
this, the ideal sequence in a textbook is not always the most adequate
one in a classroom. For instance, consider Section 5.7, which covers
the conservation principles for mechanical systems. In the text each
principle is followed by its respective example, while in the classroom it is
more efficient to present a theoretical discussion about all the principles,


x

Preface

followed by the set of examples. In this way the student is allowed to
decide which principle should be applied in each case. When the student
returns to the textbook, however, the direct association between theory
and application will always be present. This consideration is also valid
for several other topics.
The work of preparing such a textbook would not have been
possible without the invaluable help, support, and friendship of many
colleagues to whom I am immensely grateful. I would like to thank
especially Professor Arthur Palmeira Ripper Neto for reading and commenting on the text, to Professor Antonio Carlos Marques Alvim for
helping me to prepare Appendix A, and to Professor Luiz Bevilacqua
for his encouragement and optimism. I would like to thank Mrs. Elvyn
Marshall, my translator, now a close friend, for her professionalism and
sense of humor.

To complete this work, the aid of several students, who gave
hours and hours of their time taking care of many details, was essential.
Engineer Roberto Seabra dedicated himself with extraordinary competence and determination to the task of transforming my sketches and
rough diagrams into final figures stored in computer files. Most of the
book’s illustrations are his. A tragic accident deprived me of my main
collaborator and great friend. Many other students helped me and I am
very thankful to all of them.
It would not be possible to conclude without thanking the hundreds of students who, over the last years, dealt with the preliminary
versions of the text and helped me improve the book by pointing out an
endless number of errors. The remaining ones are my sole responsibility.
Comments, suggestions, and corrections will always be welcome.

Rio de Janeiro
Spring 2003

Roberto A. Tenenbaum


To the Reader

This book is divided into eight chapters, which are in turn divided into
sections, covering the main material in kinematics (Chapter 3), dynamics
of particles (Chapter 4), dynamics of systems (Chapter 5), inertia properties (Chapter 6), and dynamics of rigid bodies (Chapters 7 and 8).
An introduction to the general principles of dynamics and its general
approach (Chapter 1) and a discussion about how to handle forces and
torques (Chapter 2) are also given. There are, further, four appendices.
Appendix A presents a short review of linear algebra, just to help the
reader with vector operations and tensor interpretation. Appendix B
shows some linkage modeling, being a complement to Chapter 2. Appendix C gives a reasonably complete table of areas, volumes, centroids,
and moments and products of inertia for the most usual geometries. It

furnishes a helpful support to Chapter 6. Appendix D reveals the answers for almost all the exercices at the end of each chapter. Last, there
is a valuable index.
Each section is identified by two numbers separated by a period, the first number being a reference to the chapter and the second to
the section itself. Section 4.7 is therefore the seventh section of Chapter 4. The equations are also identified by two numbers separated by a
period, the first number indicating the section and the second indicating
sequential numbering within that section. For instance, when Eq. (3.11)
is mentioned in the text, a reference is being made to that equation
in the same chapter. When a reference must be made to an equation


xii

To the Reader

present in a chapter other than the one in which the reference is made,
it will consist of three numbers, separated by two periods, where the
first number refers to the chapter. As an example, if the reader finds
a reference to Eq. (3.3.11) in Chapter 4, a reference is being made to
Eq. (3.11) in Section 3.3 of Chapter 3. Figures are also numbered in
sequence within each section; when referred to successively in the same
section the figure is not reproduced and the reader must search for the
page where it was first introduced. When referred to in another section,
the figure is then reproduced and in this case is given a new number.
Examples are also numbered in sequence within a section. The font used
is smaller and the alignment is indented, so that they stand out clearly
from the rest of the text. Finally, exercises are given at the end of each
chapter. They are organized in series, corresponding to the topic covered
in one section or in a group of sections, and are numbered sequentially
within the series.
For the English version of this book a set of animations for

several of the examples given in the text was prepared. The main purpose of the animations is to give much more information about the
motion than that explained in the text. Also, for the examples that deal
with nonlinear equations, the numerical integration is provided showing the actual behavior of the particle, system, or body. Since the
animations are interactive, the reader may modify parameters or initial conditions to get a deeper insight into the example. Noninteractive
video files showing strictly the motion for a prescribed condition are
also provided. The animations are available on Springer’s Web site at:
www.springeronline.com/038700887X.
Students must be reminded that reading a textbook or following
the corresponding lectures, or both, is not enough for learning dynamics.
They must be supported also by the third leg of this structure, that
is, working the exercises. A fairly large set of exercises is proposed
throughout the book. Working each series by himself at the end of the
corresponding sections is the best way to consolidate the material and
to verify if it was actually well understood.
The exercises are an important part of the text. Try to work
each of them and do not give up if you do not succeed for the first
time. Try again and again. And always keep in mind the Zen aphorism:
To know but not to do is not yet knowing.


Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
To the Reader . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Chapter 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Brief Historical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Mechanical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 The Laws of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4 Mass Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.5 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.6 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Exercise Series #1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Chapter 2

Vectors and Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.1 Free, Sliding, and Bound Vectors. . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Vector Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Equivalent Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Central Axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Forces and Torques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercise Series #2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter 3
3.1

28
30
35
43
49
60
66
76

Kinematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Differentiation of Vectors and Reference Frames . . . . . . . . . . 92


Contents

xiv

3.2 Angular Velocity of a Rigid Body . . . . . . . . . . . . . . . . . . . . . . . . 99
3.3 Use of Different Reference Frames . . . . . . . . . . . . . . . . . . . . . . 106
3.4 Angular Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
3.5 Position, Velocity, and Acceleration . . . . . . . . . . . . . . . . . . . . . 118
3.6 Kinematic Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
3.7 Motion of Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
3.8 Rigid Body Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
3.9 Rolling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
3.10 Mechanical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
Exercise Series #3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
Exercise Series #4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
Exercise Series #5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
Chapter 4

Dynamics of Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

4.1 Dynamic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Newton’s Second Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Plane Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Work and Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Work and Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.7 Impulse and Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8 Conservation Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercise Series #6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter 5

Dynamics of Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

5.1 Dynamic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Force Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Continuous Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Work and Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6 Work and Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.7 Conservation Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.8 Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercise Series #7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter 6

214
221
237
242
247
257
264
276
287
298
317
328

342
350
361
367
379
387

Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400

6.1 Mass and Mass Center. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401
6.2 Inertia Properties of a Particle . . . . . . . . . . . . . . . . . . . . . . . . . . 410
6.3 Inertia Properties of Systems and Bodies. . . . . . . . . . . . . . . . 418


Contents

xv

6.4 Cartesian Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5 Transfer of Axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.6 Principal Directions of Inertia . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercise Series #8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercise Series #9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter 7

Dynamics of the Rigid Body . . . . . . . . . . . . . . . . . . . . . . . . . 484

7.1 Dynamic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3 Work on a Rigid Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.4 Work and Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5 Plane Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercise Series #10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercise Series #11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter 8

485
498
510
520
527
542
552

Advanced Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563

8.1 Motion with a Fixed Point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Gyroscopic Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3 General Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.4 Impulse and Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercise Series #12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix A

429
439
447
461
470

564

574
601
619
626

Linear Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635

A.1 Scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.2 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.3 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.4 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercise Series #13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

636
637
648
656
661

Appendix B

Linkages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663

Appendix C

Properties of Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668

C.1 Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.2 Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.3 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

C.4 Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix D

672
673
682
684

Answers to the Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 691

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707


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Introduction
Chapter 1

The subject called dynamics covers a wide range of topics. Even though
it possesses a basic theory that is trim and compact, the applications are
very numerous and far-reaching. In fact, topics such as the motion of
a material particle, draining of a fluid, kinematics of a mechanism, dynamics of a gyroscope, or analysis of a mechnical arm, to mention a few
known examples, all belong to the domain of this subject’s applications.
This chapter discusses a few introductory topics in the study
of dynamics. Section 1.1 presents a very short summary of the history
of the subject’s origins, briefly summarizing work done before the 17th
century, with comments on Galileo’s findings, discussing the establishment of the foundations of classical mechanics, dwelling on Newton’s
formidable work, and analyzing the later contributions of Euler, Lagrange, and D’Alembert, who formalized the mechanics we know today.
Section 1.2 introduces the mechanical models, that is, the fundamental

concepts employed by dynamics, such as that of force, particle, and body,
among many others. Section 1.3 deals with Newton’s laws, which will
permeate the study of dynamics in its entirety. The aim is to informally
introduce the laws, which will then be effectively used in subsequent
chapters. Section 1.4 handles the concept of mass center. The objective
here is not to enable its practical determination, a subject examined in
greater detail in Chapter 6, but to provide only an introduction to this
important concept, present throughout the text. Section 1.5 discusses
the methodology employed in the resolution of problems on dynamics.


1. Introduction

2

Perhaps the reader will find this treatment premature, which in fact it
is, but this approximation will permit a wider panoramic view of dynamics. Finally, Section 1.6 discusses the issue of notation. This is an
important topic, and in that section the structure common to the entire
notational system adopted in this book will be discussed.

1.1 Brief Historical Background
The origin of mechanics goes far into the distant past. From the very
beginning, in a continuous effort to conquer the environment, human
beings searched for explanations of the origin of phenomena surrounding
them. The first phenomena to challenge the human mind must certainly
have included free fall, the effort necessary to move objects, and the effect
of impact, all of which are of a mechanical nature.
The first more systematic reflections on the motion of bodies
and their origin occurred many centuries later, among the Greeks. The
Greek architects certainly had enough knowledge about statics to erect

safe monuments, although there are few records of such knowledge. Aristotle1 believed that the concept of force involved the idea of something
that pulls (or pushes) to maintain a body in motion, an idea known
today to be incorrect. In all likelihood, the notion of a force as a causal
element in the generation of motion is quite old, its origins probably lying in very primitive concepts that assumed that deities moved the sun,
the moon, and the stars. From this point of view, motion needed an
agent to produce it. Aristotle therefore defended the idea that a force
was necessary for the maintenance of motion, or, in other words, that
for a body to move at a constant velocity the presence of a force was
necessary. The notion of the variation of velocity, that is, of acceleration, was only to appear many centuries later, when it was perfectly
understood and formulated by Galileo.
Leonardo da Vinci2 was a man of multiple interests, having left
a large number of notes on several questions relating to the context of
mechanics of his time. His lack of methodology, however, did not lead
1
2

Aristotle, Greek philosopher, 384–322 B.C.
Leonardo da Vinci, Italian artist and thinker, 1452–1519.


1.1 Brief Historical Background

3

him to any result worthy of consideration. Although he is considered by
some to be the forerunner of Galileo and Newton, statements of his such
as “motion is an accident resulting from the inequality between weight
and force” or also, “force is the cause of motion; motion is the cause of
force” do not appear to lend to his investigations a sufficiently scientific
character.

As an architect, he studied the resistance of pillars, beams, and
arches. For example, he proposed that the resistance of a beam should
be proportional to the area of its cross section. It is suspected, however,
that this rule was already well known to the builders of the Parthenon.
The scientific concept of force was apparently introduced by
Kepler,3 who distinguished himself by formulating three laws that govern the movement of the planets around the sun. That was an epoch
when cosmogonic conceptions agitated the scientific and also the religious worlds, with the heliocentric conception of Copernicus4 opposing
the geocentric conception of Ptolemy.5 It is therefore natural that Kepler’s attention should have been centered on celestial mechanics.
Galileo6 made an important contribution to the creation of the
modern theory of classical mechanics. Even though historians disagree
in their evaluation of his importance in the history of physics, and of
mechanics in particular, there is no doubt about his prominence in this
field of human knowledge. His most important work, the Discorsi,7 consolidated the knowledge of mechanics at the time. Among other findings,
he discovered the parabolic nature of the trajectory of missiles; demonstrated experimentally that the earth’s gravitational acceleration is the
same for all bodies; conceived and clearly formulated the concept of the
reference frame, which is still used today in nonrelativistic mechanics;
explored with great insight the concept of physical similitude; discovered the laws that govern the motion of the simple pendulum (for small
oscillations); and, most important of all, formulated the laws of motion,
although in a somewhat imprecise manner. In fact, Newton himself,
3
4
5
6
7

Johann Kepler, German astronomer, 1571–1630.
Nicolau Copernicus, Polish astronomer, 1473–1543.
Ptolemy, Greek astronomer, second century A.D.
Galileo Galilei, Italian philosopher and mathematician, 1564–1642.
Discorsi e Dimostrazione Matemat. intorno a

` due nuove Scienze, 1638.


1. Introduction

4

naturally in a modest fashion, attributed to Galileo the conception of
his first two laws.
It was Galileo who effectively formulated the law of inertia. He
understood perfectly that in the absence of applied forces the velocity of
a body should remain unchanged. In the Discorsi, this law appears as
follows: “Whatever the degree is of velocity of an object, it will remain
indestructibly imprinted, provided that the external causes of acceleration or deceleration are removed.” As to the second law of motion, there
is no doubt that it must be credited to Newton. Galileo experimented
with the sloping plane and with the motion of projectiles, where the
force due to weight was always present as the cause of the motion of
bodies. Consequently he did not conceive of forces not proportional to
mass and the notion of the momentum did not occur to him. Newton
would state that the variation of the quantity of motion “is proportional
to the applied force and takes place in the direction in which the force
is applied.”
It was undoubtedly Newton8 who made the most important
contributions to mechanics, and in particular to dynamics, and for this
reason is considered the father of classical mechanics. Newton performed
a noteworthy revision of the scientific knowledge of his time, consolidating into fundamental laws what had been loosely stated by his predecessors. For example, he showed that Kepler’s three laws of planetary
motion could be reduced to a single law of universal gravitation and that
free-falling bodies were also governed by the same law, thus creating the
first and most important synthesis of celestial and terrestrial mechanics.
Newton’s most significant contributions in the realm of mechanics are described in the monumental work known as the Principia,9 which

brings together in three volumes countless discoveries made over many
years. The most important result obtained by Newton was, without
question, his second law, known today as the cornerstone of dynamics.
Newton’s discoveries will be discussed in more detail in Section 1.3.
Euler10 was another leading figure in the construction of dynamics. He made important contributions in several fields of mathe8
9
10

Isaac Newton, English scientist, 1642–1727.
Philosophiæ Naturalis Principia Mathematica, 1687.
Leonhard Euler, Swiss mathematician, 1707–1783.


1.1 Brief Historical Background

5

matics and physics and was responsible for formulating Newton’s second law in its currently most used form, namely, that of the product
of the mass and the acceleration being equal to the resultant applied
force. Going even further, Euler published this law in 1752, stating
that it is equally applicable to a finite or infinite mass, making way
for the generalization of the law, which includes fluids as well as rigid
bodies. His restless spirit was not satisfied with this finding, which was
rigorously not very innovative with respect to Newton. As a result he
started to study the problems concerning the motion of the rigid body,
which required a more careful approach. In this analysis appeared the
six scalars, referred to today as the components of the inertia tensor,
and the differential equations that govern the rotation of a rigid body
about a fixed point, currently known as Euler’s dynamic equations. It
was therefore Euler who developed the concept of the inertial rotation

of a body, having published in 1776 laws applicable to any body, or part
of a body, rigid or deformable. The laws are as follows:
1. The principle of momentum, or of the linear momentum: The
total force acting upon a body is equal to the rate of change of
the momentum;
2. The principle of moment of momentum, or of the angular momentum: The total torque acting upon a body is equal to the rate of
change of the angular momentum, where both are measured with
respect to the same fixed point.
These laws, known as Euler’s laws of mechanics, naturally encompass Newton’s second law; they are the equations that govern the
motion of bodies in general systems and are still used today in so-called
Newtonian mechanics.
Classical mechanics was given a new stimulus with the works
of D’Alembert11 and Lagrange.12 D’Alembert’s Trait´e de Dynamique
rejects the concept of Newtonian force and also introduces the forces of
inertia, reducing, in a way, the problems of dynamics to static situations.
D’Alembert also made an attempt, albeit not very successful, to deduce
all of mechanics from the laws of collision.
11
12

Jean Le Rond D’Alembert, French mathematician, 1717–1783.
Joseph-Louis Lagrange, French physicist and mathematician, 1736–1813.


1. Introduction

6

But it was Lagrange who formulated the variational principle,
valid for the vast majority of mechanical systems, in his M´echanique

Analitique (1788). It is known today, curiously, as D’Alembert’s principle. In a more precise manner, some authors refer to this formulation as
the Lagrangian form of D’Alembert’s principle, thus restoring the real
paternity of the dynamical equations within analytical mechanics. It
was also Lagrange who introduced generalized coordinates.
Analytical mechanics has become an extremely useful and powerful tool for the formulation of the equations of a mechanical system,
introducing shortcuts on the way to solving and suppressing linkage
forces. But, as Truesdell says:13 It cannot be said, from Lagrange’s
equations, whether a system does or does not have a momentum; Euler’s equations at least show this, and this comes from the fact that the
integrals of momentum appear naturally in approaches based on Euler’s
equation. Anyhow, Lagrange’s equations are relevant only for certain
types of mechanical systems, and are less general than Euler’s laws.
When Newton said about his discoveries that “If I have seen
further [than others] it is because I stood upon the shoulders of giants,”
he was clearly acknowledging the work done by his predecessors and was
also describing one important aspect of the evolution of science. Newton’s observation reminds us of the Catalan tradition of human towers,
whereby very strong individuals form a circle, other such individuals
climb upon their shoulders, and so on. The construction of the edifice of
science proceeds in a similar fashion, slowly and surely upwards. Each
new stage requires another courageous step. (But, unlike the human
towers, the tower of scientific knowledge does not occasionally collapse,
although it may suffer significant damage due to certain revolutionary
discoveries.)

1.2 Mechanical Models
Engineers may be defined as specialists in modeling. In fact, the main
task confronting an engineer is that of solving problems. This implies a
search for the understanding of a usually complex physical reality, start13

C. Truesdell, Essays in the History of Mechanics, Springer-Verlag, 1968.

1.2 Mechanical Models

7

ing from simple models that approach reality. Models are indispensable
tools, for they introduce simplifications that make problems solvable.
We are confronted indeed with a difficult task: On the one hand, we
must adopt models that are sufficiently complete (and complex) to effectively and fairly closely represent the situation under consideration;
on the other hand, we should use models that are simple enough for
us to easily reach a solution. The engineer’s task is then to discriminate and select, sometimes quite subtly, the most appropriate models
for a specific kind of problem, and to evaluate the results that can be
expected from these models. It is worth pointing out that technological
progress changes our perception of what constitutes an adequate model.
In fact, due to the decreasing costs of complex computational tools and
the recent availability of numerical simulation, successively more complex models can be adopted. As increasingly more powerful tools become
available for their solution, models can become increasingly more sophisticated. Examples of such tools include faster computers, new numerical
methods for the integration of equations, software for algebraic manipulation, among others. The fundamental models of mechanics, however,
never change.
When the methods associated with a specific theory are used
to solve an engineering problem, we are appropriating certain models
that are the basis of that theory, whether we are aware of it or not.
In this process formal mathematics is constructed on a deductive basis.
In other words, it is not introduced to us as an experimental science,
in which results are accepted because they are in accordance with the
observations produced by the experiment, but as a structure of fundamental concepts, axioms, theorems, and inference rules. Fundamental
concepts are defined as notions that are of universal use or based on
common sense, and that are therefore accepted without the need for formal definitions. Axioms, on the other hand, are statements of formulae
taken to be true without the need for proof. Let us give an example
from Euclidean geometry, a discipline with which the reader is likely to

be familiar: the statement that one and only one straight line passes
through two points is an axiom, and the notion of a point and a straight
line are fundamental concepts, and therefore undefined. Theorems, on
the other hand, are statements or formulae based on the axioms and can


8

1. Introduction

be deduced using inference rules. The famous Pythagorean theorem, for
instance, is a theorem because it can be proved based on the axioms of
Euclidean geometry. Finally, rules of inference are elements of mathematical logic that allow theorems to be proved based on axioms and
other previously proven theorems.
When dealing with an applied science, such as, for instance,
mechanics, we are quite distant from the almost absolute formality of
mathematics, but the latter’s main elements are still present, as shall be
seen. Therefore dynamics, as a branch of physics, or more specifically of
classical mechanics, is also regarded as belonging to the realm of applied
mathematics. This is so because the subject of mechanics contains a
consistent structure of fundamental concepts, principles (axioms), and
practical formulae (theorems) that approximate it to pure mathematics,
even though it deals with the elements of the physical world, such as
bodies and their motion and interactions. The remainder of this section
attempts to precisely relate these four categories present in mathematical
formalism with their corresponding mechanical equivalents.
In dynamics we will therefore find so-called models, which are
nothing more than fundamental concepts accepted without a definition.
They are referred to in this manner because they are the result of modeling, or of an idealization of the real physical world or of the world
as we see it. Examples of this category include the notion of particles,

systems, and force, among others.
We define a particle to be a very small body, when compared
with the distance it moves. Clearly this definition is not precise, like all
others that will follow, and is therefore not formal. It is an approximation of a concept that is, to be more exact, admittedly intuitive. We
will make other attempts to approach the concept of a particle. For
instance, let us say that a particle is a material point, that is to say, a
point with no dimension, but that possesses finite mass. A particle is
thus always identified by a point in Euclidean space and is associated
with a real number, its mass m. The particle is a fundamental model in
classical mechanics, from which principles (axioms) will be formulated,
as shall be seen in Section 1.3.
On the other hand, we define an infinitesimal mass element to
be a body of infinitesimal extent, the mass of which is also infinitesimal.