Principles of Engineering Mechanics: Kinematics — The Geometry of Motion

413 Pages·1986·31.46 MB·English

Principles of
Engineering Mechanics
Volume 1
Kinematics-The Geometry of Motion
MATHEMATICAL CONCEPTS AND METHODS
IN SCIENCE AND ENGINEERING
Series Editor: Angelo Miele
Mechanical Engineering and Mathematical Sciences
Rice University
Recent volumes in this series:
22 APPLICATIONS OF FUNCTIONAL ANALYSIS IN ENGINEERING
• J. L. Nowinski
23 APPLIED PROBABILITY • Frank A. Haight
24 THE CALCULUS OF VARIATIONS AND OPTIMAL CONTROL: An
Introduction • George Leitmann
25 CONTROL, IDENTIFICATION, AND INPUT OPTIMIZATION
• Robert Kalaba and Karl Spingarn
26 PROBLEMS AND METHODS OF OPTIMAL STRUCTURAL DESIGN
• N. V. Banichuk
27 REAL AND FUNCTIONAL ANALYSIS, Second Edition
Part A: Real Analysis • A. Mukherjea and K. Pothoven
28 REAL AND FUNCTIONAL ANALYSIS, Second Edition
Part B: Functional Analysis • A. Mukherjea and K. Poth oven
29 AN INTRODUCTION TO PROBABILITY THEORY WITH STATISTICAL
APPLICATIONS • Michael A. Golberg
30 MULTIPLE-CRITERIA DECISION MAKING: ConceJtts, Techniques,
and Extensions • Po-Lung Yu
31 NUMERICAL DERIVATIVES AND NONLINEAR ANALYSIS
• Harriet Kagiwada, Robert Kalaba, Nima Rasakhoo, and Karl Spingarn
32 PRINCIPLES OF ENGINEERING MECHANICS
Volume 1: Kinematics-The Geometry of Motion • Millard F. Beatty, Jr.
33 PRINCIPLES OF ENGINEERING MECHANICS
Volume 2: Dynamics-The Analysis of Motion • Millard F. Beatty, Jr.
34 STRUCTURAL OPTIMIZATION
Volume 1: Optimality Criteria • Edited by M. Save and W. Prager
A Continuation Order Plan in available for this series. A continuation order will bring delivery of
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For further information please contact the publisher.
Principles of
Engineering Mechanics
Volume 1
Kinematics-The Geometry of Motion
Millard F. Beatty, Jr.
University of Kentucky
Lexington, Kentucky
Springer Science+Business Media, LLC
Library of Congress Cataloging in Publication Data
Beatty, Millard F.
Principles of engineering mechanics.
(Mathematical concepts and methods in science and engineering; 32-
Includes bibliographies and index.
Contents: v. 1. Kinematics-the geometry of motion. 1. Mechanics, Applied. 2.
Kinematics. I. Title. II. Series.
TA350.B348 1985 620.1 85-24429
ISBN 978-1-4899-7287-3 ISBN 978-1-4899-7285-9 (eBook)
DOI 10.1007/978-1-4899-7285-9
© 1986 Springer Science+Business Media New York
Originally published by Plenum Press, New York in 1986.
Softcover reprint of the hardcover I st edition 1986
All rights reserved
No part of this book may be reproduced, stored in a retrieval system, or transmitted
in any form or by any means, electronic, mechanical, photocopying, microfilming,
recording, or otherwise, without written permission from the Publisher
To my wife and best friend,
NADINE CHUMLEY BEATTY,
and to our children,
LAURA, ANN, and SCOTT
Preface
This book is a vector treatment of the principles of mechanics written
primarily for advanced undergraduate and first-year graduate students of
engineering. However, a substantial part of the material on
kinematics--exclusive of special advanced topics clearly identified within the
text-much of the content dealing with particle dynamics, and some selected
topics from later chapters, have been used in a first course of a lower level.
This introductory course usually is taken by junior students prepared in
general physics, statics, and two years of university mathematics through dif
ferential equations, which may be studied concurrently. The reader is assumed
to be familiar with elementary vector methods, but the essentials of vector
calculus are reviewed in the applications and separately in a brief Appen
dix A, in case this familiarity is occasionally inadequate.
The arrangement of the book into two parts--Volume 1: Kinematics and
Volume 2: Dynamics-has always seemed to me the best approach. I have
found that students who first master the kinematics have little additional dif
ficulty when finally they reach the free body formulation of the dynamics
problem. In fact, this book was conceived initially from a less intensive two
term sequence of introductory courses in kinematics and dynamics that I first
taught to beginning undergraduate mechanical engineering students at the
University of Delaware in 1963. From this beginning, the: current structure
has grown from both elementary and intermediate level mechanics courses I
have taught for several years at the University of Kentucky.
When used at the beginning graduate level, I envision that both parts
may be covered in a single semester course; however, the instructor who
prefers to move at a slower pace may use these volumes in consecutive
semesters or quarters. In this case, however, I recommend that this material
be supplemented by use of selected papers and books that treat the more
traditional topics in kinematics of mechanisms, more advanced topics in
Lagrangian mechanics, and possibly some elements of conltinuum mechanics.
This advanced pair of courses also should include a variety of meaningful,
computer-oriented problems. The limitations of space and my desire to
vii
viii Preface
present a fresh development at an intermediate level force the exclusion of
these other subjects.
Naturally, the presentation is influenced by my personal interests and
background in mechanical engineering, engineering science, and mechanics.
Consequently, the approach I have chosen is somewhat more sophisticated
and mathematical than is often found in traditional textbooks on engineering
mechanics. In keeping with this approach, the aforementioned prerequisite
mathematics, largely that of the eighteenth century and earlier, is used
without apology. Nevertheless, aware that many readers may not have
mastered these prerequisite materials, I have exercised care to reinforce the
essential tools indirectly in the illustrations and problems selected for study.
Unusual mathematical topics, such as singularity functions and some
elements of tensor analysis, are introduced within the text; and the elements of
matrices, nowadays studied by most engineering students, are reviewed in
Appendix B. Parenthetical reference to use of these tools is provided along the
way, with careful indication where such materials may be omitted from a first
course, without loss of continuity. Some elements of set notation are used in
Volume 2, but the student usually is familiar with these simple applications.
In any event, where familiarity may be lacking, comprehension of the ideas
may be readily inferred from the context. Otherwise, the teacher is expected to
elaborate upon remedial mathematical topics peculiar to his or her special
needs by expanding upon these few areas and by building upon the many
examples and problems provided.
The mathematical development and the numerous companion examples
are structured to place emphasis on the predictive value of the methods and
principles of mechanics, rather than on the often empty and less interesting
computational aspects, but not to the exclusion of numerical examples that
illustrate the various operations and definitions. In addition, some meaningful
introductory computer applications are provided in the problems. Examples
have been selected for their instructive value and to help the student achieve
understanding of the various concepts, principles, and analytical methods
presented. In some instances, experimental data, factual situations, and
applications or designs that confirm analytical predictions are described.
Numerous assignment problems, ranging from easy and straightforward
extensions or reinforcements of the subject matter to more difficult problems
that test the creative skills of better students, are given at the end of each
chapter. To assist the student in his studies, some answers to the odd-num
bered problems are provided at the end.
It is axiomatic that physical intuition or insight cannot be taught. On the
other hand, competence in mathematics and physical reasoning may be
developed so that these special human qualities may be intelligently cultivated
through study of physical applications that mirror the world around us and
through practice of the rational process of reasoning from first principles.
With these attributes in mind, one objective of this book is to help the
Preface ix
engineering student develop confidence in transforming problems into
appropriate mathematical language that may be manipulated to derive sub
stantive and useful physical conclusions or specific numerical results. I intend
that this treatment should provide a more penetrating look at the elements of
classical mechanics and their applications to engineering problems; therefore,
the book is designed to deepen and broaden the student's understanding and
to develop his or her mastery of the fundamentals. However, to reap a harvest
from the seeds sown here, it is important that the student work through many
of the problems provided for study. The mere understanding that one may
apply theoretical concepts and formulas to solve a particular problem is not
equivalent to possession of the knowledge and skills required to produce its
solution. These talents grow only from experience in dealing repeatedly with
these matters. My view of the importance of solving a lot of problems is
expressed further at the beginning of the problem set for Chapter 1, and the
attitude emphasized there is echoed throughout the text.
It is unfortunate that the subject of mechanical design analysis has suf
fered such considerable neglect in engineering curricula in this country. I shall
not speculate on the reasons for this decay. On the contrary, it is pleasant to
see in some schools rejuvenation of the role of mechanics and mathematics,
and innovation of the use of the computer in mechanical design curricula. It is
only in recent years that these ingredients have begun to restore life to this
important and exciting area of engineering.
But I feel that more needs to be accomplished. Various aspects of
mechanical, electrical, and structural design, for example, should be
introduced in certain pilot courses, and the content of these premier courses
taken earlier must be integrated into the various design sequences. I perceive
no reason why problems in mechanical design analysis, for instance, ought
not to be introduced as examples in courses prerequisite for a major course in
this subject. If this plan were followed, advanced problems and computer
aided applications could be studied in a more carefully planned design
curriculum that draws materials from virtually every previous fundamental
course in the student's program, namely, statics and dynamics, solid and fluid
mechanics, thermodynamics and heat transfer, vibrations and controls, cir
cuits and fields, and so on. Consequently, I have chosen for illustration several
examples and problems that illustrate simple introductory applications of
kinematics and dynamics in analysis of some problems in mechanical design.
It is my hope that this book may provide engineering students with solid
mathematical and mechanical foundations for future advanced study of topics
in mechanical design analysis, advanced kinematics of mechanisms and
analytical dynamics, mechanical vibrations and controls, electomagnetics and
acoustics, and continuum mechanics of solids and fluids.
X Preface
The Contents of This Volume
In any treatment of a classical subject like mechanics, it is difficult to
know with certainty what may be new or what is simply an unfamiliar,
rediscovered result. On the other hand, a fresh approach usually is not hard
to spot; and I believe the reader may find many fresh developments within
these pages. The division into kinematics and dynamics, though not unique to
this text, is uncommon. Yet this division surely is as logical and pedagogically
natural as the separation of statics from dynamics found in virtually every
elementary book I have seen.
Volume 1: Kinematics, concerns the geometry of motion from its basic
definition for a particle through the general theory of motion of a rigid
body and of a particle referred to a moving reference frame. Rectilinear
motion, commonly covered in general physics and elementary mechanics,
is reviewed only indirectly by illustrations in the text and in assignment
problems, so the work herein begins with the spatial description of motion in
three dimensions. The reader will find here a consistent, logical, and gradual
building of well-known kinematical concepts, theorems, and formulas, begin
ning from the definitions of motion, velocity, and acceleration of a particle in
Chapter 1: Kinematics of a Particle, and extending to the beautiful general
relations for the velocity and acceleration of a material point referred to a
moving reference frame presented in Chapter 4. And there is much in between
that is novel.
The use of singularity functions appears often these days in a good first
course in the mechanics of deformable solids, and certainly the subject is
useful in courses in mechanical vibrations and electrical circuits, for example.
However, I know of no source that provides a thorough and elementary
introduction to singularity functions with applications to problems in
kinematics. These topics are presented at the close of Chapter 1. Illustrated by
several elementary examples, this treatment provides the student with power
ful tools to treat discontinuous motions common to many mechanical
systems. Therefore, this study shows another useful and important application
of singularity functions at an elementary level.
In Chapter 2: Kinematics of Rigid Body Motion, the construction begins
with the derivation of the finite rotation of a rigid body about a line and leads
ultimately to the fundamental equations for the velocity and acceleration of a
rigid body point in terms of the translation and rotation of the body. This
unusual approach, in my opinion, provides the clearest and most natural way
to arrive at the proper description of the angular velocity and angular
acceleration vectors for a rigid body. The chapter includes many worked
examples and applications; and it closes with a discussion of the theory of
instantaneous screws, including a description of the graphical method of
instantaneous centers for a rigid body.