Mechanical design handbook measurement, analysis, and control of dynamic systems

MECHANICAL
DESIGN
HANDBOOK
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MECHANICAL
DESIGN
HANDBOOK
Measurement, Analysis, and Control
of Dynamic Systems
Harold A. Rothbart Editor
Dean Emeritus
College of Science and Engineering
Fairleigh Dickinson University
Teaneck, N.J.
Thomas H. Brown, Jr. Editor
Faculty Associate
Institute for Transportation Research and Education
North Carolina State University
Raleigh, N.C.
Second Edition
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Copyright © 2006, 1996 by The McGraw-Hill Companies, Inc. All rights reserved. Manufactured in
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v
CONTENTS
Contributors vii
Foreword ix
Preface xi
Acknowledgments xiii
Part 1 Mechanical Design Fundamentals
Chapter 1. Classical Mechanics 1.3
Chapter 2. Mechanics of Materials 2.1
Chapter 3. Kinematics of Mechanisms 3.1
Chapter 4. Mechanical Vibrations 4.1
Chapter 5. Static and Fatigue Design 5.1
Chapter 6. Properties of Engineering Materials 6.1
Chapter 7. Friction, Lubrication, and Wear 7.1
Part 2 Mechanical System Analysis
Chapter 8. System Dynamics
8.3
Chapter 9. Continuous Time Control Systems
9.1
Chapter 10. Digital Control Systems

10.1
For more information about this title, click here
vi CONTENTS
Chapter 11. Optical Systems 11.1
Chapter 12. Machine Systems 12.1
Chapter 13. System Reliability 13.1
Part 3 Mechanical Subsystem Components
Chapter 14. Cam Mechanisms 14.3
Chapter 15. Rolling-Element Bearings 15.1
Chapter 16. Power Screws 16.1
Chapter 17. Friction Clutches 17.1
Chapter 18. Friction Brakes 18.1
Chapter 19. Belts 19.1
Chapter 20. Chains 20.1
Chapter 21. Gearing 21.1
Chapter 22. Springs 22.1
Appendix A. Analytical Methods for Engineers A.1
Appendix B. Numerical Methods for Engineers B.1
Index follows Appendix B
CONTRIBUTORS
William J. Anderson Vice President, NASTEC Inc., Cleveland, Ohio (Chap. 15, Rolling-
Ellement Bearings)
William H. Baier Director of Engineering, The Fitzpatrick Co., Elmhurst, Ill. (Chap. 19, Belts)
Stephen B. Bennett Manager of Research and Product Development, Delaval Turbine
Division, Imo Industries, Inc., Trenton, N.J. (Chap. 2, Mechanics of Materials)
Thomas H. Brown, Jr. Faculty Associate, Institute for Transportation Research and Education,
North Carolina State University, Raleigh, N.C. (Co-Editor)
John J. Coy Chief of Mechanical Systems Technology Branch, NASA Lewis Research Center,
Cleveland, Ohio (Chap. 21, Gearing)
Thomas A. Dow Professor of Mechanical and Aerospace Engineering, North Carolina State

University, Raleigh, N.C. (Chap. 17, Friction Clutches, and Chap. 18, Friction Brakes)
Saul K. Fenster President Emeritus, New Jersey Institute of Technology, Newark, N.J. (App. A,
Analytical Methods for Engineers)
Ferdinand Freudenstein Stevens Professor of Mechanical Engineering, Columbia University,
New York, N.Y. (Chap. 3, Kinematics of Mechanisms)
Theodore Gela Professor Emeritus of Metallurgy, Stevens Institute of Technology, Hoboken, N.J.
(Chap. 6, Properties of Engineering Materials)
Herbert H. Gould Chief, Crashworthiness Division, Transportation Systems Center, U.S.
Department of Transportation, Cambridge, Mass. (App. A, Analytical Methods for Engineers)
Bernard J. Hamrock Professor of Mechanical Engineering, Ohio State University, Columbus,
Ohio (Chap. 15, Rolling-Element Bearings)
John E. Johnson Manager, Mechanical Model Shops, TRW Corp., Redondo Beach, Calif.
(Chap. 16, Power Screws)
Sheldon Kaminsky Consulting Engineer, Weston, Conn. (Chap. 8, System Dynamics)
Kailash C. Kapur Professor and Director of Industrial Engineering, University of Washington,
Seattle, Wash. (Chap. 13, System Reliability)
Robert P. Kolb Manager of Engineering (Retired), Delaval Turbine Division, Imo Industries,
Inc., Trenton, N.J. (Chap. 2, Mechanics of Materials)
Leonard R. Lamberson Professor and Dean, College of Engineering and Applied Sciences,
West Michigan University, Kalamazoo, Mich. (Chap. 13, System Reliability)
Thomas P. Mitchell Professor, Department of Mechanical and Environmental Engineering,
University of California, Santa Barbara, Calif. (Chap. 1, Classical Mechanics)
Burton Paul Asa Whitney Professor of Dynamical Engineering, Department of Mechanical
Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, Pa. (Chap. 12,
Machine Systems)
vii
Copyright © 2006, 1996 by The McGraw-Hill Companies, Inc. Click here for terms of use.
J. David Powell Professor of Aeronautics/Astronautics and Mechanical Engineering, Stanford
University, Stanford, Calif. (Chap. 10, Digital Control Systems)
Abillo A. Relvas Manager––Techical Assistance, Associated Spring, Barnes Group, Inc., Bristol,

Conn. (Chap. 22, Springs)
Harold A. Rothbart Dean Emeritus, College of Science and Engineering, Fairleigh Dickenson
University, Teaneck, N.J. (Chap. 14, Cam Mechanisms, and Co-Editor)
Andrew R. Sage Associate Vice President for Academic Affairs, George Mason Universtiy,
Fairfax, Va. (Chap. 9, Continuous Time Control Systems)
Warren J. Smith Vice President, Research and Development, Santa Barbara Applied Optics, a
subsidiary of Infrared Industries, Inc., Santa Barbara, Calif. (Chap. 11, Optical Systems)
David Tabor Professor Emeritus, Laboratory for the Physics and Chemistry of Solids,
Department of Physics, Cambridge University, Cambridge, England (Chap. 7, Friction, Lubrication,
and Wear)
Steven M. Tipton Associate Professor of Mechanical Engineering, University of Tulsa, Tulsa,
Okla. (Chap. 5, Static and Fatigue Design)
George V. Tordion Professor of Mechanical Engineering, Université Laval, Quebec, Canada
(Chap. 20, Chains)
Dennis P. Townsend Senior Research Engineer, NASA Lewis Research Center, Cleveland,
Ohio (Chap. 21, Gearing)
Eric E. Ungar Chief Consulting Engineer, Bolt, Beranek, and Newman, Inc., Cambridge, Mass.
(Chap. 4, Mechanical Vibrations)
C. C. Wang Senior Staff Engineer, Central Engineer Laboratories, FMC Corporation, Santa
Clara, Calif. (App. B, Numerical Methods for Engineers)
Erwin V. Zaretsky Chief Engineer of Structures, NASA Lewis Research Center, Cleveland, Ohio
(Chap. 21, Gearing)
viii CONTRIBUTORS
ix
FOREWORD
Mechanical design is one of the most rewarding activities because of its incredible
complexity. It is complex because a successful design involves any number of individual
mechanical elements combined appropriately into what is called a system. The word
system came into popular use at the beginning of the space age, but became somewhat
overused and seemed to disappear. However, any modern machine is a system and

must operate as such. The information in this handbook is limited to the mechanical
elements of a system, since encompassing all elements (electrical, electronic, etc.)
would be too overwhelming.
The purpose of the Mechanical Design Handbook has been from its inception to pro-
vide the mechanical designer the most comprehensive and up-to-date information on
what is available, and how to utilize it effectively and efficiently in a single reference
source. Unique to this edition, is the combination of the fundamentals of mechanical
design with a systems approach, incorporating the most important mechanical subsystem
components. The original editor and a contributing author, Harold A. Rothbart, is one of
the most well known and respected individuals in the mechanical engineering community.
From the First Edition of the Mechanical Design and Systems Handbook published over
forty years ago to this Second Edition of the Mechanical Design Handbook, he has con-
tinued to assemble experts in every field of machine design—mechanisms and linkages,
cams, every type of gear and gear train, springs, clutches, brakes, belts, chains, all manner
of roller bearings, failure analysis, vibration, engineering materials, and classical mechanics,
including stress and deformation analysis. This incredible wealth of information, which
would otherwise involve searching through dozens of books and hundreds of scientific and
professional papers, is organized into twenty-two distinct chapters and two appendices. This
provides direct access for the designer to a specific area of interest or need.
The Mechanical Design Handbook is a unique reference, spanning the breadth and
depth of design information, incorporating the vital information needed for a mechanical
design. It is hoped that this collection will create, through a system perspective, the level of
confidence that will ultimately produce a successful and safe design and a proud designer.
Harold A. Rothbart
Thomas H. Brown, Jr.
Copyright © 2006, 1996 by The McGraw-Hill Companies, Inc. Click here for terms of use.
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xi
PREFACE
This Second Edition of the Mechanical Design Handbook has been completely reorga-

nized from its previous edition and includes seven chapters from the Mechanical Design
and Systems Handbook, the precursor to the First Edition. The twenty-two chapters con-
tained in this new edition are divided into three main sections: Mechanical Design
Fundamentals, Mechanical System Analysis, and Mechanical Subsystem Components. It
is hoped that this new edition will meet the needs of practicing engineers providing the
critical resource of information needed in their mechanical designs.
The first section, Part I, Mechanical Design Fundamentals, includes seven chapters
covering the foundational information in mechanical design. Chapter 1, Classical
Mechanics, is one of the seven chapters included from the Second Edition of the
Mechanical Design and Systems Handbook, and covers the basic laws of dynamics and
the motion of rigid bodies so important in the analysis of machines in three-dimensional
motion. Comprehensive information on topics such as stress, strain, beam theory, and
an extensive table of shear and bending moment diagrams, including deflection equa-
tions, is provided in Chap. 2. Also in Chap. 2 are the equations for the design of
columns, plates, and shells, as well as a complete discussion of the finite-element
analysis approach. Chapter 3, Kinematics of Mechanisms, contains an endless number
of ways to achieve desired mechanical motion. Kinematics, or the geometry of
motion, is probably the most important step in the design process, as it sets the stage
for many of the other decisions that will be made as a successful design evolves.
Whether it’s a particular multi-bar linkage, a complex cam shape, or noncircular gear
combinations, the information for its proper design is provided. Chapter 4, Mechanical
Vibrations, provides the basic equations governing mechanical vibrations, including an
extensive set of tables compiling critical design information such as, mechanical
impedances, mechanical-electrical analogies, natural frequencies of basic systems, tor-
sional systems, beams in flexure, plates, shells, and several tables of spring constants
for a wide variety of mechanical configurations. Design information on both static and
dynamic failure theories, for ductile and brittle materials, is given in Chap. 5, Static and
Fatigue Design, while Chap. 6, Properties of Engineering Materials, covers the issues
and requirements for material selection of machine elements. Extensive tables and charts
provide the experimental data on heat treatments, hardening, high-temperature and low-

temperature applications, physical and mechanical properties, including properties for
ceramics and plastics. Chapter 7, Friction, Lubrication, and Wear, gives a basic overview
of these three very important areas, primarily directed towards the accuracy requirements
of the machining of materials.
The second section, Part II, Mechanical System Analysis, contains six chapters, the
first four of which are from the Second Edition of the Mechanical Design and Systems
Handbook. Chapter 8, Systems Dynamics, presents the fundamentals of how a complex
dynamic system can be modeled mathematically. While the solution of such systems will
be accomplished by computer algorithms, it is important to have a solid foundation on
Copyright © 2006, 1996 by The McGraw-Hill Companies, Inc. Click here for terms of use.
how all the components interact—this chapter provides that comprehensive analysis.
Chapter 9, Continuous Time Control Systems, expands on the material in Chap. 8 by
introducing the necessary elements in the analysis when there is a time-dependent
input to the mechanical system. Response to feedback loops, particularly for nonlinear
damped systems, is also presented. Chapter 10, Digital Control Systems, continues
with the system analysis presented in Chaps. 8 and 9 of solving the mathematical
equations for a complex dynamic system on a computer. Regardless of the hardware
used, from personal desktop computers to supercomputers, digitalization of the equa-
tions must be carefully considered to avoid errors being introduced by the analog to
digital conversion. A comprehensive discussion of the basics of optics and the passage
of light through common elements of optical systems is provided in Chap. 11, Optical
Systems, and Chap. 12, Machine Systems, presents the dynamics of mechanical sys-
tems primarily from an energy approach, with an extensive discussion of Lagrange’s
equations for three-dimensional motion. To complete this section, Chap. 13, System
Reliability, provides a system approach rather than addressing single mechanical elements.
Reliability testing is discussed along with the Weibull distribution used in the statistical
analysis of reliability.
The third and last section, Part III, Mechanical Subsystem Components, contains nine
chapters covering the most important elements of a mechanical system. Cam layout and
geometry, dynamics, loads, and the accuracy of motion are discussed in Chap. 14 while

Chap. 15, Rolling-Element Bearings, presents ball and roller bearing, materials of construc-
tion, static and dynamic loads, friction and lubrication, bearing life, and dynamic analysis.
Types of threads available, forces, friction, and efficiency are covered in Chap. 16, Power
Screws. Chapter 17, Friction Clutches, and Chap. 18, Friction Brakes, both contain an
extensive presentation of these two important mechanical subsystems. Included are the
types of clutches and brakes, materials, thermal considerations, and application to various
transmission systems. The geometry of belt assemblies, flat and v-belt designs, and belt
dynamics is explained in Chap. 19, Belts, while chain arrangements, ratings, and noise are
dealt with in Chap. 20, Chains. Chapter 21, Gearing, contains every possible gear type,
from basic spur gears and helical gears to complex hypoid bevel gears sets, as well as the
intricacies of worm gearing. Included is important design information on processing and
manufacture, stresses and deflection, gear life and power-loss predictions, lubrication, and
optimal design considerations. Important design considerations for helical compression,
extension and torsional springs, conical springs, leaf springs, torsion-bar springs, power
springs, constant-force springs, and Belleville washers are presented in Chap. 22, Springs.
This second edition of the Mechanical Design Handbook contains two new appen-
dices not in the first edition: App. A, Analytical Methods for Engineers, and App. B,
Numerical Methods for Engineers. They have been provided so that the practicing
engineer does not have to search elsewhere for important mathematical information
needed in mechanical design.
It is hoped that this Second Edition continues in the tradition of the First Edition,
providing relevant mechanical design information on the critical topics of interest to
the engineer. Suggestions for improvement are welcome and will be appreciated.
Harold A. Rothbart
Thomas H. Brown, Jr.
xii PREFACE
ACKNOWLEDGMENTS
Our deepest appreciation and love goes to our families, Florence, Ellen, Dan, and Jane
(Rothbart), and Miriam, Sianna, Hunter, and Elliott (Brown). Their encouragement, help,
suggestions, and patience are a blessing to both of us.

To our Senior Editor Ken McCombs, whose continued confidence and support has
guided us throughout this project, we gratefully thank him. To Gita Raman and her
wonderful and competent staff at International Typesetting and Composition (ITC) in
Noida, India, it has been a pleasure and honor to collaborate with them to bring this
Second Edition to reality.
And finally, without the many engineers who found the First Edition of the
Mechanical Design Handbook, as well as the First and Second Editions of the
Mechanical Design and Systems Handbook, useful in their work, this newest edition
would not have been undertaken. To all of you we wish the best in your career and
consider it a privilege to provide this reference for you.
Harold A. Rothbart
Thomas H. Brown, Jr.
xiii
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MECHANICAL
DESIGN
HANDBOOK
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A
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1
MECHANICAL DESIGN
FUNDAMENTALS

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1.3
CHAPTER 1
1.1 INTRODUCTION 1.3
1.2 THE BASIC LAWS OF DYNAMICS 1.3
1.3 THE DYNAMICS OF A SYSTEM OF
MASSES 1.5
1.3.1 The Motion of the Center of Mass 1.6
1.3.2 The Kinetic Energy of a System 1.7
1.3.3 Angular Momentum of a System
(Moment of Momentum) 1.8
1.4 THE MOTION OF A RIGID BODY 1.9
1.5 ANALYTICAL DYNAMICS 1.12
1.5.1 Generalized Forces and d’Alembert’s
Principle 1.12
1.5.2 The Lagrange Equations 1.14
1.5.3 The Euler Angles 1.15
1.5.4 Small Oscillations of a System near
Equilibrium 1.17
1.5.5 Hamilton’s Principle 1.19
CLASSICAL MECHANICS
Thomas P. Mitchell, Ph.D.
Professor
Department of Mechanical and Environmental Engineering
University of California
Santa Barbara, Calif.
The aim of this chapter is to present the concepts and results of newtonian dynamics
which are required in a discussion of rigid-body motion. The detailed analysis of par-
ticular rigid-body motions is not included. The chapter contains a few topics which,

while not directly needed in the discussion, either serve to round out the presentation
or are required elsewhere in this handbook.
1.1 INTRODUCTION
The study of classical dynamics is founded on Newton’s three laws of motion and on
the accompanying assumptions of the existence of absolute space and absolute time.
In addition, in problems in which gravitational effects are of importance, Newton’s
law of gravitation is adopted. The objective of the study is to enable one to predict,
given the initial conditions and the forces which act, the evolution in time of a
mechanical system or, given the motion, to determine the forces which produce it.
The mathematical formulation and development of the subject can be approached in two
ways. The vectorial method, that used by Newton, emphasizes the vector quantities force
and acceleration. The analytical method, which is largely due to Lagrange, utilizes the
scalar quantities work and energy. The former method is the more physical and generally
possesses the advantage in situations in which dissipative forces are present. The latter is
more mathematical and accordingly is very useful in developing powerful general results.
1.2 THE BASIC LAWS OF DYNAMICS
The “first law of motion” states that a body which is under the action of no force
remains at rest or continues in uniform motion in a straight line. This statement is also
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1.4 MECHANICAL DESIGN FUNDAMENTALS
known as the “law of inertia,” inertia being that property of a body which demands
that a force is necessary to change its motion. “Inertial mass” is the numerical measure
of inertia. The conditions under which an experimental proof of this law could be carried
out are clearly not attainable.
In order to investigate the motion of a system it is necessary to choose a frame of refer-
ence, assumed to be rigid, relative to which the displacement, velocity, etc., of the system are
to be measured. The law of inertia immediately classifies the possible frames of reference
into two types. For, suppose that in a certain frame S the law is found to be true; then it must
also be true in any frame which has a constant velocity vector relative to S. However, the law
is found not to be true in any frame which is in accelerated motion relative to S. A frame of

reference in which the law of inertia is valid is called an “inertial frame,” and any frame in
accelerated motion relative to it is said to be “noninertial.” Any one of the infinity of inertial
frames can claim to be at rest while all others are in motion relative to it. Hence it is not
possible to distinguish, by observation, between a state of rest and one of uniform motion in
a straight line. The transformation rules by which the observations relative to two inertial
frames are correlated can be deduced from the second law of motion.
Newton’s “second law of motion” states that in an inertial frame the force acting on
a mass is equal to the time rate of change of its linear momentum. “Linear momentum,”
a vector, is defined to be the product of the inertial mass and the velocity. The law can
be expressed in the form
dրdt(mv) ϭ F (1.1)
which, in the many cases in which the mass m is constant, reduces to
ma ϭ F (1.2)
where a is the acceleration of the mass.
The “third law of motion,” the “law of action and reaction,” states that the force with
which a mass m
i
acts on a mass m
j
is equal in magnitude and opposite in direction to
the force which m
j
exerts on m
i
. The additional assumption that these forces are
collinear is needed in some applications, e.g., in the development of the equations govern-
ing the motion of a rigid body.
The “law of gravitation” asserts that the force of attraction between two point
masses is proportional to the product of the masses and inversely proportional to the
square of the distance between them. The masses involved in this formula are the

gravitational masses. The fact that falling bodies possess identical accelerations leads,
in conjunction with Eq. (1.2), to the proportionality of the inertial mass of a body to
its gravitational mass. The results of very precise experiments by Eotvös and others
show that inertial mass is, in fact, equal to gravitational mass. In the future the word
mass will be used without either qualifying adjective.
If a mass in motion possesses the position vectors r
1
and r
2
relative to the origins
of two inertial frames S
1
and S
2
, respectively, and if further S
1
and S
2
have a relative
velocity V, then it follows from Eq. (1.2) that
r
1
ϭ r
2
ϩ Vt
2
ϩ const
(1.3)
t
1

ϭ t
2
ϩ const
in which t
1
and t
2
are the times measured in S
1
and S
2
. The transformation rules Eq. (1.3),
in which the constants depend merely upon the choice of origin, are called “galilean
transformations.” It is clear that acceleration is an invariant under such transformations.
The rules of transformation between an inertial frame and a noninertial frame are
considerably more complicated than Eq. (1.3). Their derivation is facilitated by the
application of the following theorem: a frame S
1
possesses relative to a frame S an angular
velocity ␻ passing through the common origin of the two frames. The time rate of change
of any vector A as measured in S is related to that measured in S
1
by the formula
(dAրdt)
S
ϭ (dAրdt)
S
1
ϩ ␻ ϫ A (1.4)
The interpretation of Eq. (1.4) is clear. The first term on the right-hand side accounts

for the change in the magnitude of A, while the second corresponds to its change in
direction.
If S is an inertial frame and S
1
is a frame rotating relative to it, as explained in the
statement of the theorem, S
1
being therefore noninertial, the substitution of the posi-
tion vector r for A in Eq. (1.4) produces the result
v
abs
ϭ v
rel
ϩ ␻ ϫ r (1.5)
In Eq. (1.5) v
abs
represents the velocity measured relative to S, v
rel
the velocity relative
to S
1
, and ␻ ϫ r is the transport velocity of a point rigidly attached to S
1
. The law of
transformation of acceleration is found on a second application of Eq. (1.4), in which
A is replaced by v
abs
. The result of this substitution leads directly to
(d
2

rրdt
2
)
S
ϭ (d
2
rրdt
2
)
S
1
ϩ ␻ ϫ (␻ ϫ r) ϩ ␻
и
ϫ r ϩ 2␻ ϫ v
rel
(1.6)
in which ␻
и
is the time derivative, in either frame, of ␻. The physical interpretation of
Eq. (1.6) can be shown in the form
a
abs
ϭ a
rel
ϩ a
trans
ϩ a
cor
(1.7)
where a

cor
represents the Coriolis acceleration 2␻ ϫ v
rel
. The results, Eqs. (1.5) and
(1.7), constitute the rules of transformation between an inertial and a nonintertial
frame. Equation (1.7) shows in addition that in a noninertial frame the second law of
motion takes the form
ma
rel
ϭ F
abs
− ma
cor
− ma
trans
(1.8)
The modifications required in the above formulas are easily made for the case in which
S
1
is translating as well as rotating relative to S. For, if D(t) is the position vector of the
origin of the S
1
frame relative to that of S, Eq. (1.5) is replaced by
V
abs
ϭ (dDրdt)
S
ϩ v
rel
ϩ ␻ ϫ r

and consequently, Eq. (1.7) is replaced by
a
abs
ϭ (d
2
Dրdt
2
)
S
ϩ a
rel
ϩ a
trans
ϩ a
cor
In practice the decision as to what constitutes an inertial frame of reference depends
upon the accuracy sought in the contemplated analysis. In many cases a set of axes rigidly
attached to the earth’s surface is sufficient, even though such a frame is noninertial to the
extent of its taking part in the daily rotation of the earth about its axis and also its yearly
rotation about the sun. When more precise results are required, a set of axes fixed at the
center of the earth may be used. Such a set of axes is subject only to the orbital motion of
the earth. In still more demanding circumstances, an inertial frame is taken to be one
whose orientation relative to the fixed stars is constant.
1.3 THE DYNAMICS OF A SYSTEM OF MASSES
The problem of locating a system in space involves the determination of a certain
number of variables as functions of time. This basic number, which cannot be reduced
without the imposition of constraints, is characteristic of the system and is known as
CLASSICAL MECHANICS 1.5
its number of degrees of freedom. A point mass free to move in space has three
degrees of freedom. A system of two point masses free to move in space, but subject

to the constraint that the distance between them remains constant, possesses five
degrees of freedom. It is clear that the presence of constraints reduces the number of
degrees of freedom of a system.
Three possibilities arise in the analysis of the motion-of-mass systems. First, the
system may consist of a small number of masses and hence its number of degrees of
freedom is small. Second, there may be a very large number of masses in the system,
but the constraints which are imposed on it reduce the degrees of freedom to a small
number; this happens in the case of a rigid body. Finally, it may be that the constraints
acting on a system which contains a large number of masses do not provide an appreciable
reduction in the number of degrees of freedom. This third case is treated in statistical
mechanics, the degrees of freedom being reduced by statistical methods.
In the following paragraphs the fundamental results relating to the dynamics of mass sys-
tems are derived. The system is assumed to consist of n constant masses m
i
(i ϭ 1, 2, . . ., n).
The position vector of m
i
, relative to the origin O of an inertial frame, is denoted by r
i
. The
force acting on m
i
is represented in the form
(1.9)
in which F
i
e
is the external force acting on m
i
, F

ij
is the force exerted on m
i
by m
j
, and
F
ii
is zero.
1.3.1 The Motion of the Center of Mass
The motion of m
i
relative to the inertial frame is determined from the equation
(1.10)
On summing the n equations of this type one finds
(1.11)
where F
e
is the resultant of all the external forces which act on the system. But
Newton’s third law states that
F
ij
ϭ −F
ji
and hence the double sum in Eq. (1.11) vanishes. Further, the position vector r
c
of the
center of mass of the system relative to O is defined by the relation
(1.12)
in which m denotes the total mass of the system. It follows from Eq. (1.12) that

(1.13)
and therefore from Eq. (1.11) that
F
e
ϭ m d
2
r
c
րdt
2
(1.14)
mv
c
ϭ
a
n
iϭ1
m
i
v
i
mr
c
ϭ
a
n
iϭ1
m
i
r

i
F
e
ϩ
a
n
iϭ1

a
n
jϭ1
F
ij
ϭ
a
n
iϭ1
m
i

dv
i
dt
F
i
e
ϩ
a
n
jϭ1

F
ij
ϭ m
i

dv
i
dt
F
i
ϭ F
e
i
ϩ
a
n
jϭ 1
F
ij
1.6 MECHANICAL DESIGN FUNDAMENTALS
which proves the theorem: the center of mass moves as if the entire mass of the system
were concentrated there and the resultant of the external forces acted there.
Two first integrals of Eq. (1.14) provide useful results [Eqs. (1.15) and (1.16):
(1.15)
The integral on the left-hand side is called the “impulse” of the external force.
Equation (1.15) shows that the change in linear momentum of the center of mass is
equal to the impulse of the external force. This leads to the conservation-of-linear-
momentum theorem: the linear momentum of the center of mass is constant if no
resultant external force acts on the system or, in view of Eq. (1.13), the total linear
momentum of the system is constant if no resultant external force acts:

(1.16)
which constitutes the work-energy theorem: the work done by the resultant external
force acting at the center of mass is equal to the change in the kinetic energy of the
center of mass.
In certain cases the external force F
i
e
may be the gradient of a scalar quantity V
which is a function of position only. Then
F
e
ϭ −∂V/∂r
c
and Eq (1.16) takes the form
(1.17)
If such a function V exists, the force field is said to be conservative and Eq. (1.17) provides
the conservation-of-energy theorem.
1.3.2 The Kinetic Energy of a System
The total kinetic energy of a system is the sum of the kinetic energies of the individual
masses. However, it is possible to cast this sum into a form which frequently makes
the calculation of the kinetic energy less difficult. The total kinetic energy of the masses
in their motion relative to O is
but r
i
ϭ r
c
ϩ␴
i
where ␴
i

is the position vector of m
i
relative to the
system center of mass C (see Fig. 1.1).
Hence
T ϭ
1
2

a
n
iϭ1
m
i
r
.
2
c
ϩ
a
n
iϭ1
m
i
r
.
c

#
␴

.
i
ϩ
1
2

a
n
iϭ1
m
i
␴
.
i
2
T ϭ
1
2

a
n
iϭ1
m
i
v
2
i
c
1
2

mv
2
c
ϩ Vd
2
1
ϭ 0
Ύ
2
1
F
e
#
r
c
ϭ
1
2
mv
2
c
d
2
1
Ύ
t
2
t
1
F

e
dt ϭ mv
c
st
2
d Ϫ mv
c
st
1
d
CLASSICAL MECHANICS 1.7
0
r
i
m
i
σ
i
r
c
C
FIG. 1.1