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NOMENCLATURE
Matrices are in boldface upper-casc characters
Column matrices, algebraic vectors, and arrays are
in boldface lower-casc characters.
Scalars arc in lightfaçe characters.
Column vector (array)
Row vector (array)
Matrix
Element of matrix A in ith sow and jk column
Zero vector
Zero Enull) matrix
OVERSCORES
Geometric vector
3 x 3 skew-symmetric matrix
4 X 4 skew-symmetric matrix containing a negative
3 X 3 skew-symmetric matrix
4 X 4 skew-symmerric matrix containing a positive
3 X 3 skew-symmctrie matrix
First derivative with respect to time
Second derivative with respect to time
SUPERSCRIPTS
Matrix inverse
ith time step
Matrix or vector transpose
Type of constraint or force
Components of a vector in a body-fixed
coordinate system
Components of a vector or matrix
in Euter-parameter space
SUBSCRIPTS
ith body in a system
Projection of a vector along a known axis
Orr bas ba
6,5
do;
d
€gs Pis Co, Es
&
SYMBOLS
Vector of righl-hand side of acceleration
equations
Angle between two vectors
Vector of Lagrange multipliers
Polar moment of inertia for body é
Local (body-fixed) Cartesian coordinate
system
Radius of a circle
Lagrange multiplier associated with the
constraint on ps
Angle of rotation
Bryant angles
Euler angles
Angular velocity vector for body 1
Global components of à;
Local components of à,
One constraint; vector of constraints
Jacobian matrix of constraints
Number of bodies
Vector containing quadratic velocity
terms for body é
Vector of quadratic velocity terms
Vector with its ends on two different
bodies
Global components of d
Euler parameters
Vector of three Euler parameters €1, é, 3
for body é
Force acting on body
Global components of f;
Vector of forces for body / containing f,
and n;
Vector of forces for a system
Vector of constraint reaction forces
Veloeity vector for body é containing E,
and of
Vector of velocities for a system
Number o! degrees of freedom (DOF)
Vector wilh its ends on two different
bodies
Global components of
Number of constraint equations
ter
nt
J,
J
grrqãa
qa
Mass of a particle
Mass of body i
Number of coordinates
Moment acting on body à |
Global components of à,
Local components of à;
Components of à, in fout-dimensional |
space
Vector of four Euler parameters
ea €1, 3, €3 for body à
Vector of coordinates for body 1
Vector of coordinates for a system
Translational position vector for body é
Global coordinates of 7;
Vector with both ends on body é
(constant magnitude)
Global components of 3;
Local components of 5;
Time
Initial time
Final (nd) time
Unit vector
Global components of à; vector of
dependent coordinates
Vector of independent coordinates
Global Cartesian coordinate system
Vector of integration variables
Rotationul transformation matrix for
body é
3 X 4 transformation matrix for body i
3 x 3 or gencral identity matrix
4X 4 identity matrix
Global incrtia tensor for body i
Local (constant) inertia tensor for body é
4 X 4 inertia tensor
Lower triangular matrix
3 X 4 transtormation matrix for body é
6 X 6 mass matrix for body i containing
N, and J;
Mass matrix for a system
3 X 3 diagonal mass matrix for body é
Upper triangular matrix
Potential energy
Computer-Aided
Analysis
of
Mechanical Systems
PARVIZ E. NIKRAVESH
Aerospace and Mechanical
Engineering Department
University of Arizona
PRENTICE HALL, Englewood Cliffs, New Jersey, 07632
Nikrayesh, Parviz E.
Computer-aided analysis of mechanical systems / by Parviz E.
Niravesh
em.
Biblio: p.
Includes index.
TSBN 0-13-164220-0
1. Machinery, Kinematics of-- Data processing. 2. Machinery,
Dymamies of — Data processing. 1. Title.
TITS ,N52 1987 s7-22908
621.811 —deto cr
Editorial/production supervision
and interior design: Elena Le Pera
Cover design: Photo Plus Art
Manufacturing buyer: Cindy Grant
To the memory of my sister, Henriette.
& 1988 by Prentice-Hall, Inc.
A Division of Simon & Schuster
Englewood Cliffs, New Jersey 07632
Aly rights reserved. No part of this book may be
reproduced, in any form or by any means,
without permission in writing from the publisher.
Printed in the United States of Amcrica
0987654321
ISBN 0-19-1k4220-0 025
Prentice-Hall International (UK) Limited, London
Prentice-Hall of Australia Pty. Limited, Sydney
Prentice-Hall Canada Inc., foronto
Prentice-Hall Hispanoamericana, S.A., Mexico
Prentice-Hall of India Private Limited, New Delhi
Prentice-Hall of Japan, Inc., Tokyo
Simon & Schuster Asia Pte, Ltd., Singapore
Editora Prentice-Hall do Brasil, Lida., Rio de Janeiro
Contents
Preface
Note on Unit System
INTRODUCTION
11
1.2
1.3
1.4
Computers in Design and Manufacturing
111 Computer-Aided Analysis 2
Multibody Mechanical Systems 3
Branches of Mechanics 6
131 Methods of Analysis 6
Computational Methods 9
141 Efficiency versus Simplicity 10
142 A Genera-Purpose Program 14
VECTORS AND MATRICES
21
2.2
2.3
Geometric Vectors 19
Matrix and Algebraic Vectors 21
22.1 Matrix Operations 21
2.2.2 Algebraic Vector Operations 24
Vector and Matrix Differentiation 28
2.3.1 Time Derivatives 28
2.3.2 Partial Derivatives 29
Problems 33
1
xiii
19
iv Contents
BASIC CONCEPTS AND NUMERICAL METHODS
IN KINEMATICS 35
3.1 Definitons 35
3,1] Classification of Kinematic Pairs 37
3.12 Vector of Coordinates 38
313 Degreesofkreedom 40
314 Constraint Eguations 41
3.1.5 Redundant Constraints 41
3.2 Kinematic Analysis 42
32.1 Coordinate Partitioning Method 43
3.2.2 Method of Appended Driving
Constraints 48
3.3 Linear Algebraic Equations 50
3.3.1 Gaussian Methods 51
3.32 Pivoting 53
3.3.3 LU Factorization 56
334 LU Factorization with Pivoting 61
3.3.5 Subroutines for Lincar Algebraic
Equations 63
3.4 Nonlinear Algebraic Equations 66
34.1 Newton-Raphson Method for One Equation
in One Unknown 66
342 Newton-Raphson Method for n Equations in
nUnknowns 67
343 A Subroutine for Nonlinear Algebraic
Equations 70
Problems 72
PLANAR KINEMATICS 77
4.1 Cartesian Coordinates 77
4.2 Kinematic Constraints 80
42.1 Revolute and Transitional Joints
(tp) 81
42.2 Composite Soinis (LP) 84
4.2.3 Spur Gears and Rack and Pinion (HP) 86
424 Curve Representation 89
42.5 Cam-Followers (HP) 93
4.2.6 Point-Follower (HP) 97
4.2.7 Simplificd Constraints 98
428 Driving Links 100
4.3 Position, Velocity, and Acceleration Analysis 101
4.3.1 Systematic Generation of Some Basic
elements 103
4.4 Kincmatic Modeling 105
44.1 Slider-Crank Mechanism 105
442 Quick-Return Mechanism 110
Problems 115
Contents v
A FORTRAN PROGRAM FOR ANALYSIS
OF PLANAR KINEMATICS 119
5.1 Kinematic Analysis Program (KAP) 119
5.1.1 Model-Description Subrowines 123
5.12 Kinematic Analysis 127
5.13 Function Evaluation 130
5.14 Input Prompis 134
5.2 Simple Examples 134
5.2.1 Four-Bar Lintage 135
5.2.2 Slider-Crank Mechanism 137
523 QuickReturn Mechanism 139
5.3 Program Expansion 140
Problems 140
EULER PARAMETERS 153
6.1 Coordinates of A Body 153
6.1.1 Euler's Theorem on the Motion ofa
Body 157
6.1.2 Active and Passive Points of View 157
6.1.3 Euter Parameters 158
6.14 Determination of Euler Parameters 160
6.15 Determination of the Direction
Cosines 164
6.2 Identities with Euler Parameters 166
6.2.1 Identities with Arbitrary Vectors 170
6.3 The Concept of Angular Velocity 172
6.3.1 Time Derivatives of Euler
Parameters 174
6.4 Semirotating Coordinate Systems 176
6.5 Relative Axis of Rotation 177
6.5.1 Intermediate Axis of Rotation 180
6.6 Finite Rotation 180
Problems 181
SPATIAL KINEMATICS 186
7.1 Relative Constraints between Two Vectors 186
74! Two Perpendicular Vectors 188
7.12 Two Parallel Vectors 188
7.2 Relative Constraints between Two Bodies 189
7.2.1 Spherical, Universal, and Revolute Joints
(LP) 190
7.22 Cylindrical, Translational, and Screw Joints
(tr) 192
7.2.3 Composite doints 196
724 Simplified Constraints 199
vi Contents
7.3 Position, Velocity, and Acceleration Analysis 200
7.3.1 Modified Jacobian Matrix and Modified
Vecory 201
Problems 204
8 BASIC CONCEPTS IN DYNAMICS 208
8.1 Dynamics of a Particle 208
8.2 Dynamics of a System of Particles 209
8.3 Dynamics ofaBody 211
8.3.1 Moments and Couples 212
83.2 Rotational Equations of Motion 215
8.3.3 The lnertia Tensor 217
8.3.4 An Unconsirained Body 219
8.4 Dynamics of a System of Bodies 221
8.4.1 A System of Unconstrained Bodies 221
842 A System of Constrained Bodies 222
8.4.3 Constraint Reaction Forces 223
8.5 Conditions for Planar Motion 224
9 PLANAR DYNAMICS 227
9.1 Equations of Motion 227
9.2 Vectorof Forces 229
9.2.1 Gravitutional Force 229
92.2 Single Force or Moment 229
9.2.3 Transiational Acruators 231
9.24 Transtational Springs 232
9.2.5 Translational Dampers 234
9.2.6 Rotational Springs 236
9.27 Rotational Dampers 237
9.3 Constraint Reaction Forces 237
9.3.1 Revoine Joint 237
9.3.2 Revolwe-Revolute Joint 240
9.33 Transtational Joiy 242
9.4 System of Planar Equations of Motion 242
9.5 Static Forces 244
9.6 Static Balance Forces 245
9.7 Kinctostatic Analysis 247
Problems 248
10 A FORTRAN PROGRAM FOR ANALYSIS
OF PLANAR DYNAMICS é 253
10.1 Solving the Equations of Motion 253
10.2 Dynamic Analysis Program (DAP) 254
1021 Model-Description Subroutines 258
11
12
Contents
10.2.2 Dynamic Analysis 260
10.23 Function Evaluation 263
10.24 Force Evaluation 263
102.5 Repórtiny 265
102.6 Static Analysis 266
102.7 input Prompes 267
10.3 Simple Examples 268
103.1 Four-Bar Linkage 268
10.32 Horizontal Platform 269
10.3.3 DumpTruck 273
10.4 Time Step Selection 277
Problems 281
SPATIAL DYNAMICS
11.1 Vectorof Forces 289
Hit Conversion of Moments 289
11.2 Equations of Motion for an Unconstrained
Body 291
11.3 Equations of Motion for a Constrained Body
11.4 System of Equations 293
H1.4.1 Unconstrained Bodies 294
11.42 Constrained Bodies 296
11.5 Conversion of Kinematic Equations 297
Problems 299
NUMERICAL METHODS FOR ORDINARY
DIFFERENTIAL EQUATIONS
12.1 Initial-Value Problems 301
12.2 Taylor Series Algorithms 302
122.1 Runge-Kutta Algorithms 303
222 A Subroutine for a Runge-Kutta
Algorithm 304
12.3 Polynomial Approximation 307
12.3.4 Explicit Mutistep Algorithms 308
42.3.2 Implicii Mulistep Algorithms 308
12.3.3 Predictor-Corrector Algorithms 309
12.34 Methods for Starting Multistep
Algorithms 309
12.4 Algorithms for Stiff Systems 310
12.5 Algorithms for Variable Order and Step Size
Problems 311
292
311
vii
289
301
viii
13 NUMERICAL METHODS IN DYNAMICS
13.1
13.2
13.3
13.4
Integration Atrays 313
Kinematically Unconstrained Systems 314
13.2.1 Mathematical Constraints 315
1322 Using Angular Velocities 317
Kinematically Constrained Systems 318
133.1 Constraint Violation Stabilization
Method 319
13.32 Coordinate Partitioning Method 321
13.3.3 Automatic Partitioning of the
Coordinates 324
13.34 Stiff Differential Equation Method 327
Joint Coordinate Method 330
1344 Open-Chain Systems 331
1342 ClosedLoop Systems 334
Problems 335
14 STATIC EQUILIBRIUM ANALYSIS
14.1
14.2
14.3
14.4
Appendix A.
Al
AZ
Appendix B.
B.l
B.2
Appendix C.
An Iterative Method 339
4,11 Coordinare Partitioning 340
Potential Energy Function 341
142.1 Minimization of Potential Energy 342
Fictitious Damping Method 344
Joint Coordinates Method 345
EULER ANGLES AND BRYANT ANGLES
Euler Angles 347
Ali Time Derivatives of Euler Angles 349
Bryant Angles 351
A2.1 Time Derivatives of Bryant Angles 352
RELATIONSHIP BETWEEN EULER PARAMETERS
AND EULER ANGLES
Euler Parameters in Terms of Euler Angles 353
Euler Angles in Terms of Euler Parameters 354
COORDINATE PARTITIONING
WITH L-U FACTORIZATION
REFERENCES
BIBLIOGRAPHY
INDEX
Contents
313
339
347
353
355
357
359
363
Preface
This book is designed to introduce fundamental theories and numerical methods for use
in computational mechanics. These theories and methods can be used to develop com-
puter programs for analyzing the response of simple and complex mechanical systems.
In such programs the equations of motion are formulated systematicaily, and then solved
numerically, Because they are rélatively easy to use, the book focuses on Cartesian co-
ordinates for formulating the equations of motion, After the reader has become familiar
with this method of formulation, it can serve as a stepping stone to formulating the
equations of motion in other seis of coordinates. The numerical algorithms that are dis-
cussed in this book can be applied to the equations of motion when formulated in any
coordinate system.
Organization of the Book
The text is organized in such a way that it can be used for teaching or for self-
study, The concepts and numerical methods used in kinematics are systematically treated
before the concepts and numerical methods used in dynamics are introduced. Separate
chapters on cach of these tópics allow the text to be used for the study of each topic
separately or for some desired combination of topics. Furthermore, the text first treats
the less complex problems of pfanar kinematic and dynamic analysis before it discusses
spatial kinematic and dynamic analysis.
With the exception of the first two chapters and the last chapter, the text can be
divided into two subjects —kinematics and dynamics. Chapter 1 gives an introduction to
the subject of computational methods in kinematics and dynamics. Simple examples
ilustrate how à problem can be formulated using different coordinate systems. Chapter 1
also explains why Cartesian coordinates provide a simple tool, if not necessarily the
most computationally efficient one. Chapter 2 presents a revicw of vector and matrix
ix
x Preface
algebra, with an emphasis on the kind of formulation that lends itself to implementation
in computer programs.
Chapters 3 through 7 deal with kinematics. Chapter 3 introduces the basic con-
cepts in kinematics that are applicable to both planar and spatial systems. Algebraic con-
straint equations, the various coordinate systems, and the idea of degrees of freedom are
presented as a foundation for both the analytical and the numerical aspects of kinematic
analysis. Position, velocity, and acceleration analysis techniques are presented and illus-
trated through the solution of simple mechanisms. Numerical methods for solving the
associated kincmatic equations are presented and illustrated, These include methods for
solving sets of linear and nonlinear algebraic equations. À comprehensive treatment of
planar kinematics using Cartesian coordinates is presented in Chapter 4. In that chapter,
a library of kinematic constraints is defined and the governing algebraic constraint equa-
tions are derived.
Chapter 5 contains a FORTRAN progtam for planar kinematic analysis. The pro-
gram is developed and explained as a collection of subroutines that carry out the func-
tions of kinematic analysis. The problems at the end of Chapter 5 provide guidelines for
the extensions that allow for the expansion of the program to treat broader classes of
planar kinematic systems.
Chapter 6 presents a set of spatial rotational coordinates known as Euler parameters.
The physical properties of Euler parameters and the development of their algebraic prop-
crties are introduced to allow the reader to become comfortable with and confident in their
use. Also, velocity relationships — including the definition of angular velocity — and
other identities are developed that are necessary for the formulation of spatial kinematic
and dynamic analysis.
Chapter 7 presents a unified formulation of spatial kinematics using Cartesian co-
ordinates and Euler parameters. Vector relationships that are required for the definition
of kinematic joints are first presented and then applied to derive the governing equations
for a library of spatial kinematic joints. Although this book does not provide a source
listing for a spatial kinematic analysis program, the computer program in Chapter 5 and
the constraint formulations in Chapter 7 provide all the information that the reader needs
to develop a computer program.
Chapters 8 through 13 deal with dynamics. Basic concepts in dynamics arc pre-
sented in Chapter 8. Discussion begins with familiar concepts of the dynamics of à parti-
cle and progresses to the dynamics of systems of particles and, finally, to the dynamics
of rigid bodies. By means of a building block formulation, the complete theory of the
dynamics of systems of rigid bodies is developed in a systematic and understandable
way. The Newton-Euler equations of motion are derived and used as a fundamental tool
in the dynamic analysis of systems of rigid bodies that are connected by kinematic
joints. The Lagrange multiplier formulation for constrained systems is developed, and
the reaction forces between the joints are derived in terms of the Lagrange multiplicrs,
Chapter 9 discusses the planar dynamics of systems of constrained rígid bodies,
drawing upon the kinematics theory discussed in Chapter 4 and the basic dynamics
theory discussed in Chapter 8. Even though the numerical methods for solving the differ-
ential equations of motion arc discussed in detail in Chapters 12 and 13, a FORTRAN
program for planar dynamic analysis is presented in Chapter 10. This program, which is
Preface xi
a collection of subroutines used to implement a variety of computations required in the
formulation and solution of equations of motion, builds upon the kinematic analysis
program in Chapter 5. The computer program is demonstrated through the solution of
simple examples, and extensions to the program are included as problems at the end of
the chapter.
Chapter 11 presents the formulation of spatial system dynamics using Cartestan
coordinates and Euler parameters. The eguations of motion of kinematically constrained
systems of rigid bodies are derived and developed in a form suitable for computational
implementation. Chapter 12 presents a brief overview of numerical methods for solving
ordinary differential equations. A FORTRAN listing of a fourth-order Runge-Kutta
algorithm illustrates the implementation of these numerical methods along with some
examples. Chapter 13 presents a number of advanced numcrical methods for multibody
dynamics. Alternate techniques and algorithms for the solution of mixed systems of dif-
ferential and algebraic equations that arise in system dynamics are presented.
Jn the analysis of multibody mechanical systems, it may be necessary to go beyond
kinematics and dynamics and find the static equilibrium state of a system. Chapter 14
discusses several computation-based methods for static equilibrium analysis.
Level of Courses
The book can be covered in two successive courses. The student is required
to know the fundamentals of kinematies and dynamics, to have a basic knowledge of
numerical methods, and to know computer programming, preferably FORTRAN.
The first course — a senior undergraduate or a first-year graduate course — could
cover Chapters 1 through 5, 9, and 10, on planar motion; if students do not have the
proper background in numcrical methods in ordinary differential equations, Chapter 12
should also be covered to the extent necessary. The course could be project-oriented:
students could be assigned to find existing medium- to large-scale mechanical systems
and analyze them using the computer programs that are provided in the book. The second
course would then cover Chapters 6 through 8 and 11 thtough 14, on spatial motion; this
would be quite suitable as a graduate-level course. Students, divided into groups, should
be able to develop a spatial-motion dynamic analysis program.
Another possibility would be one course, covering Chapters 1 through 7, on the
subject of kinematics, and a second course, covering Chapters 8 through 14, on the sub-
ject of dynamics.
Exercises
Problem assignments can be found at the end of most chapters. The problems are.
designed to clarify certain points and to provide ideas for program development and
analysis techniques. However, by no means do these problems represent the ultimate
flexibility and power of the formulations and algorithms that are stated in the book.
Most realistic multibody problems that arise in engineering practice can be treated by
employing similar techniques and ideas.
xii Preface
Computer Programs
Two FORTRAN programs called KAP and DAP, for planar kinematic and dy-
namic analysis, respectively, are developed and listed in the book. Other programs, for
static equilibrium analysis, or for spatial kinematic and dynamic analysis, can be devel-
oped by the reader by following the formulations and algorithms that are discussed in
various chapters. Source codes for KAP, DAP, and other complementary programs can
be obtained on a floppy disk from the publisher.
ACKNOWLEDGMENTS
1 began working on the first version of this manuscript in 1980 at the University of lowa
while 1 was teaching two ncwly developed courses on this subject. Some of the material
that was taught in these courses grew out of research collaboration with two of my
colieagues, Dr. Edward J, Haug and Dr, Roger A. Wehage. I would like to express my
appreciation to Ed for his encouragement and comments on the earlier versions of the
manuscript. 1 am deeply gruteful to Roger for many stimulating discussions over the
ycars. Without his initiative, creativity, and support this book would not have been
possible.
1 would like to thank all of the graduate students who assisted me in many ways.
They checked many of the formulas, found many errors, and generated many ideas. Tn
particular, 1 would like to express my thanks to Mr. Hamid M. Lankarani for his assis-
tance during the past two years and to Mr. Jorge A.C. Ambrosio for his effort in gener-
ating the computer graphic images.
Finally, 1 would like to thank my wife, Agnes, for her support and patience, and
for not getting upset about the many evenings, weekends, and holidays that 1 spent
working on this book,
University of Arizona Parviz Nikravesh
Tucson
Note on Unit System xiii
NOTE ON UNIT SYSTEM
“Fhe system of units adopted in this book is, unless otherwise stated, the international
system of units (SI). In most examples and problems, the variables are organized as the
elements of arrays suitable for programming purposes. These variables usually represent
various different quantitics and therefore have different units. Hf the unit of each element
of an array were to bc stated, it would cause notational confusion. Therefore, in order to
eliminate this problem, the units of the variables are not stated in most parts of the text.
The reader must assign the correct unit to each variable. The unit of degree or radian
alonc is stated for variables representing angular quantities.
SI Units Used in This Book
Quentity Unit SI Symbol
(Base Units)
Length meter m
Mass kilogram kg
Time second 5
(Derived Units)
Acceleration, translational meter/second? m/s?
Acceleration, angular radian! /second? rad/8
Damping cocfficient newton-second/meter N.s/m
Force newton N(=kgm/8)
Moment of force newton-meter Nm
Moment of Tnertia, mass kilogram-meter” kgm?
Pressure pascal Pa (=N/m?)
Spring constant newton/meter N/m
Velocity, translational meter/second m/s
Volocity, angular radian” /second rad/s
Tor degree
—— 1 ——
Introduction
The major goal of the engineering profession is to design and manufacture marketable
produets of high quality. Today's industries are utilizing computers in every phase of the
design, management, manufacture, and storage of their products. The process of design
and manufacture, beginning with an idea and ending with a final product, is a closed-loop
process. Almost every link in the loop can benefit from the power of digital computers.
1.1 COMPUTERS IN DESIGN AND MANUFACTURING
Factory automation is one of the major objectives of modem industry. Although there is
no onc plan for factory automation, a general configuration is presented in Fig. 1.1. In
this configuration, all branches of the factory communicate and exchange information
through a central data base, Various parts of the product are designed in the computer-
aided engineering (CAE) branch, and then the design is sent to the computer-uided manu-
faeturing (CAM) branch [or parts manufacturing and final assembly. Two of the major
subbranches of CAE are computer-aided product design and computer-aided manufac-
turing design.
The computer-sided product design branch, better known as computer-aíded design
(CAD),* may considor the design of single parts or it may concem itself with the final
product as an assembly of those parts. Computerized product design requires such capa-
bilities as computer-aided analysis, computer-aided draíting, design sensitivity analysis,
or optimization. The computer-aided analysis capability serves as part of the design proc-
*The abbroviation CAD is commonly used for both computer-aidod drafting and computer-aided design. Most
of the CAD systems available today are intelligent computerized drafting systems with limited design capability.
1
2 Introduction Chap. 1
Warehouse
and
shipping
Management
CAM
parts and
assembly
Computer-sided
product design
Computer-aided
drafting
Computer-sided
analysis
Computer-aided
manufacturing
design
Optimization
techniques
Figure 1.1 Automated desiga and manufacturing.
ess and is also used as a model simulator for the finished manufactured product. Analy-
sis may be considered especially appropriate for a product whose initial design has to be
modified several times during the manufacturing process. Thus computer-aided analysis
can bg used as a substitute for laboratory or field tests in order to reduce the cost.
The computer-aided manufacturing design branch is concerned with the design
of the manufacturing process. This branch considers the manufacturability of newly
designed parts and employs techniques to improve the manufacturing process, in addi-
tion to on-linc control of the manufacturing process.
1.1.1 Computer-Aided Analysis
The computer-sided analysis process (CAA) allows thc engineer to simulate the behav-
ior of a product and modify its design prior to actual production. In contrast, prior to the
introduction of CAA, the manufacturer had to construct and test a series of prototypes, a
process which was not only time-consuming but also costly. Most optimal design tech-
niques require repetitive analysis processes. Although one of the major goals of an auto-
mated factory is computer-aided design, computer-aided analysis techniques must be
developed first.
Sec. 1.2 Muitibody Mechanical Systems 3
Computer-aided analysis techniques may be applied to the study of electrical and
electronic circuits, structures, or mechanical systems. The development of algorithms
for analyzing electrical circuits began in the early days of electronic computers. Similar
techniques were also employed to develop computer programs for struetural analysis.
Today, these programs, known as finite-element techniques, have become highly
advanced and are used widely in various fields of engineering.
Tt was not until the early 1970s that computational techniques found their way into
the field of mechanical engineering. One of the areas of mechanical engineering where com-
putational techniques can be employed is the analysis of multibody mechanical systems.
1.2 MULTIBODY MECHANICAL SYSTEMS
Pendulym
ta)
A mechanical system is defined as a collection of bodies (or links) in which some or all
of the bodies can move relative to one another. Mechanical systems may range from
the very simplé to the very complex. An example of a simple mechanical system is the
single pendulum, shown in Fig. 1.2(a). This system contains two bodies — the pendu-
lum and the ground. Examples of more complex mechanical systems are the four-bar
linkage and the slider-crank mechanism, shown in Fig. 1.2(b) and (c), respectively. The
four-bar linkage is the most commonly uscd mechanism for motion transmission. he
slider-crank mechanism finds its greatest application in the internal-combustion engine.
While the motion of the systems in Fig. 1,2 is planar (two-dimensional), other
mechanical systems may experience spatial (three-dimensional) motion. For example,
the suspension and the steering system of an automobile, shown in Fig. 1.3, contain
several spatial mechanisms. This system as a whole has several degrees of freedom.
While the kinematics of the individual linkages in this vehicle are more.complicated than
those of the mechanisms shown in Fig. 1.2, the concept remains the same.
A cascade of simple planar linkage systems can be put together to perform rather
complex tasks. The deployable satellite antenna shown in Fig. 1.4 contains such a cas-
cade of six four-bar linkages.º Before deployment, the panels of the antenna are folded
in order to occupy the minimum space. Once the satellite is in orbit the panels are
unfolded in a predefined sequence, as shown in Fig. 1.5. When the unfolding process is
completed, the four-bar linkages become a truss structure to support the panels.
Connecting
rod
ELink 2 quink3)
(Link 4)
tb) te)
Figure 1.2 Examples of simple mechenical systems: (a) a single pendulum, (0) a four-bar mecha-
nism, and (c) a slider-crank mechanism.
Introduction Chap. 1
Figure 1,3. The suspension system and the stecring mechanism of am automobite.
Figure 1.4 A deployed satellite antenna.
ta) tb) toy
Figure 1.5 Unfolding process of the antenna in orbit: (4) folded panela; (b-c) unfold-
ing process.
Sec. 1.2 Multibody Mechanical Systems 5
Figure 1.5 (continued)
Another example of a mechanical system is a robotic device. A robot can be fixed
to a stationary base or to a movable base, as shown in Fig. 1.6. The motion and the posi-
tion of the end effector of a robot are controlled through force actuators located about
each joint connecting the bodies that-make up the robot.
fa) tbh
Figure 1.6 Examples of robots with (a) stationary basc and (b) movable base.
Any mechanical system can be represented schematically as a multibody system in
the manner shown in Fig. 1.7. The actual shape or outlinc of a body may not be of imme-
diate concern in the process of analysis. Of primary importance is the connectivity of the
bodies, the inertial characteristics of the bodies, the type and the location of the joints,
and the physical characteristics of the springs, dampets, and other elements in the system.
6 Introduction Chap. 1
Vigure 1.7 Schematie representation of a
multibody system.
1.3 BRANCHES OF MECHANICS
There are two different aspects to the study of a mechanical system: analysis and design.
When a mechanical system is acted on by a given excitation, for example, an external force,
the system exhibits a certain response. The process which allows an engineer to study
the response of an already existing system to a known excitation is called analysis. This
requires a complete knowledge of the physical characteristics of the mechanical system,
such as material composition, shape, and arrangement of paris. The process of determin-
ing which physical characteristics are necessary for a mechanical system to perform a
ptescribed task is called design or synthesis. The design process requires the application
of scientific techniques along with the engincer's judgment. The scientific techniques in
the design process ate merely tools to be used by the engineer. These are mainly analy-
sis techniques and optimization methods. Although thesc techniques can be employed in
a systematic manner in the design process, the overall process hinges on the judgment of
the design enginecr. Since the scientific aspect of the design process requires analysis
teclmiques as à tool, it is important to leam about methods of analysis prior to design.
The branch of analysis which studies motion, time, and forces is called mechanics.
Tt consists of two parts —statics and dynamics. Statics considers the analysis of station-
ary systems — systems in which time is not a factor. Dynamics, on the other hand, deals
with systems that are nonstationary — systems that change their response with respect to
time. Dynamics is divided into two disciplines — kinematics and kinetics. Xinematics is
the study of motion regardiess of the forces thut produce ths motion. More explicitly,
Kinematics is the study of displacement, velocity, and acceleration. Kinetics, on the other
hand, is the study of motion and its relationship with the forces that produce that motion.
The focus of this book is on the dynamics of mechanical systems, with an emphasis
on computational methods. In addition, one chapter às devoted to computational methods
in static equilibrium analysis, since this may be needed prior to dynamic analysis for
certain mechanical systems.
1.3.1 Methods of Analysis
Before wc analyze the motion of any mechanical system, we must make some simplify-
ing assumptions. For example, if the overall acceleration of a vehicle under the applicd
Sec. 1.3 Branches of Mechanics 2
load of the engine is to be determined, then the vibrational motions of certain parts of the
vehicle are of no significance. Hf one decides to consider the vibration and local deforma-
tion of every part of the vehicie, then determining the response of the system becomes
highly complicated, if not impossible. Therefore, these simplifying assumptions serve
two purposes: to make the problem solvable and to climinate the expenditure of effort on
unnecessary or insignificant responses.
Classical methods of analysis in mechanics have relicd upon graphical and often
quite complex techniques. These techniques are based on geometrical interpretations of
the system under consideration. As an example, consider the slider-crank mechanism
shown in Fig. 1.8. The crank is rotating with a constant angular velocity. The objective
is to find the velocity ol the slider. A graphical solution to this problem can be achieved
rather easily. The velocity of point A, 54, has a magnitude of vº = (1.0)(0.1) = 0.1 m/s
and is perpendicular to the crank OA, as shown in Figure 1.9(a). The velocity of point
B, V?, is in the direction of the motion of the slider, and the velocity of point B relative
to point A, denoted by vector 7”, is perpendicular to the connecting rod AB, A vector
expression relating these velocities is given as
pt= 04 + 0 D
A vector diagram (velocity polygon) corresponding to this expression is shown in
Fig. 1.9(b). From this diagram the magnitude and the direction of »? can be found.
Z Figure 1,8 A slider-crank mechanism.
veia
Á vs
A f
TA Tm /
o e f 01 mis veia
om Det sys
uv
B va
A
v
ta to)
Figure 1,9. Graphical solution.
Although a graphical solution to this problem is rather simple, its accuracy is
limited. The graphical approach can yield more accurate results if some trigonometric
formulas and geometric relations are introduced into the process. For example, for the
slider-crank mechanism, since the angle à and the lengths of the crank and the connect-
ing tod are known, other geometric information for this system can be found easily, as
depicted in Fig. 1.J0(a). Then a vector diagram can be constructed with complete details
as shown in Fig. 1.10(b). From this diagram, vê can be calculated from the elementary

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