Computer-aided analysis of mechanical systems, Notas de estudo de Engenharia Elétrica
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Computer-aided analysis of mechanical systems, Notas de estudo de Engenharia Elétrica

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TO TEM a) TM E REM QU ia ND ENTAD AINDA VINTON SYSTEMS Parviz E. Nikravesh A E E EE TRE HEART —28, + ini, — S6y8, Dr RE Rar E RD EE ER CD E CR E RE PR a E JO TR [e o)Lo)+ il [5] 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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