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Mechanism Design Enumeration of Kinematic Structures According to Function © 2001 by CRC Press LLC Mechanical Engineering Series Frank Kreith - Series Editor Published Titles Entropy Generation Minimization Adrian Bejan Finite Element Method Using MATLAB Young W. Kwon & Hyochoong Bang Fundamentals of Environmental Discharge Modeling Lorin R. Davis Intelligent Transportation Systems: New Principles and Architectures Sumit Ghosh & Tony Lee Mathematical & Physical Modeling of Materials Processing Operations Olusegun Johnson Ileghus, Manabu Iguchi & Walter E. Wahnsiedler Mechanics of Composite Materials Autar K. Kaw Mechanics of Fatigue Vladimir V. Bolotin Mechanism Design: Enumeration of Kinematic Structures According to Function Lung-Wen Tsai Nonlinear Analysis of Structures M. Sathyamoorthy Practical Inverse Analysis in Engineering David M. Trujillo & Henry R. Busby Thermodynamics for Engineers Kau-Fui Wong Viscoelastic Solids Roderic S. Lakes Forthcoming Titles Distributed Generation: The Power Paradigm for the New Millennium Anne-Marie Borbely & Jan F. Kreider Engineering Experimentation Euan Somerscales Energy Audit of Building Systems: An Engineering Approach Moncef Krarti Finite Element Method Using MATLAB, 2nd Edition Young W. Kwon & Hyochoong Bang Introduction Finite Element Method Chandrakant S. Desai & Tribikram Kundu Mechanics of Solids & Shells Gerald Wempner & Demosthenes Talaslidis Principles of Solid Mechanics Rowland Richards, Jr. © 2001 by CRC Press LLC Mechanism Design Enumeration of Kinematic Structures According to Function Lung-Wen Tsai Presidential Chair Professor Department of Mechanical Engineering Bourns College of Engineering University of California, Riverside CRC Press Boca Raton London New York Washington, D.C. ON THE COVER: A 4-speed automatic transmission. (Courtesy of General Motors, Warren, MI.) Library of Congress Cataloging-in-Publication Data Tsai, Lung-Wen. Mechanism design : enumeration of kinematic structures according to function / Lung-Wen Tsai p. cm.--(Mechanical engineering series) Includes bibliographical references and index. ISBN 0-8493-0901-8 1. Machinery, Kinematics of. 2. Machine design. I. Title. II. Advanced topics in mechanical engineering series. TJ175 .T78 2000 621.8′11--dc21 00-056415 This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe. © 2001 by CRC Press LLC No claim to original U.S. Government works International Standard Book Number 0-8493-0901-8 Library of Congress Card Number 00-056415 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper Preface This textbook has evolved from class notes used for a course in systematic design of mechanisms that the author has taught for over a decade. Although it is written primarily for senior and first-year graduate level students in engineering, it is equally valuable for practicing engineers, particularly for mechanism and machine designers. Traditionally, mechanisms are created by the designer’s intuition, ingenuity, and experience. This ad hoc approach, however, cannot ensure the identification of all feasible design alternatives, nor does it necessarily lead to an optimum design. Two approaches have been developed to alleviate the problem. The first involves the development of atlases of mechanisms grouped according to function for use as a primary source of ideas. The second makes use of a symbolic representation of the kinematic structure and the combinatorial analysis as a tool for enumeration of mechanisms. This textbook introduces a systematic methodology for the creation and classification of mechanisms. The approach is partly analytical and partly algorithmic. It is based on the idea that, during the conceptual design phase, some of the functional requirements of a desired mechanism can be transformed into structural characteristics that can be employed for systematic enumeration of mechanisms. The kinematic structure of a mechanism contains the essential information about which link is connected to which other link by what type of joint. Using graph theory, combinatorial analysis, and computer algorithms, kinematic structures of the same nature, i.e., the same the number of degrees of freedom, type of motion (planar or spatial), and complexity can be enumerated in an essentially systematic and unbiased manner. Then each mechanism structure is sketched and evaluated with respect to the remaining functional requirements. This results in a class of feasible mechanisms that can be subject to dimensional synthesis, kinematic and dynamic analyses, design optimization, and design detailing. This textbook is organized as follows: Chapter 1 provides a brief review of the design process and a systematic methodology for creation of mechanisms. Some terminologies related to the kinematics of mechanism are defined. Mechanisms are classified according to the nature of motion into planar, spherical, and spatial mechanisms. © 2001 by CRC Press LLC Chapter 2 is concerned with the basic concepts of graph theory, which is essential for structural analysis and structural synthesis of mechanisms. This material is extremely important since the design methodology employs graphs to represent the mechanism structure and mechanism structures are enumerated with the aid of graph theory. Chapter 3 introduces several methods of representation of the kinematic structure of mechanisms. The kinematic structure, which contains the essential information about which link is connected to which other links by what types of joint, will be used for enumeration of mechanisms. Chapter 4 examines the structural characteristics of mechanisms. The correspondence between graph and mechanism is established, from which several important mechanism structural characteristics are derived. The degrees of freedom of a mechanism, the loop-mobility criterion, the concept of structural isomorphism, and various methods of identification of structural isomorphism are described. Chapter 5 deals with the enumeration of graphs of kinematic chains. Systematic algorithms for the enumeration of contracted and conventional graphs are presented. Atlases of contracted graphs and conventional graphs are developed. Using these atlases, an enormous number of mechanisms can be developed. Chapter 6 describes a general procedure for the enumeration and classification of mechanisms. Planar bar linkages, geared mechanisms, cam mechanisms, spherical mechanisms, and spatial mechanisms are enumerated and classified according to the number of degrees of freedom, the number of independent loops, etc. Chapter 7 covers the enumeration and classification of epicyclic gear trains (EGTs). The structural characteristics of EGTs are identified. Various methods of enumeration including Buchsbaum and Freudenstein’s method, the genetic graph approach, and the parent bar linkage method are discussed. Furthermore, the theory of fundamental circuits is introduced for the speed-ratio analysis of EGTs. Chapters 8 and 9 offer several conceptual design examples to demonstrate the power of the methodology. Chapter 8 concentrates on the enumeration of automotive mechanisms, whereas Chapter 9 involves the enumeration of robotic mechanisms. Atlases of parallel manipulators and robotic wrist mechanisms are developed. Appendix A presents an algorithm for solving a system of m linear equations in n unknowns. A nested do-loops algorithm serves as the basis for systematic enumeration of mechanisms. Appendix B provides an atlas of contracted graphs having two to four independent loops. Appendix C is comprised of an atlas of graphs of kinematic chains having up to three independent loops and eight links. Appendix D offers an atlas of planar linkages with one, two, and three degrees of freedom. Appendix E contains an atlas of spatial one-dof, single-loop kinematic chains. Appendix F includes an atlas of epicyclic gear trains classified according to the number of degrees of freedom, the number of independent loops, and the vertex degree listing. Appendix G furnishes the schematic diagrams and clutching sequences of some commonly used epicyclic transmission gear trains. Prerequisites for readers of this textbook include the basic concepts of combinatorial analysis, graph theory, matrix theory, and the kinematics of mechanisms that are usually taught at the undergraduate level. Thomas Edison said, “genius is one percent © 2001 by CRC Press LLC inspiration and ninety-nine percent perspiration.” Inspiration can occur more readily when perspiration is properly directed and focused. The methodology presented in this book is intended to help designers better organize the perspiration so that the inspiration can take place early in the design process. For those who are willing to try, the rewards should be well worth it. The author wishes to express his sincere appreciation to Dr. Bernard Roth, his former Ph.D. advisor at Stanford University, and Dr. Ferdinand Freudenstein, Professor Emeritus at Columbia University, for their lifelong advice and encouragement. A major portion of the material presented in this textbook is derived from Dr. Freudenstein and his former students’ research results. Others are taken from the author’s research in collaboration with professional colleagues, Ting Liu and Roland Maki, and with his former students, Sun-Lai Chang, Goutam Chatterjee, Dar-Zen Chen, Hsin-I Hsieh, Chen-Chou Lin, Richard Stamper, and Farhad Tahmasebi. Their efforts are greatly appreciated. Lastly, the author appreciates the patience and sacrifice of his family members, Lung-Chu Tsai, Jule Ann Tsai, and David Jeanchung Tsai, over the past few years while the textbook was being written. Lung-Wen Tsai Riverside, California © 2001 by CRC Press LLC The Author Lung-Wen Tsai is a Presidential Chair Professor in the Department of Mechanical Engineering at the University of California in Riverside. He obtained his B.S. degree in mechanical engineering from the National Taiwan University in Taipei, Taiwan; M.S. degree in engineering science from the State University of New York (SUNY) in Buffalo, New York; and Ph.D. in mechanical engineering from Stanford University in Stanford, California. From 1973 to 1978, Dr. Tsai was a research and development engineer for Hewlett Packard responsible for the design of instrumentation tape recorders and X-Y plotters. From 1978 to 1986 he was a senior staff research engineer for General Motors and led projects in the development of variable-stroke engine mechanisms, variable-valve timing mechanisms, active engine balancing devices, automatic transmission mechanisms, and kinematics of robot manipulators. His most recent position was with the University of Maryland in College Park from 1986 to July 2000 where he established a nationally recognized research and education program in mechanisms and machine theory, automotive engineering, and robot manipulators. Dr. Tsai joined the Department of Mechanical Engineering at the University of California at Riverside in the Fall of 2000. Dr. Tsai is a registered professional engineer in California, a Fellow of the ASME, and a member of the SAE. He is Chief Editor for the ASME Journal of Mechanical Design and Chairman of the 2000 ASME International Design Engineering Technical Conferences and the Computer in Engineering Conference. Dr. Tsai has published one book on robot analysis (Robot Analysis: The Mechanics of Serial and Parallel Manipulators, John Wiley & Sons, New York, 1999) and more than 100 journal and © 2001 by CRC Press LLC conference proceedings papers. He is the recipient of numerous awards, including the 1984 ASME Mechanism Committee best paper award, 1989 and 1991 AMR Procter & Gamble Awards, 1985 ASME Melville Medal, 1986 GM John Campbell Award, 1988 SAE Arch Colwell Merit Award, and 1993 AMR South-Pointing-Chariot Rotating Trophy. © 2001 by CRC Press LLC Contents 1 Introduction 1.1 Introduction 1.2 A Systematic Design Methodology 1.3 Links and Joints 1.4 Kinematic Chains, Mechanisms, and Machines 1.5 Kinematics of Mechanisms 1.6 Planar, Spherical, and Spatial Mechanisms 1.7 Kinematic Inversions 1.8 Summary References 2 Basic Concepts of Graph Theory 2.1 Definitions 2.1.1 Degree of a Vertex 2.1.2 Walks and Circuits 2.1.3 Connected Graphs, Subgraphs, and Components 2.1.4 Articulation Points, Bridges, and Blocks 2.1.5 Parallel Edges, Slings, and Multigraphs 2.1.6 Directed Graph and Rooted Graph 2.1.7 Complete Graph and Bipartite 2.1.8 Graph Isomorphisms 2.2 Tree 2.3 Planar Graph 2.4 Spanning Trees and Fundamental Circuits 2.5 Euler’s Equation 2.6 Topological Characteristics of Planar Graphs 2.7 Matrix Representations of Graph 2.7.1 Adjacency Matrix 2.7.2 Incidence Matrix 2.7.3 Circuit Matrix 2.7.4 Path Matrix 2.8 Contracted Graphs © 2001 by CRC Press LLC