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Volume 1 Mechanism Design Analysis and Synth�1;;:;� 1 ..: .._:.;· �'....·-� . , ... .,::J\::, Fourth Edition ... ;::-\?• ,t f)�,.. - � � Web Enhanced ARTHUR G. ERDMAN Morse Alumni Distinguished Teaching Professor of Mechanical Engineering University of Minnesota GEORGE N. SANDOR Research Professor Emeritus of Mechanical Engineering University of Florida SR/DHAR KOTA Professor of Mechanical Engineering University of Michi�ga�n ��- . •, - - ,P-: 0!J� 1I ..f.. ,\.. ., .: ·-::.,,.;.r:_-<i. .. -.�/,·., ... .-. _\: ..·.1·,:·_ ·...-i r•- J"i-.".4 A. .e.: •., ..I. t· �_ r:. ...••. �/· .. -/.l .... o'""'•- 2.) -•- : l:•. ., "'v',,i., I· · .··.,· · ·· · · · · · · · · .) •.) .• .. ., \ >i: l.:· Prentice Hall Upper Saddle River, New Jersey 07458 Llbrarv 0JC011gress Cataloging·in-Publican·on Data ERDMAN. ARTHL'R G. Mechanism design: analysis a.nd synthesis I Anhur G. Erdman. George N. Sandor. Sridhar K<'ta- p. cm. Includes bibliographical references .:ind index. ISBN 0-13-0.0872-7(,·. I) l. Machine-Design. L Sandor. George N. LL Kota, Sridhar Ill. Tille. CIPDATA AVAILABLE. ClP Vice President and Editorial Director of ECS: MARCL.\ HORTON Acquistions Editor: L..\llRA CURLESS Editorial Assistant: ERL'< KRA TCHMAR Vice President and Director of Production and Manufacruring, ESJ\1: 0..\ VID W. RICCARDI Executive Managing Editor: VINCE o·sRJEN I Managing Editor: OA no A. GEORGE Production Editor: IRWIN ZUCKER Director of Creative Services: PAUL BELf .. \.NTI Creative Director: CAROLE ANSON Cover Design: BRUCE K£NSELAAR An Editor: ADAM VELT HAUS Manufacru.ring Manager: TRUDY PISCIOTTI I Manufacturing Buyer: P.-.T BROWN Marketing Manager: HOLLY STARK An Erdman George Sandor Sridhar Kata and Marketing Assisumt: K..\.REN '.\IOON dedicates this work dedicates this work Art Erdman dedicate o 2001 by Arthur G. Erdman. George N. Sandor. and Sridhar Kola to his wife Mary to his wife Magdi. this work to the. � 1997, I 99 l. l 984 by Arthur G. Erdman and George '<. Sandor Jo, daughters memory of • Published by Prentice-Hall, Inc. Kristy and Kari Professor Upper Saddle River. New Jersey 07458 and son Aaron. Athmaram (Abe) All rights reserved. No pan of this book may be reproduced. in any format or by any means. without permission in writing He thanks the Lord H. SoniJor his from the publisher for blessing him lifelong contributions and enabling him to the engineering The author and publisher of this book have used their best efforts in preparing this book. These efforts include the develop to contribute to com1111111i1y. ment, research. and testing of the theories and programs to determine their effectiveness. . The author and publisher make no this book. warranty of any kind. expressed or implied. with regard 10 these programs or the documemarion contained in this book. The author and publisher ;h31J not be liable in any event for incidental or consequential damages in connection with. or arising our of. the furnishing. performance, or use of these programs. TRADEMARK 11'ff0Rlv1AT ION: ADAMS (Automatic Dynamic Analysis of Mechanical Systems) is 3 trademark of Mechanical Dynamics Inc. DADS a trademark of CADS! Inc. Working Model software a trademark of Know ledge Revolution. Mechanica is a trademark of'Rasna Corp. Printed in the United States of America !098765432 ISBN 0·13·040872-7 Prentice-Hall Jnrernaiional (UK) Limited. London Prentice-Hall of Australia Pty. Limited. Sydney 'Loo Prentice-Hall Canada lnc., Toronto Prentice-Hall Hispanoarncricana, S.A., Mexico Prentice-Hall oflndia Private Limited. New Delhi I l Prcnucc-Hall of Japan, lnc., Tokyo Pearson Education Asia Pie. Ltd., Singapore Editora Prentice-Hall do Brasil. Ltda., Rio de Janeiro I About the Cover Contents Front Cover IN MEMORY ix The cover depicts the computer model of a three-fingered Universal Robotic Gripper that PREFACE xi can grasp objects of any shape. The design was based on a single-input, three-output dif ferential mechanism that allows all three fingers to exert same force regardless of their position. Such single-input, plural-output differential mechanisms were invented by 1 INTRODUCTION TO KINEMATICS S. Kota and S. Bidare (U.S. patents 5,423,726 and 5,435,790). The particular embodiment AND MECHANISMS 1 shown on the cover was developed by Dr. Mary Frecker. Penn State University, as a grad uate student at the University of Michigan in 1994. The computer model was created by 1.1 Inrroduction l Dr. Zhe Li, University of Michigan, using ADAMS software. 1.2 Motion I 1.3 The Four-Bar Linkage 2 Back Cover 1.4 Relative Motion 9 1.5 Kinematic Diagrams 9 Top right: A snapshot of cam synthesis program. called CAMSYN, developed in MAT 1.6 Six-Bar Chains 14 LAU by Dr. Zhe Li and S. Kota 1.7 Degrees of Freedom 21 1.8 Analysis versus Synthesis 39 Middle left: A page from Module I of the companion web site showing ADAMS simula 1.9 Mechanism Design Example: Variable Speed tion of a sheet-metal feeding mechanism, its kinematic diagram and computation of de Transmission 30 grees of freedom. Problems 40 Bottom right: A page from Module IO of the companion web-site showing computer sim ulations of four (among numerous others) different types of mechanical grippers. 2 MECHANISM DESIGN PROCESS 96 2.1 Introduction 96 2.2 The Seven Stages of Computer-Aided Engineering Design 96 2.3 How the Seven Stages Relate to This Text 101 2.4 A Need for Mechanisms I 02 I 2.5 Design Categories and Mechanism Parameters I 07 v 2.6 Troubleshooting Guide: Symptoms. Causes, 5.7 Discussion of the Superposition and Matrix and Sources of Assistance 113 Approach to Kinetosratics 330 2.7 History of Computer-Aided Mechanism 5.8 Time Response to Mechanisms 330 Design I 16 5.9 Dynamic Simulation of Mechanisms 346 Appendix: Commercial Software Programs 354 Problems 358 3 DISPLACEMENT AND VELOCITY ANALYSIS 119 3.1 Displacement Analysis: Useful Indices for Position 6 CAM DESIGN 373 Analysis of Linkages 119 3.2 Displacement Analysis: Graphical Method 13 I 6.1 Introduction 373 3.3 Displacement Analysis: Analytical Method 135 6.2 Cam and Follower Types 374 3.4 Concept of Relative Motion l 37 6.3 Cam Synthesis 378 3.5 Velocity Analysis: Graphical Method 139 6.4 Displacement Diagrams: Graphical 3.6 Velocity Analysis. Analytical Method 149 Development 380 3.7 Instant Centers 152 6.5 Displacement Diagrams: Analytical 3.8 Velocity Analysis Using Instant Centers 160 Development 388 3.9 Mechanical Advantage 165 6.6 Advanced Cam Profile Techniques 394 3.JO Analytical Method for Velocity and Mechanical 6.7 Graphical Cam Profile Synthesis 408 Advantage Determination 176 6.8 Analytical Cam Profile Synthesis 410 3.11 Computer Program for the Kinematic: Analysis 6.9 Cam Synthesis for Remote Follower 425 of a Four-Bar Linkage 181 6.10 Cam-Modulated Linkages 4'.!6 Appendix: Review of Complex Numbers 183 Problems 435 Problems 192 Exercises 232 7 GEARS AND GEAR TRAINS 447 4 ACCELERATION ANALYSIS 233 7.1 lntroducrion 447 7.2 Gear Tooth Nomenclature 452 4.1 Introduction 233 7.3 Forming of Gear Teeth 456 4.2 Acceleration Difference 234 7.4 Gear Trains 458 4.3 Relative Acceleration 239 7.5 Planetary Gear Trains 465 4.4 Coriolis Acceleration 243 7.6 The Formula Method 473 4.5 Mechanisms with Curved Slots and l Iighcr-Pair 7.7 The Tabular Method 480 Connecrions 263 7.8 The Instant Center Method (or Tangential Velocity Problems 268 Method) 484 .. 7.9 Tooth Loads and Power Flow in Branching 5 INTRODUCTION TO DYNAMICS Planetary Gear Systems 490 OF MECHANISMS 291 Problems 498 5.1 Introduction 291 S.2 Inertia Forces in Linkages 296 8 INTRODUCTION TO KINEMATIC SYNTHESIS: 5.3 Kinetostatic Analysis of Mechanisms 299 GRAPHICAL AND LINEAR ANALYTICAL METHODS 514 5.4 The Superposition Method (Graphical and Analytical) 301 8.1 Introduction 514 5.5 Design Example: Analysis of a Variable-Speed 8.2 Tasks of Kinematic Synthesis 516 Drive 309 8.3 Type Synthesis 526 5.6 The Matrix Method 3 l 8 8.4 Tools of Dimensional Synthesis 539 vii vi Contents Contents 8.5 Graphical Synthesis-Motion Generation: Two Prescribed Positions 539 8.6 Graphical Synthesis-Motion Generation: Three Prescribed Positions 542 In Memory 8.7 Graphical Synthesis for Path Generation: Three Prescribed Positions 543 8.8 Path Generation with Prescribed Timing: Three Prescribed Positions 544 8.9 Graphical Synthesis for Path Generation (without Prescribed Timing): Four Positions 546 8.10 Function Generator: Three Precision Points 548 8.11 The Overlay Method 553 8.12 Analytical Synthesis Techniques 554 8.13 Introduction to Analytical Synthesis 555 8.14 The Standard Dyad Form 562 8.15 Number of Prescribed Positions versus Number of Free Choices 566 8.16 Three Prescribed Positions for Motion, Path. and Function Generation 568 8.17 Three-Precision-Point Synthesis Examples 574 We arc all saddened with the passing of Dr. George N. Sandor during the preparati0n of 8.18 Circle-Point and Center-Point Circles 580 the third edition of this book. George was a world renowned professor, engineer, a great 8.19 Ground-Pivot Specification 588 friend and major contributor to the kinematic community. At tbe age of 84 he was a re 8.20 Extension of Three-Precision-Point Synthesis tired Research Professor Emeritus and past Director of the Mechanical Engineering De to Multiloop Mechanisms 591 sign Laboratory at the University of Florida, Gainesville. Dr. Sandor formerly taught at 8.21 Freudenstein's Equation for Three-Point Function Rensselaer Polytechnic Institute and at Yale and Columbia Universities. He was t�e Generation 595 ALCOA Foundation Professor of Mechanisms Design from 1966 to 1975. He worked in 8.22 Loop-Closure-Equation Technique 598 U.S. industry for 21 years before starting his graduate work at Columbia. During th_at 8.23 Order Synthesis: Four-Bar Function time, he made numerous contributions including designing the first color press for life Generation 60 I Magazine. . . . . 8.24 Three-Precision-Point Synthesis: Analytical versus Dr. Sandor received his Doctorate in Engineering Science at Columb1:.i University Graphical 604 in 1959 and, in 1986, was honored with Doctor Honoris Causa in Mechanical Engineer Appendix: Case Study-Type of Synthesis of ing at the Technological University, University of Budapest. Hungary. He had become the Casement Window Mechanisms 604 first mechanical engineer in the previous 19 years to receive this honor. Dr. Sandor was Problems 624 also elected Honorary Member of the Hungarian Academy of Sciences. . Dr. Sandor wrote over 140 technical. scientific and educational papers. He invented he or co-invented six issued patents. In all, advised more than 50 master's and doctor's ANSWERS TO SELECTED PROBLEMS 647 :ork graduates. Dr. Sandor was a Life Fellow of ASME and a member of the New Ac�d erny of Science. I le received numerous honors including the ASME Ma�hme Design REFERENCES 650 Award and the OSU Applicc.l Mechanisms Award. He is one of the Outsrandmg Educators in America and is listed in Who's Who in America and American Men and Women of INDEX 661 Science. Dr. Sandor held many engineering, administrative, executive and board positions in machinery design, manufacture, and research and development. This book has .the benefit of these experiences which include the Hungarian Rubber Co. (affiliated with Dunlop Ltd.), Babcock Printing Press Corp., H.W. Facber Corp., and TTME Inc. Hew� a m�m ber of the Board of Directors at Huck Co., from 1963-70 and held P.E. licenses in Flonda, New York, North Carolina, and New Jersey. ix viii Contents . Dr. Sandor w�s �n avid �ier, _sa!lor, musician, and family poet laureate who spoke sev_en l�oguages. His interest in aV1a1100 spanned over 50 years. While a student at the University �f Polytechnics in Budapest, Hungry, he helped design an open-cockpit, two pas�enger biplane. for an engineering course project. Unlike many student projects. San Preface dor s staggered-wing prototype flew perfectly the first try. . George is well remembered by his kindness to all, his wisdom and unbound curios- 1t?' for the field of kinematics. His contributions 10 the science and application of rnecha msm_s are many and are evident in this book. His enthusiasm for life and research is poss1b!y unmatched. George is now with the Lord, continuing to uncover the secrets be yond hfe. The original two-volume work. consisting of Volume I. :-.rech:mism Design: Analysis and Synthesis. and Volume 2. Advanced Mechanism Design: Analysis and Synthesis. was de veloped over a 15-year period chiefly from the teaching. research, and consulting practice or the authors, with contributions from their working associates and with adapraiions of published papers. This work represented the culmination of research toward a general method of kinematic. dynamic, and synthesis, starting with the dissertation of Dr. G.N. Sandor under the direction of Dr. Freudenstein at Columbia University. The authors acknowledge many colleagues who made contributions to the first edi tion: John Gustafson. Lee Hunt. Tom Carlson. Ray Giese. Bill Dahlof. Sem Heng Wang. Dr. Tom Chase, Dr. Sanjay G. Dhandi, Dr. Patrick Starr. Dr. William Carsen. Dr. Charles F. Reinholtz. Dr. Manuel Hernandez, Manin 01 Girolamo, Xirong Zhuang, and others. The second edition of Volume I was based on feedback that came from over a hun dred institutions in the United States and abroad. including the authors· own universities. Several chapters were reorganized and over 50 new problems and examples were added. Also new to this edition was an 113M disk which supplemented chapters 3,4.6 and 8. Readers were able to design four-bar linkages for three design positions and then analyze the synthesized mechanism. Also a cam design module illustrated the concepts outlined in Chapter 6. The authors acknowledge many colleagues who made contributions to the second edition: Dr. Sridhar Kora, Dr. Tom Chase, John Titus, Dr. Donald Riley, Dr. Alben C. Esterline Dr. Suren Dwivdei, and Dr. Harold Johnson. Other contributors include Chris Huber, Ralph Peterson. Mike Lucas, Jon Thoreson, Elizabeth Logan, Greg Vetter, and Gary Bisrram, for photography. The third edition of Volume I was a result of further improvement to the text. Over 60 new problems and examples were added - taken from industry, from patents or solu tions to practical needs. Several chapters were modified with the objective of simplifying the teaching of the materials. For example, in Chapter 2, a building block approach to mechanism design was added based on input from Dr. Sridhar Kora. In Chapter 7, the xi x In Memory planetary gear train section was improved with the help of Dr. Frank Kelso. A major Making easier to study change to the third edition was the CD-ROM which included more than 90 animation's of Motion of the Linkage Body real and computer-generated mechanisms. How they move in plane and Three Dee The authors thank the following individuals for their contribution to this third edi Makes it clear and learning easy! tion: Dr. Tom Chase, Dr. Jenny Holte, and Prof. Daryl Logan at the University of \Vis consin, Planeville, as well as Dr. Raed Rizq, David Wulfman, Tim Berg, Jim Warren. That's the goal of this one writer Dr. Boyang Hong, James Holroyd, Nick Gamble, Phil Schlanger, and Stephanie Clark. Other author even brighter! We are very pleased to introduce the fourth edition which continues the tradition of So, we wish you happy reading innovative approaches to teaching mechanism design. The CD-ROM has been replaced May your study earn high grading! by a web-accessible set of over 200 mechanism simulations, many of which are full 3-D models created in ADAMS™ (Automated Dynamic Analysis of Dynamic Systems). Dr. Highland, North Carolina, May 9, 1994 Sridhar Kora, who has been a significant contributor to previous editions of this book, has George N. Sandor been brought on as a coauthor. He and Dr. Zhe Li at the University of Michigan have gen erated all of the new Web-page material, available at hnp://www.prenhall.com/erdman. A large number of the mechanisms in the book are now fully modeled and ani mated. Thus, students may actually see kinematic and dynamic motions rather than at tempt to envision movement. ln addition, ADAi\1S models of selected problems will be available on the web. In some cases students can modify design parameters in order to test systems response. There are many helpful tutorials and case studies on the Web page which allows the instructor to teach a course in mechanism design almost entirely from the web connection, including homework assignments. Chapters 5 and 6 have been revised to reflect the web-enhanced fourth edition, A compilation of student design projects will be regularly updated on the web site. Several new design examples of type synthesis and applications of symmetrical coupler curves. cognates, and paralJel motion mechanisms are included on the web ..A n extensive compi lation of simulations of robotic grippers is also included. A new general purpose CAM design module has been added and new material on type synthesis. path curvature. and ro botic grippers are on die Web site. t The authors wish to thank Dr. Ycsb Singh from UTSA and Dr. John Lenox of De f sign Excellence, Inc. for their helpful input to this new addition. The authors thank Alyssa i Burger for her help with the manuscript. As before, the authors acknow ledge numerous i students and colleagues from within and external to their universities for continued feed back. encouragement, and influence that helped generate this book. f j I Anhur G. Erdman George N. Sandor Sridhar Kota i l This book deals with Kinematics 1 Synthetics and Analytics i Written with love of the Science Keeping in mind Srudent Clients! \ t xii Preface Sec. 1.1 Introduction xiii 1 Introduction to Kinematics and Mechanisms 1. 1 INTRODUCTION Engineering is based on the fundamental sciences of mathematics, physics, and chemistry: In most cases. engineering involves the analysis of the conversion cf energy from some source to one or more outputs, using one or more of the basic principles of these sciences. Solid mechanics is one of the branches of physics which. among others. comains three major subbranches: kinematics, which deals with the study of relative motion: statics. which is the study of forces and moments. apart from motion: and kinetics. which deal� with the action of forces on bodies. The combination of kinematics and kinetics is re ferrcd 10 as dynamics. This text describes the appropriate mathematics, kinematics, ar.d dynamics required to accomplish mechanism design. A mechanism is a mechanical device that has t.i� purpose of transferring motitJon and/or force from a source to an ourput. A lmku:;c consists of links tor bars) (see Tabk l. l ), generally considered rigid, which arc connected by joints (Sec Table 1.2). such a.'£ pins (or rcvolutes), or prismatic joints, to form open or closed chains (or loops). Sud'.l kinematic chains. with at least one link fixed. become (I) mechanisms if at least two oiher links retain mobility. or (2) structures if no mobility remains. In other words, a mccha nism permits relative motion between its "rigid" links; a structure does not. Since linkages make simple mechanisms and can be designed to perform complex tasks, such as nonlirs car motion and force transmission, thev will receive much attention in this book. Some 01f the linkage design techniques presented here arc the result of a resurgence in the theory ovf mechanisms based on the availability of the computer. Many of the design methods wer..e discovered before the 1960s, but long, cumbersome calculation discouraged any further development at that time. MOTION A large majority of mechanisms exhibit motion such that aJI the links move in paralleil planes. This text emphasizes this type of motion, which is called two-dimensional, plane; or planar motion. . Planar rigid-body motion consists of rotation about axes perpendiculan' 11 to the plane of motion and translation=-wuere all points in the body move along parallel p Path Tracer Point str�1ght _o� plan": cun:itinear paths and all lines embedded in the body remain parallel to --- their original orientation. Spatial mechanisms, introduced in Chap. 6 of Vol. 2, allow movement in thr�e di_mensions. Co�binations of rotation around up to three nonparallel CCouurpvlee- r ,( / " axe� an� translations 111 .u� to three different directions are possible depending on the con stramts imposed by the Joints between links (spherical, helical, cylindrical, etc.: see Table 6.1, Vol. 2). B ln these discussions, all links are assumed lo be rigid bodies. In the second volume (Chap. 5) of this text, this rigid-body assumption is relaxed. and it is assumed that the links have elastic properties. But for now, lei us retain our rigid-body assumption for mechanism links. 1.3 THE FOUR-BAR LINKAGE Mech?nisms _are used in a great variety of machines and devices. The simplest closed loop linkage is the four-bar, which has three moving links (plus one fixed link)" and four "revolore," "pivoted." or "pin" joints (sec Fig. I. I a). The link that is connected to the Figure I.la Four-bar linkage notation. power s�urce _or prime mover is called the i11p11t link (A01l). The follower link connects the movmg pivot B 10 ground pivot 80. The coupler or floating link connects the two tracer point about Q m long. Since there is a hook at the path tracer point that holds a wire moving pivots. A and B, thereby "coupling" the input link to the output link. Points on the rope (which will always hang vertically). the orientation of the coupler link is not impor coupler li_nk (called path tracer points) generally trace out sixth-order algebraic coupler tant. Thus. this is clearly a path generation task. c_urvcs. figure I. I b is taken from [89)t. in which very different coupler curves (dashed Figure 1.2b is a drive linkage for a lawn sprinkler. which is adjustable to obtain dif lines) can be generated by using different path tracer points (the small solid circles). ferent ranges of oscillation of the sprinkler head. This adjustable linkage can be used to The four-bar linkage is the most basic chain of pin-connected links that allows rcla vary the angle of rotation of the sprinkler head by using the clamping screw to change the ti.ve motion be:wecn the links. (Three links pinned together is a structure.) Although a point of attachment of the coupler and follower links. The rcl�tive rotati?ns be_twcen the simple mechan ism, the four-bar is very versatile and is used in thousands of applications. input and follower links of this mechanism accomplish the desired task ot function gener The examples shown in Figs. I .2 through 1.6 illustrate a wide range of uses for the four ation. bar. Even though these applications arc quite different, the linkages shown in the exam Fiuure l.2c shows a four-bar automobile hood linkage design. The linkage controls ples (as well as all mechanisms) can be classified into three categories depending on the the relative orientation between the hood and the car frame. The hood must not interfere t�sk that _th.e linkage �crforms: ji.111c1io11 generation. path generation, and motion genera with the frame of the car as it opens and must tit flush into the cavity in the car in the tton (or rigid-body guidance). A.function generator (Figs. I .2b, I .4a, and 1.5) is a linkaue in which the relative motion (or forces) between links connected to ground is of interest. . pIn� tfhu ngcetnioenr agteionne r(aFtiiogns., t1h.2ea t aasnkd dtohees fnooutr -rbeqaru iproe rati opna tho ftr Faicge.r 1p.o3i)n, tw oen athree ccoonucpelernr elidn ko.n lIyn ,/ , ,,,,,. ... --, .. , \\ ,; , - ... ' w1th_ the p�th of a tracer point and not with the rotation of the coupler link. In morion gen ' '' I'I ',' I era11011 (�1gs. 1.2c an� I .6). the entire motion of the coupler link is of concern: the path Il A / ::�------""' tracer point x, y coordinates, and the angular orientation of the coupler link. These tasks I ', are also discussed in Chaps. 2 and 8. \ __ -, ,..,,,..,,,. Figure 1.2 shows a different four-bar that has been used to accomplish each task. The level luffing crane of Fig. l .2a is a special type of four-bar that generates approximate Bo straight-line morion of the path tracer point (point P). Cranes of this type can be rated at A0A = 1 AB = 2 B0B = 3 A0B0 = 3 AoA = 1 AB= 3 B0B = 2 Ao Bo= 3 50 tons capacity and typically have an approximate straight-line travel of the coupler Figure l.lb Sample pages from the atlas of four-bar coupler curves by Hrones and Nel son (89). Jn [89). lengths of dashes of the curves indicate I 0° increments of crank rota • A linkage with one link fixed is a mechanism. tions. Here the lengths of dashes are not to scale. Solid circles are different path tracer tNumbcrs in square brackets pertain to References at the end of this book. points. 2 Introduction to Kinematics and Mechanisms Chap. 1 Sec. 1.3 The Four-Bar Linkage 3 Input Link (al (b) � Sucker Rod (a) lb) (c) f'igure 1.2 Demonstration of four-bar tasks. !cl ldl closed position. The x. y locations of a path tracer point on the end of the hood as well as figure 1.3 (a and b) A scale model of the "Minnesota" oil pump. These fi�res sho,� the m��ha the angle of the hood with respect to the car are critical. Thus this a case of motion gener ism near the limits of the straight-line portion of the path tracer point. (e) An imermediare posiuon ation. :� the four-bar ponion produced by the Lineages� sofiware. (d) The ro.ur·bar along wit_h the driving Figure 1.3 shows another example of a four-bar mechanism generating an approxi I1\0\, O th· eli1 n cko ucphlaeirn li"ndky a4d. "T (hleinske sa 5ll oawnd a d6j)u. sNtmoteicnet othfa tth eth setrreo kaer er asnevgeer. atlh cepreuboyn cs bf aonrg ci· onngn teh ce t iIn·cg n getht his o dfy tahde mate straight-line path. In this case, the objective is to replace the standard "horse head" straight-line of the coupler curve.© University of Minnesota, type of oil pumping mechanism shown in Fig. 1.4 with a design in which a cam (horse ohbeajedc)t iivse nso btu rte aqruei rcelda.s sTifhieed f obuyr -dbifafre rmeenct htaasnkissm. Tsh seh ostwannd ainr dt hAemsee rtwicoa nf iPgeutrreosle huamv eI nssimtitiulater *�J It·.J'! " • �icon in margin indicates reference 10 CD·ROM m• back of book. 4 Introduction to Kinematics and Mechanisms Chap. 1 Sec. 1.3 The Four-Bar Linkage 5