Sandor, Arthur G. Erdman-Advanced Mechanism Design_ Analysis and Synthesis Vol. II (1984)

LibraryofCongress Cataloging in Publication Data(Revised for vol. 2)

ERDMAN . ARTH UR G . (da te)

Mechanism design.

Vol. 2 has title: Advanced mechanism design:analysis and synthesis.

Auth ors' names in reverse order in v. 2.Includes bibliographi es and indexes.I. Machinery-Design. 2. Machinery , Dynam ics of.

3. Machinery, Kinematics of. I. Sandor , George N.II. Title. Ill. Title: Advanced mechanism design.TJ230.E67 1984 621.8 ' 15 83-3148ISBN (}'13-572396-5 (v. 1)ISBN ().13-0I 1437-5 (v. 2)

1-

Editorial/production supervisionand interior design: Karen Skrable

Manufa cturing buyer: Anthony CarusoCover design: Photo Plus Art, Celine Brandes

@ 1984 by George N. Sandor and Arthur G. Erdman

Volume 1 published under the title Mechanism Design:Analysis and Synthes is, Vol. 1.

All rights reserved. No part of this bookmay be reproduced in any form orby any means without permission in writingfrom the publisher.

Printed in the United States of America

10 9 8 7 6 5 4 3 2

ISBN 0-13-011437-5 01

PRENTICE-HALL INTERNATIONAL, INC., LondonPRENTICE-HALL OF AUSTRALIA PTY. LIMITED, SydneyEDITORA PRENTICE-HALL DO BRASIL, LTDA., Rio de JaneiroPRENTICE-HALL CANADA INC., TorontoPRENTICE-HALL OF INDIA PRIVATE LIMITED, New DelhiPRENTICE-HALL OF JAPAN, INC., TokyoPRENTICE-HALL OF SOUTHEAST ASIA PTE. LTD., SingaporeWHITEHALL BOOKS LIMITED , Wellington, New Zealand

George Sandordedicates this work

to his wifeMagdi

Art Erdmandedicates this work

to his daughtersKristy and Kari

-

i Contents

I

PREFACE ix

1

2

INTRODUCTION TO KINEMATICS AND MECHANISMS

1.1 Introduction 11.2 Motion 21.3 The Four-Bar Linkage 21.4 The Science of Relative Motion 51.5 Kinematic Diagrams 51.6 Six-Bar Chains 101.7 Degrees of Freedom 161.8 Analysis Versus Synthesis 24

Problems 25

INTRODUCTION TO KINEMATIC SYNTHESIS:GRAPHICAL AND LINEAR ANALYTICAL METHODS

1

49

2.1 Introduction 492.2 Tasks of Kinematic Synthesis 522.3 Number Synthesis : The Associated Linkage Concept 642.4 Tools of Dimensional Synthesis 752.5 Graphical Synthesis - Motion Generation: Two Prescribed Positions 762.6 Graphical Synthesis - Motion Generation: Three Prescribed Positions 782.7 Graphical Synthesis for Path Generation: Three Prescribed Positions 79

v

vi Contents

2.8 Path Generation with Prescribed Timing : Three Prescribed Positions 812.9 Graphical Synthesis for Path Generation (without Prescribed Timing) :

Four Positions 822.10 Function Generator: Three Precision Points 852.11 The Overlay Method 872.12 Analytical Synthesis Techniques 892.13 Complex Number Modeling in Kinematic Synthesis 902.14 The Dyad or Standard Form 922.15 Number of Prescribed Positions versus Number of Free Choices 942.16 Three Prescribed Positions for Motion, Path,

and Function Generation 972.17 Three-Precision-Point Synthesis Program for Four-Bar Linkages 1032.18 Three-Precision-Point Synthesis: Analytical versus Graphical 1102.19 Extension of Three-Precision-Point Synthesis to Multiloop Mechanisms 1122.20 Circle-Point and Center-Point Circles 1142.21 Ground-Pivot Specification 1222.22 Freudenstein's Equation for Three-Point Function Generation 1272.23 Loop-Closure-Equation Technique 1302.24 Order Synthesis: Four-Bar Function Generation 133

Appendix: Case Study - Type Synthesis of Casement WindowMechanisms 136Problems 157

KINEMATIC SYNTHESIS OF LINKAGES: ADVANCED TOPICS 177

3.1 Introduction 1773.2 Motion Generation with Four Prescribed Positions 1773.3 Solution Procedure for Four Prescribed Positions 1803.4 Computer Program for Four Prescribed Precision Points 1843.5 Four Prescribed Motion-Generation Positions :

Superposition of Two Three-Precision-Point Cases 1883.6 Special Cases of Four-Position Synthesis 1913.7 Motion Generation: Five Positions 1993.8 Solution Procedure for Five Prescribed Positions 2023.9 Extensions of Burmester Point Theory: Path Generation

with Prescribed Timing and Function Generation 2043.10 Further Extension of Burmester Theory 2113.11 Synthesis of Multiloop Linkages 2163.12 Applications of Dual-Purpose Multiloop Mechanisms 2183.13 Kinematic Synthesis of Geared Linkages 2303.14 Discussion of Multiply separated Position Synthesis 239

Appendix A3.1: The LINCAGES Package 251Appendix A3.2 261Problems 263

6

viii

5.17 Balancing - Appendix A: The Physical Pendulum 4725.18 Balancing - Appendix B: The Effect of

Counterweight Configuration on Balance 4825.19 Analysis of High-Speed Elastic Mechanisms 4835.20 Elastic Beam Element in Plane Motion 4865.21 Displacement Fields for Beam Element 4825.22 Element Mass and Stiffness Matrices 4905.23 System Mass and Stiffness Matrices 4935.24 Elastic Linkage Model 4955.25 Construction of Total System Matrices 4975.26 Equations of Motion 5015.27 Damping in Linkages 5025.28 Rigid-Body Acceleration 5055.29 Stress Computation 5055.30 Method of Solution 506

Problems 530

SPATIAL MECHANISMS-WITH AN INTRODUCTION TOROBOTICS

Contents

543

6.1 Introduction 5436.2 Transformations Describing Planar Finite Displacements 5526.3 Planar Finite Transformations 5526.4 Identity Transformation 5556.5 Planar Matrix Operator for Finite Rotation 5556.6 Homogeneous Coordinates and Finite Planar Translation 5566.7 Concatenation of Finite Displacements 5586.8 Rotation about an Axis not through the Origin 5616.9 Rigid-Body Transformations 563

6.10 Spatial Transformations 5646.11 Analysis of Spatial Mechanisms 5846.12 Link and Joint Modeling with Elementary Matrices 5906.13 Kinematic Analysis of an Industrial Robot 6036.14 Position Analysis 6196.15 Velocity Analysis 6236.16 Acceleration Analysis 6246.17 Point Kinematics in Three-Dimensional Space 6276.18 Example: Kinematic Analysis of a Three-Dimensional Mechanism 6306.19 Vector Synthesis of Spatial Mechanisms 635

Problems 641Exercises 653

REFERENCESINDEX

666685

4

5

Contents

CURVATURE THEORY

4.1 Introduction 3014.2 Fixed and Moving Centrodes 3014.3 Velocities 3054.4 Accelerations 3134.5 Inflection Points and the Inflection Circle 3154.6 The Euler-Savary Equation 3204.7 Bobillier's Constructions 3274.8 The Collineation Axis 3304.9 Bobillier's Theorem 331

4.10 Hartmann's Construction 3324.11 The Bresse Circle 3364.12 The Acceleration Field 3384.13 The Return Circle 3404.14 Cusp Points 3424.15 Crunode Points 3434.16 The p Curve 3434.17 The Cubic of Stationary Curvature or Burmester's

Circle-Point and Center-Point Curves for FourInfinitesimally Close Positions of the Moving Plane 345

4.18 The Circle-Point Curve and Center-Point Curve for Four ISPs 3524.19 Ball's Point 354

Problems 356Exercises 362

DYNAMICS OF MECHANISMS: ADVANCED CONCEPTS

5.1 Introduction 3665.2 Review of Kinetostatics Using the Matrix Method 3675.3 Time Response 3775.4 Modification of the Time Response of Mechanisms 3875.5 Virtual Work 3905.6 Lagrange Equations of Motion 3965.7 Free Vibration of Systems with One Degree of Freedom 4115.8 Decay of Free Vibrations 4155.9 Forced Vibrations of Systems with One Degree of Freedom 418

5.10 Rotor Balancing 4285.11 Introduction to Force and Moment Balancing of Linkages 4355.12 Optimization of Shaking Moments 4365.13 Shaking Moment Balancing 4465.14 Effect of Moment Balance on Input Torque 4635.15 Other Techniques for Balancing Linkages 4695.16 Computer Program for Force and Moment Balance 472

vii

301

366

i Preface

I

~(

This two-volume work, consisting of Volume 1, Mechanism Design: Analysis andSynthesis. and Volume 2, Advanced Mechanism Design: Analysis and Synthesis. wasdeveloped over a IS-year period chiefly from the teaching, research, and consultingpractice of the authors, with contributions from their working associates and withadaptations of published papers. The authors represent a combination of over 30years of teaching experience in mechanism design and collectively have renderedconsulting services to over 3S companies in design and analysis of mechanical systems.

The work represents the culmination of research toward a general method ofkinematic, dynamic, and kineto-elastodynamic analysis and synthesis, starting withthe dissertation of Dr. Sandor under the direction of Dr. Freudenstein at ColumbiaUniversity, and continuing through a succession of well over 100 publications.The authors' purpose was to present texts that are timely, computer-oriented, andteachable, with numerous worked-out examples and end-of-chapter problems.

The topics covered in these two textbooks were selected with the objectives ofproviding the student, on one hand, with sufficient theoretical background to understand contemporary mechanism design techniques and, on the other hand, of developing skills for applying these theories in practice. Further objectives were for thebooks to serve as a reference for the practicing designer and as a source-work forthe researcher. To this end, the treatment features a computer-aided approach tomechanism design (CAD). Useful and informative graphically based techniques arecombined with computer-assisted methods, including applications of interactive graphics, which provide the student and the practitioner with powerful mechanism designtools. In this manner, the authors attempted to make all contemporary kinematicanalysis and synthesis readily available for the student as well as for the busy practicing

ix

x Preface

designer, without the need for going through the large number of pertinent papersand articles and digesting their contents .

Many actual design examples and case studies from industry are included inthe books. These illustrate the usefulness of the complex-number method, as wellas other techniques of linkage analysis and synthesis. In addition, there are numerousend-of-chapter problems throughout both volumes: over 250 multipart problems ineach volume, representing a mix of SI and English units .

The authors assumed only a basic knowledge of mathematics and mechanicson the part of the student. Thus Volume I in its entirety can serve as a first-leveltext for a comprehensive one- or two-semester undergraduate course (sequence) inKinematic Analysis and Synthesis of Mechanisms. For example, a one-semester,self-contained course of the subject can be fashioned by omitting Chapters 6 and 7(cams and gears). Volume 2 contains material for a one- or two-semester graduatecourse. Selected chapters can be used for specialized one-quarter or one-semestercourses. For example, Chapter 5, Dynamics of Mechanisms-Advanced Concepts,with the use of parts from Chapters 2 and 3, provides material for a course thatcovers kinetostatics, time response, vibration, balancing, and kineto-elastodynamicsof linkage mechanisms, including rigid-rotor balancing.

The foregoing are a few examples of how the books can be used. However,due to the self-contained character of most of the chapters, the instructor may useother chapters or their combinations for specialized purposes . Copious referencelists at the end of each book serve as helpful sources for further study and researchby interested readers. Each volume is a separate entity, usable without referenceto the other, because Chapters I and 8 of Volume I are repeated as Chapters Iand 2 of Volume 2.

The contents of each volume may be briefly described as follows. In VolumeI, the first chapter, Introduction to Kinematics and Mechanisms, is a general overviewof the fundamentals of mechanism design. Chapter 2, Mechanism Design Philosophy,covers design methodology and serves as a guide for selecting the particular chapter(s)of these books to deal with specific tasks and problems arising in the design of mechanisms or in their actual operation. Chapter 3, Displacement and Velocity Analysis,discusses both graphical and analytical methods for finding absolute and relativevelocities, joint forces, and mechanical advantage; it contains all the necessary information for the development of a complex-number-based computer program for the analysis of four-bar linkages adaptable to various types of computers. Chapter 4, Acceleration Analysis, deals with graphical and analytical methods for determining accelerationdifferences, relative accelerations and Coriolis accelerations; it explains velocity equivalence of planar mechanisms, illustrating the concept with examples. Chapter 5 introduces dynamic and kinetostatic analyses with various methods and emphasizes freebody diagrams of mechanism links. Chapter 6 presents design methods for bothsimple cam-and-follower systems as well as for cam-modulated linkages. Chapter7 acquaints the student with involute gears and gear trains, including the velocityratio, as well as force and power-flow analysis of planetary gear trains. The closingchapter of Volume I, Chapter 8, is an introduction to dimensional synthesis of planar

Preface xi

mechanisms using both graphical and closed-form linear analytical methods basedon a "standard-form" complex -number approach. It treats the synthesis of singleand multiloop mechanisms as function, path, and motion generators, with first- andhigher-order approximations.

Volume 2 starts with the same introductory chapter, Chapter I, as the firstvolume. Chapter 2 of Volume 2 is the same as Chapter 8 of Volume I. Chapter3 extends planar analytical kinematic synthesis to greater than three-condition precision, accomplished by way of closed-form nonlinear methods, including Burmestertheory, and describes a computer package, "LINCAGES," to take care of the computational burden. Cycloidal-crank and geared linkages are also included. Chapter 4presents a new computer-oriented, complex-number approach to planar path-curvaturetheory with new explicit forms of the Euler-Savary Equation (ESE) and describesall varieties of BobiIIier's Construction (BC), demonstrating the equivalence of theESE and BC methods. Chapter 5 is a comprehensive treatment of the dynamics ofmechanisms. It covers matrix methods, the Lagrangian approach, free and dampedvibrations, vibration isolation, rigid-rotor balancing, and linkage balancing for shakingforces and shaking-moments, all with reference to computer programs. Also coveredis an introduction to kineto-e1astodynamics (KED), the study of high-speed mechanisms in which the customary rigid-link assumption must be relaxed to account forstresses and strains in elastic links due to inertial forces. Rigid-body kinematicsand dynamics are combined with elastic finite-element techniques to help solve thiscomplex problem. The final chapter of Volume 2, Chapter 6, covers displacement,velocity, and acceleration analysis of three-dimensional spatial mechanisms, includingrobot manipulators, using matrix methods. It contains an easily teachable, visualizabletreatment of Euler-angle rotations. The chapter, and with it the book, closes withan introduction to some of the tools and their applications of spatial kinematic synthesis, illustrated by examples .

In view of the ABET accreditation requirements for increasing the design contentof the mechanical engineering curriculum, these books provide an excellent vehiclefor studying mechanisms from the design perspective. These books also fit in withthe emphasis in engineering curricula placed on CAD/CAM and computer-aidedengineering (CAE). Many computer programs are either included in the texts asflow charts with example input-output listings or are available through the authors.

The complex-number approach in this book is used as the basis for interactivecomputer programs that utilize graphical output and CRT display terminals. Thedesigner , without the need for studying the underlying theory, can interface withthe computer on a graphics screen and explore literally thousands of possible alternatives in search of an optimal solution to a design problem. Thus, while the burdenof computation is delegated to the computer, the designer remains in the " loop" ateach stage where decisions based on human judgment need to be made.

The authors wish to express their appreciation to the many colleagues andstudents, too many to name individually, who have made valuable contributions duringthe development of this work by way of critiques, suggestions, working out and/orchecking of examples, and providing first drafts for some of the sections. Among

xii Preface

the latter are Dr. Ashok Midha (prepared KED section) , Dianne Rekow (Balancingsection), Dr. Robert Williams (Spatial Mechanisms), and Dr. Donald R. Riley, whotaught from the preliminary versions of the texts and offered numerous suggestionsfor improvements. Others making significant contributions are John Gustafson, LeeHunt, Tom Carlson, Ray Giese, Bill Dahlof, Tom Chase, Sern Hong Wang, Dr.Sanjay G. Dhande, Dr. Patrick Starr, Dr. William Carson, Dr. Charles F. Reinholtz,Dr. Manuel Hernandez, Martin Di Girolamo, Xirong Zhuang, Shang-pei Yang, andothers.

Acknowledgment is also due to the Mechanical Systems Program, Civil andMechanical Engineering Division, National Science Foundation, for sponsoring Research Grant No. MEA-80258l2 at the University of Florida, under which parts ofthe curvature chapter were conceived and which led to the publication of severaljournal articles. Sources of illustrations and case studies are acknowledged in thetext and in captions. Other sponsors are acknowledged in many of the authors'journal papers (listed among the references), from which material was adapted forthis work.

The authors and their collaborators continue to develop new material towardpossible inclusion in future editions. To this end, they will appreciate commentsand suggestions from the readers and users of these texts.

George N. SandorArthur G. Erdman

e • •

I~inematic Synthesis

of Linl",ages:

Advanced Topics

•II•I•I•

-I"3 -_..._.1._._._. .... _._-I.,

3.1 INTRODUCTION

In Chap. 2, a variety of approaches to the synthesis of linkages were introduced.Motion, path, and function generation for three prescribed positions was emphasizedbecause that number of precision points corresponds to the maximum number ofprescribed positions for which a four-bar linkage may be synthesized by linear methodsusing the standard dyad form (see Table 2.1). This chapter expands on the complexnumber technique and the standard dyad form introduced in Chap. 2 to investigatethe four- and five-precision-point cases. Other methods of kinematic synthesis forthese cases will not be pursued here since most planar linkages may be designedwith the standard-form algorithm. It will be shown that in the four-precision-pointcase, the infinity of solutions (Table 2.1) may be surveyed at a glance by viewingcomputer graphics routines designed to take advantage of the capabilities of interactivedisplays.

3.2 FOUR PRESCRIBED POSITIONS: MOTION GENERATION

Figures 2.35 to 2.38 presented a geometric interpretation of synthesizing dyads forthe two- and three-prescribed-position coplanar motion-generation case. Figure 3.laand b show a moving plane sr in two and three positions . The notation in thischapter corresponds to the kinematics literature: ground pivots are designated by m(for the German "Mittelpunkt" meaning "center point"), while moving pivots ofground-pivoted binary links are signified as k (for "Kreispunkt" meaning "circlepoint").

177

178 Kinematic Synthesis of Linkages: Advanced Topics Chap. 3

(a) (b)

(c)

Figure 3.1 (a) two coplanar prescribed positions of moving plane 1T; (b) three prescribed positions; (c)four prescribed positions: perpendicular bisectors (k~. and k~.) constructed for the second and four positionsdo not pass through ground pivots m' and m 2 obtained from first three positions.

Recall that for two prescribed positions, there are three infinities of k and mpoint pairs, since k i (the moving pivot in its first position) may be located anywherein the first position of plane tr , and m anywhere along the perpendicular bisectorof k i and k 2, the first and second corresponding positions of k. For example, inFig. 3.1a, two of these k-m point pairs are shown: k~mi and kim2. For threepositions (Fig. 3.lb), the location of k~ represents two infinities of choices, but theintersection of perpendicular bisectors of k~k~ and k~k; yield only one center point,mi.

Figure 3.lc shows the three previously prescribed positions of a plane ('Trio'Tr2, and 'Tr3) plus an additional position, tt4. Also shown are the two ground pivots

Sec. 3.2 Four Prescribed Positions: Motion Generation 179

(m l and m 2) corresponding to the moving pivots (k l and k 2) in the first three prescribed positions. Perpendicular bisectors k~4 and k~4 of k~, k~ and k~, k~ arealso shown. Notice that these perpendicular bisectors do not pass through m 1 andm». This means that neither k l nor k 2 are acceptable moving pivots for these fourpositions. But Table 2.1 indicates an infinite number of solutions in general-isthis a paradox? No. The question should read : Are there any points k in body 71'

whose corresponding positions lie on a circle of the fixed plane for the four arbitrarilyprescribed positions of 71'? This same question was asked (and answered in the affirmative) by Burmester (1876). The following Burmester theory development by meansof complex numbers will parallel what he discovered by geometric means.

The standard-form dyad expression (see Figs. 2.56 and 2.57) will be derivedhere again. Figure 3.2 shows a moving plane 71' in two prescribed positions, 71'1

and 71'j. Positions of the plane are defined by locations of the embedded tracerpoint P and of a directed line segment Pa, also embedded in the moving plane.Positions P, and Pj may be located with respect to an arbitrary fixed coordinatesystem by R, and Rj respectively. A path displacement vector i)j = Rj - R, locatesPj with respect to Pl' The rotation of the plane from position 1 to j is equal tothe rotation of the directed line segment Pa, signified as aj .

Let point k, (embedded in the moving plane) be the unknown location of apossible circle point and let point m be the corresponding unknown center point(embedded in the fixed plane). Since both k and P are embedded in the movingplane, an unknown vector Z embedded in 71'h may be drawn from k l to Pl' Also,we may locate the circle point k l with respect to the center point m by anotherunknown vector W. Thus, as plane 71' moves from 71'1 to 71'j, vector W rotates bythe unknown angle {3j about m, while Psa, rotates to Pjaj by the angle aj.

m

Tracer PointPath

Figure 3.2 The unknown dyad W,Z,which can guide the moving plane 1T

from the first to the jth position.Points m and k, are an unknown Burmester Point Pair.


Notice that the vectors defined above form a closed loop, including the firstand jth positions:

Weif3j+Zeiaj-8j -z-w=oCombining terms, we obtain

(3.1)

(3.2)

Note that this equation is the "standard form" [see Eq. (2.16)], since 8j and aj areknown from the prescribed positions of 7T. For four positions, there will be threeequations like Eq. (3.2), (j = 2, 3, 4):

W(eif3 2- 1)+ Z(eia2- 1) = 82

W(eif33- 1)+ Z(eia3- 1) = 83 (3.3)

W(eif34- 1)+ Z(eia4 - 1) = 84

Recall from Table 2.1 that, for four prescribed positions, one free choice must bemade in order to balance the number of equations and the number of unknowns.If one of the rotations of link W is chosen, say 132, then the system must be solvedfor six unknown reals: Z and Wand the angles 133 and 134' Equations (3.3) arenonlinear (transcendental) in 133 and 134.

3.3 SOLUTION PROCEDURE FOR FOUR PRESCRIBED POSITIONS

Let us for a minute consider Eqs. (3.3) to be a set of three complex equations linearand non-homogeneous in the two complex unknowns Z and W. In order for thisset of three equations to have simultaneous solutions-for Z and W, one of the complexequations must be linearly dependent on the other two; that is, the coefficients ofthe equations must satisfy certain "compatibility" relations. Satisfaction of theserelations will lead to the solution of the equations above.

Equation (3.3) may be expressed in matrix form as

(3.4)

The second column of the coefficient matrix on the left side of the equation as wellas the right side of the equation contain prescribed input data, while the first columncontains unknown rotations 133 and 134' This system can have a solution only ifthe rank" of the "augmented matrix" of the coefficients is 2. The augmented matrixM is formed by adding the right-hand column of system (3.4) to the coefficientmatrix of the left side. Thus it is necessary that the determinant of the augmentedmatrix of this system be equal to zero:

• A matrix has rank r if at least one of its (r X r)-order square minors is nonzero, while all[(r + I) X (r + I)] and higher-order minors are zero.

Sec. 3.3 Solution Procedure for Four Prescribed Postions 181

[

e i f32 - 1Oet M=Oet e i f33 - 1

e i f34 - 1

e i a2 - 1e ia3 - 1e i a4 - 1

(3.5)

Equation (3.5) is a complex equation (containing two independent scalar equations)and thus may be solved for the two scalar unknowns, {33 and {34' Observing thatthe unknowns appear in the first column of matrix M, the determinant is expandedabout this column:

where

a l = -a2 - a 3 - a 4

and the a j, j = 2, 3, 4, are the cofactors of the elements in the first column:

(3.6)

(3.7)

Iei a3

- 1 83

1a 2=

e i a4 - 1 84

Iei a2

- I 82

1a 3 = - . (3.8)

e,a4 - 1 84

Ieia 2

- 1 82

1a 4=

e i a3 - 1 83

The a's are known, since they contain only known input data. Equation (3.6) istermed a compatibility equation, because sets of {32' {33' and {34' which satisfy thisequation, will render system (3.3) "compatible." This means that the system willyield simultaneous solutions for Wand Z.

In the compatibility equation, the unknowns are located in the exponents ofexponentials. This transcendental equation can be simplified through a graphicalsolution procedure, adaptable for computer programming, as shown in Table 3.1(see also Fig. 3.3).

Equation (3.6) can be further simplified in notation as follows:"

(3.9)

Then, for an arbitrary choice of {32, -a as well as a 3 and a 4 are known and canbe drawn to scale as illustrated in Fig. 3.3. Notice that in Eq. (3.9) a 3 is multipliedby e i f33, which is a rotation operator. This also holds for a 4 and e i f34• Equation(3.9) tells us that when a 3 is rotated by {33 and a 4 is rotated {34' these two vectorsform a closed loop with a.

Equation (3.6) may be regarded as the "equation of closure" of a four-barlinkage, the so-called "compatibility linkage," with "fixed link" a h "movable links"aj, j = 2, 3, 4, and "link rotations" {3j, measured from the "starting position" of

• Note that this equation is the same form as derived for the "ground-pivot specification" techniquefor three precision points [Eq. (2.57)].


where Ii.} = Iii.}I and Ii. = IIi.I

TABLE 3.1 ANALYTICAL SOLUTION OF COMPATIBILITY EQUATION EQ. (3.6) FOR p} ,j = 3, 4, BASED ON GEOMETRIC CONSTRUCTION.

Computation of P3. P••lJ3' and lJ. for a given value of P. (see Fig. 3.3)

Ii. = Ii., + li..el{J.

Ii.' -Ii.' -Ii.'cos 8 = • 3

3 21i.31i.

sin 83= 1(1 - cos" 83)1/ " 1 ~ 0

Let cos 83 = x, sin 83 = Y > 0WATIV) to find 0 ::; 83 ::; tt ,

P3 = arg Ii.+ 83 - arg 1i.3

03 = 2.".- 83

lJ3= arg Ii.+03- arg 1i.3

1i."-6," -6,"cos 8 =

3•

• 26,.6,

sin 8. = 1(1 - cos" 8.)'/ " 1 ~ 0

Let cos 8. = x, sin 8. = Y > 0

P. = arg Ii. - 8. - arg Ii..

lJ. = arg Ii.+ 8. - arg Ii.. + tt

With these, use the ATAN2 function (FORTRAN IV or

Use the ATAN2 function to find 0 ::; 8. ::; .".; O. = -8•.

the compatibility linkage , defined by Eq. (3.7) as the closure equation at the start.This concept is illustrated in Fig. 3.3. Here the starting position is shown in fulllines. Regarding /12 as the "driving crank," the arbitrarily assumed /32 amounts to

/'/

//

Figure 3.3 Geometric solution of the compatibility equation [Eq . (3.6)] for the unknown angles p}. j = 3,4.

Sec. 3.3 Solution Procedure for Four Prescribed Postions 183

imparting a rotational displacement to ~2, shown in dashed lines in its new position.The corresponding displaced positions of ~3 and ~4 also appear with dashed lines.However, Eq. (3.6) can also be satisfied by the dot-dashed positions of ~3 and ~4'

characterized by the respective rotations 133, 134' Thus, in general, for each assumedvalue of {32, there will be two sets of values for {3j, j = 3, 4.

The range within which {32 may be assumed is determined by the limits ofmobility of the compatibility linkage, found either graphically according to Fig. 3.3,or analytically (see Sees. 3.1 and 3.3 of Vol. 1). Analytical expressions for the computation of {33, {34' 133, and 134 for a given value of {32 are given in Table 3.1. Anytwo e« from either of two sets, {32, {33, {34 and {32, 133, 134, may be inserted intotwo of the three standard-form equations [Eq. (3.3)]. Then Cramer's rule or anyother method of solving sets of linear equations may be used to solve for Z and W,from which the circle point

and the center point

m=k1-W

(3.10)

(3.11)

may be found .If {32 is varied in steps from 0 to 27T, each value of {32 will yield (when the

compatibility linkage closes) two sets of Burmester point pairs (BPPs) , each consistingof a circle point k i and a center point m. Note that the circle point k, is a pointof the first prescribed position of the moving plane . A plot of the center points foreach value of {32 will sweep out two branches of the center-point curve, one branchassociated with {33 and {34' the other with 133 and 134' If the "compatibility linkage"of Fig. 3.3 allows complete rotation of ~2' these two branches will meet. The circlepoints may also be plotted similarly to yield the circle-point curve. A portion of atypical center- and circle-point curve is shown in Fig. 3.4 (see Sec. 3.5, which coversanother technique for generating Burmester curves) . Every point on the center-pointcurve represents a possible ground pivot. This ground pivot can be linked with itsconjugate, the circle point in the particular BPP, and with the first prescribed pathpoint. This will form a dyad with ground pivot m, crank W, pin joint k l • floatinglink Z, and terminal point PI (Fig. 3.2). This dyad can serve as one-half of thefour-bar motion generator and it may be combined with any other similarly formeddyad to complete a four-bar linkage.

In examining the motion of a solution linkage formed by two dyads , it maybe observed that, although the moving body will assume each of the four prescribedpositions as the input crank rotates in one direction through its range of motion,they may not be reached in the prescribed order, unless certain conditions are fulfilledin choosing the BPP along the two curves [302,303,304,116,110]. Also, the resultingfour-bar may have other undesirable characteristics (poor ground and moving pivotlocations, low transmission angles, branching, etc.). Some techniques are availablefor transmission angle control [114,115,306] and branching [301-303] . The infinitenumber of solutions may be surveyed for the "best" solution by using these techniquesin conjunction with computer graphics described next.


PLl".IT (iF l {l "~: C Uf~· t )ES

M -~.--

"" -' .0-,----__,,_---,-------,-------r------,

..K\ . 0<> -+-----.,I------+------+--+---+-+-.."...-<o.........j

---<,

o2"' . 0-+----f----+----++~!--+-.::......-_i

0 .0 -+----+----+----...----H''<-------i

2 0.00 . 0-20 .0- 40 . 0

- 20 . 1) -+------r----1!--r--+--.....--+-~---f-__,,_~-60 . 0

Figure 3.4 Typical computer-generated Burmester curves. Precisionpoints: open square, P,; open circles, p•• P3 , and p.. Poles: p,.• top.... Solid line, locus of m. ground pivots or centerpoint curve; dashedline, locus of k I , moving pivots or circlepoint curve.

3.4 COMPUTER PROGRAM FOR FOUR PRESCRIBED PRECISIONPOINTS

The solution of the standard-form equations for four prescribed, finitely separatedpositions has been fully described in Sec. 3.3, which should enable the reader toprogram it for automatic computation. However, it has been programmed and ispart of the LINCAGES· package [78,83,84,218,270] (available from the second author). Rotation {32 is used as an independent parameter to generate solutions tothe synthesis equations [Eq. (3.3)] resulting in the circle-point and center-point curves .

The four-precision-point option of the LINCAGES package is introduced byway of an example in the appendix to this chapter. Many interactive subroutinesare available with graphical output to help the linkage designer to survey numerouspossibilities in order to help arrive at an optimal solution.

The following example demonstrates how sensitive the Burmester curves canbe to a slight change in input data (82, 83, 84, U2, U3, U4)'

• Copyright , University of Minnesota, 1979. This computer graphics package, developed at theUniversity of Minnesota, conta ins a Teletype, LSI-II (programmed on a TERAK microprocessor) and aTektronix 4010 Series version. Other versions are now available on other mainframe computers and onsome turn -key systems. Any planar linkage that can be composed of dyads can be synthesized for eitherthree , four, or five finitely separated precision points for motion , path, or function generation or fortheir combinations.

Sec. 3.4 Computer Program for Four Prescribed Precision Points 185

Example 3.1 [318,83]

If the boom of a front loader is pivoted about a fixed axis, as it is raised above thehorizontal position, the bucket tends to arc back in the direction of the loader cab.If the bucket must be lifted fairly high, the forward reach is reduced and if there isany spillback, it might occur near the operator. One manufacturer has overcome thisproblem by guiding the boom with pivoted links so that the boom is connected to thecoupler link of a four-bar linkage. The boom is designed to move in such a mannerthat the bucket does not arc back toward the operator.*

In Fig. 3.5, the length of boom k iP, and crank length mk, as well as the loaderframe were roughly proportioned from a photograph of the above-mentioned commercially available loader. As boom klPI is pivoted clockwise to raise the bucket, linkmk, is pivoted counterclockwise to give four positions of point P that lie approximatelyon a straight line that slopes outward from the vertical, away from the loader. Theangular positions of line kP for the four positions assumed were accurately determinedfrom the drawing (with respect to the horizontal x axis) to be

81 = 191.26°,

82 = 175.68°,

83 = 164.77°

8. = 152.04°

Portions of the center-point and circle-point curves for this example, correspondingto the four positions of the line kP (8j = P, - PI and aj = 8j - 81> j = 2, 3, 4)are shown in Fig. 3.5. One mk , combination that locates the ground pivot m in anideal position is shown. The only feasible design observed from the Burmester curveswould be one in which the fixed pivot for the other dyad would be on a bracket extendingbackward and upward from the top rear of the cab and the corresponding moving

Figure 3.5 Four assumed positions offront -loader-boom kP and portion s of thecorresponding Burmester curves. Onedyad , W"ZI, has been chosen from theBurmestercurves. No othersuitable dyadis found .

• The bucket angle is controlled by a piston pinned between the boom and the bucket.

186 Kinemat ic Synthesis of Linkages: Advanced Topics Chap. 3

e," 19 1. 2 6 "e. " 175.6B"e." 152.04"

------

......-------- ......,;"/ ,......-------..,:::....

,/ .,/ /

/ // I

(\'- -------- - - - --

Figure 3.6 Centerpoint curves forthree slightly different angular positions 83 of plane kP in position 3.

joint would be on the circle-point curve almost vertically above k- , Assuming thatno major extensions are to be built onto the existing cab structure shown, such a designwould not be satisfactory. However, it is known that the Burmester curves may shiftconsiderably with small angular changes of the line determining the four positions ofthe plane. Small angular changes of line kjPj would not appreciably affect the generaldirection in which the bucket travels . This shift of the center-point curve is illustratedin Fig. 3.6, in which three center-point curves are shown which share the same first,

0 1 ~ 191 .26 °

0 2 ~ 175.93°

0 3 ~ 164 .64°

04 ~ 152 .04°

Circl e-Point Curve

Figure 3.7 Burmester curves for theselected design positions of plane kP.Two dyads, Z' ,W' and Z2,W2, makeup the final four-bar solution.

Sec. 3.4 Computer Program for Four Prescribed Precision Points 187

second, and fourth angular positions of line kiP I • The prescribed path points are alsoidentical for each curve but the third angular position , 83, has different values of 164.77°,164.52°, and 164.27°. With the latter value of 83, the center-point curve changes froma one-part curve to a two-part curve. Additional runs were made varying 82 and 84

by small amounts. From the results of the several runs made, it was apparent that

81 = 191.26°,

82 = 175.93°,

83 = 164.64°

84 = 152.04°

might permit the choice of a fixed pivot near the front top of the cab, where it islocated on the commercially available loader previously mentioned, and at the sametime give a good location for the moving pivot. The linkage resulting from this choiceof angles is shown in Fig. 3.7. The actual center-point and circle-point loci for thiscase [318,83] are shown in Fig. 3.8.

Notice, that with each change in 8j or a j, an infinity of solutions is surveyed.The portions of the ground and moving pivot curves that are too far away are cut offby the "window" chosen on the graphics screen. The designer is now synthesizing

.'

•• 0

/CENTER - POINT CURVE

.--... ~.

~'~C LE - POINT CURV~.\".._._~~.: ., .

8.,;

00

'" ..00,.;

.. II·

0../0.....

m ...

~.OO

o

."...

3.00

0·o o'

. ..

.....o. . u.--

00

... . ....... ...3. 00 -2.00 -1 .00 8100 1.00

0 0 ,.'

0# .. ...

~,p

.i'•..'• 0

000

00 N.' I.

• 0

Figure 3.8 Portion of the plot of the final Burmester curves for the selected design positions for thefront loader of Example 3.1 [318].


infinities at one time rather than one mechanism at a time as is done graphically orwith cardboard and thumb tacks or even by computational methods without the aidof the graphics screen.

3.5* MOTION-GENERATION WITH FOUR FINITELY SEPARATEDPRESCRIBED POSITIONS: SUPERPOSITION OF TWOTHREE-PRECISION-POINT CASES

The four-position motion dyad synthesis problem with 82, 8a, 84, a2, aa, and a4prescribed can be developed from a different viewpoint [170]. Four prescribed positions can be considered as a superposition of two three-position motion-generationsubproblems, with (82, 8a, a2, aa) prescribed in one and (82, 84, a2, a4) prescribedin the other. Recall that Sec. 2.20 presented the circle-point circles for the threeposition case. For a selected value of the rotation of the ground-link (W) (see Fig.3.2), {32' a pair of M and K circles can be drawn for both three-precision-pointsubproblems; intersections, of such two M-circles and K-circles, respectively, definepoints on the m and k curves that satisfy both subproblems simultaneously.

For example, suppose that the following dyad-motion precision positions (Fig.3.2) were desired:

Problem 1:

82 = 2 + 2;, a2 = 60 0

8a = 5 + 2;, aa = 1200

Problem 2:

82 = 2 + 2;, a2 = 60 0

84 =4 + 3;, a4 = 1800

(where 82 and a2 are the same in both problems). The intersections of the M circlesand K circles in both three-point problems (i.e., for each (32), as described above,define points on the "circle-point" and "center-point" curves.

Following the graphical construction procedure of Sec. 2.20, we begin by findingthe poles P12, P13, P14, P2a, P24, P2a, P24. There will be two sets of M and Kcircles, labeled with superscripts 1 and 2 (Mi corresponds to problem 1). The centersof circles Mi , M2, Ki, K2 lie on the bisectors of PiaP2a, P14P24, PiaP2a, and P14P24,respectively, labeled as "M) axis," "K! axis," and so on, in Fig. 3.9. Note thatthe following length equalities prevail:

IP iaP2a l = IPiaP2a land

IP14P24 I= IP14P24 I

• This section may be skipped without loss of continuity.

....COCD

K2 + axis

K ' + axis

K1+

Figure 3.9 Finding points on the centerpoint and circlepoint curves(M- and K-curves) for four finitely-separated positions by constructingintersections of M- and K-circles of two separate 3-position problems:one for positions I, 2, and 3, and the other for positions I, 2, and 4(see sec. 3.5).

o = point on K curve

~ = po int on M curv e

• = pol e

+ = precision po int

K'- axis


Up to four m - and k-curve points can be found for each value of angle E of Fig.3.9 (eight including its supplement, as was done here for ease of construction), thepositive angle subtended by the respective pole distances at the corresponding circlecenters. For example, for a chosen value of E, strike an arc with radiuskIP13P23 IcsC(E/ 2) from PI3 to intersect the MH and MI- axes.* This locates thecenters of one MI+ and one MI- circle with this radius. Next , with this same valueof E, repeat the construction using IP14P24 ! and draw one M2+ and one M2- circle .The construction for the KI and K2 circles is similar, using IP I3P23 I and IP I4P24 I·Intersections of KI+, KI- . MH, and MI- with K2+, K2-, M2+, and M2-, respectively ,are possible moving pivots k and ground pivots m for the four-position problem,and thus are points on the circle-point and center-point curves. If E = 40° , theangles /32 corresponding to each intersection pair are (two circles may intersect attwo points)

M+: /32 = -E = -40°

K+ : /32 = a 2 - E= 60° - 40° = 20°

K-: /32 = a2 - (-E) = 60° + 40° = 100°

The K+ and the K- intersections apply for /32 less than a2, or for /32 greater thana2, respectively. Note, for example , that if one desired the moving pivots for /32 =40°. they would be the K + intersections for E = a2 - /32 = 20° > O.

This procedure, derived here from the three-position M and K circles, is thesame as the classical geometric Burmester curve construction based on the oppositepole quadrilateral [125]. Opposite poles are defined as two poles carrying four differentsubscripts. For this example there are three pairs of opposite poles: (PI3• P24), (P23•

P 14 ) , and (PI2, P34). An opposite pole quadrilateral has its diagonal s connectingtwo opposite poles. Here we are using the opposite pole quadrilateral (PI3• P23,

P14 • P24) for the construction of points on the M curve.Classically, the M circles are not presented as intersections of three-point-solu

tion loci, but as loci of points that subtend equal or supplementary angles at theside of the opposite-pole quadrilateral representing the chord of the circle. Theintersections of circles through opposite-pole pairs whose peripheral points subtendequal or supplementary angles at their respective chords are constructed; these pointssatisfy the theorem of Burmester, which states that the points on the M curve subtendequal or supplemental angles at opposite sides of the opposite-pole quadrilateral.Points on the K curve were generated classically by inverting the motion and repeatingthe same construction.

To summarize: Points on the M/K curve (center-point curve)/(circle-pointcurve) for four finitely separated positions may be generated as intersections of theM /K circles of two three-precision-point subproblems, for which the procedure waslaid out in Sec. 2.20.

• The plus or minus sign in the superscripts of the axes indicates one direction or the other towardinfinity.

Sec. 3.6 Special Cases of Four-Posltion Synthesis 191

3.6 SPECIAL CASES OF FOUR-POSITION SYNTHESIS

The Slider Point or Finite Ball's Point

In Chap. 4 we will discuss "Ball's point," that point of the moving plane tr whosepath has third-order contact with its path tangent. In other words , the path hasfour-point contact with its tangent in the vicinity of the position under study.In still other words, Ball' s point moves in a straight line through four infinitesimallyclose positions , or through four "infinitesimally separated positions" (ISPs) .

The counterpart of Ball's point for four finitely separated positions (FSPs) isthe slider point, which we might as well call "finite Ball's point. " It is that pointof the moving plane whose four corresponding positions fall on a straight line.It is a special circle point, whose conjugate center point is at infinity, and hence islocated in the direction of the asymptote of the center-point curve.

Kaufman [148] has shown how the slider point can be formed as a singularsolution of the compatibility equation, Eq. (3.5), by setting {3j = 0, j = 2, 3, 4 (seeFig. 3.2). We propose the following intuitive proof of Kaufman's approach. Asseen from Eq. (3.4), the vector W, which connects the unknown stationary centerpoint m to the unknown moving circle point k- : does not rotate ({3j = 0) as k l

moves by finite displacements to k». k 3 , and k 4• This is possible only if W is ofinfinite length. Since all displacements of k l (to k 2, k 3 , and k 4) must be alongpaths that are always perpendicular to the momentary position of W, that path mustbe a straight line. Therefore, this k l is the finite Ball's point sought. To signifythis we attach the superscript s (for "slider point"), thus: W s and kf.

However, {3j = 0 makes the coefficient matrix M singular, and therefore Eqs.(3.5) and (3.4) cannot be solved in their original form to locate kt. Kaufman suggestsa way around this, illustrated in Fig. 3.10. Here P, and P.i (j = 2,3,4) are arbitrarilyprescribed finitely separated positions of point P of the moving plane tr, shown inpositions 'Trl and 'Trj. Rotations of' er, a j , are also prescribed. ZS is an unknownvector embedded in tr , defined in position 'Tr}, which connects the unknown finiteBall's point Kt to Pi. From positions 1 to j, kt moves along the unknown line ofthe unknown vector S, measured from an unknown point Q. The unknown distanceof such (straight-line sliding motion) is designated by the unknown stretch ratio pjof the vector S. With these we write the following equation of closure for the polygonQkJPjPlktQ:

j = 2,3,4 (3.12)

where the unknown p/s are scalars, such that pj ¥ 1, pj ¥ Pk. The compatibilityequation for this system is

eia 2 - 1eia 3 - 1eia 4 - 1

(3.13)

Assuming an arbitrary value for P2 ¥ 1 and expanding about the first column, weobtain


iy

-------------- q---

II

II

//

I "I "

r · I "I I ,,"

I "I ,,"

/ ,, " r1

/ "I "

I "/ /"

I "1,, / _---

Figure 3.10 Kaufman's scheme for finding the "finite Ball's point" k S, for four finitely-separat ed positionsof a plane (see sec. 3.6).

a3 p 3 +a4p 4 = -at - a2 p 2 (3.14)

where the aj are defined as in Eqs. (3.7) and (3.8). To solve for P3 and P4, weseparate real and imaginary parts and form the real system

(3.15)

where aj = !:i.jr + i!:i.jy, and from which

- !:i.lI - P2!:i.2x

-!:i.ty - P2!:i.2Y

1

!:i.3x !:i.u

!:i.3Y !:i.4Y

(3.16)

Sec. 3.6 Special Cases of Four-Position Synthesis 193

(3.17)

and -alI - P2a2x-aly - P2a2Y

aax a4I Ia3Y a4y

Substituting back the set of Pi (j = 2, 3, 4) thus obtained into system (3.12) andsolving any two of its equations simultaneously for Sand zs, we obtain a sliderdyad which guides the plane 7T through its four prescribed positions . Combiningthis with a pinned dyad forms a slider-crank mechanism, whose coupler goes throughthe four prescribed positions; a suitable pinned dyad is obtained by assuming a valuefor /32 ¥- 0, /32 ¥- U2, and solving for Z and W as discussed in Sec. 3.3. Figure3~ a sketch of such a slider-crank mechanism: slide guide Q, slide S, crank(m (l)kD = Wi, and coupler k~Plkl, with sides ZS and Zl.

The slider-crank of Fig. 3.11, while it can assume the positions shown , it wouldjam in between. To avoid this, we need to seek another mlkl pair. The questionarises: how many such slider-crank mechanisms can we find for one arbitrary set offour prescribed body positions? It can be shown that no matter what value wepick for P2 ¥- 1, the solution for Z S is always the same, and that S, P3, and P4 are

-:/'

Figure 3.11 Finding a Burmester Point Pair , kl. mi. to form a slider-crank with Kau fman' sfinite Ball's point k·l .


such that the resulting kJ are the same. Thus it is seen that there is only oneslider dyad , which agrees with the fact that there is only one Ball's point for a setof four ISP, and thus only one finite Ball's point for a set of four FSP. However,the number of pinned dyads is infinite. Thus there will be a single infinity ofsolutionsfor the four-position motion generator slider-crank. Each of these has a cognate(see Sec. 3.9) which is a four-position path generator slider-crank with prescribed timing,shown in Fig. 3.12.

Figure 3.12 Slider-crank mechanism m'k:k~P and its cognate m'FIP. Observe that couplerrotations a j of the first mechanism are the crank rotat ions in the cognate .


%

Figure 3.13 Function-generation synthesis of the slider-crank mechanism forfour FSP [see Eqs. (3.18) to (3.21)).

Four-Point Function Generation with the Slider-Crank

Referring to Fig. 3.13, we write the standard-form equation of closure as follows:

W(eill j - 1) + Z(e iaj - 1) = PjS

The compatibility equation for this system is

(3.18)

eill 2 - 1eilla - 1eill 4 -1

eia2 - 1eiaa - 1eia4 - 1

(3.19)

Here aj, pj are prescribed, correlating crank rotations and slider translations in afunctional relationship. S is arbitrary, because it determines only the scale and orientation of the linkage. Expanding (3.19), we obtain

(3.20)

where

(3.21)

The rest of the solution follows the procedure of Table 3.1 to obtain compatiblesets of {3j and then solving any two of Eqs. (3.18) for Wand Z.

Four-Position Motion-Generator Turning-Block Mechanism:The Concurrency Point

Equation (3.5) is trivially satisfied when a j = {3j, j = 2, 3, 4. This means thatboth Z and W rotate with the moving body st, Since W rotates about the unknownfixed center point m, while Z is embedded in and moves with 7T', and because Zand W concur at the unknown circle point k- : a little thought will show that bothmust be of infinite length and opposite in direction. Since the tip of Wand the


tail of Z meet at the circle point k -: the latter must be at infinity, in the directionof the asymptote of the circle-point curve. This is illustrated in Fig. 3.14, where7Th 7Tj, Pi, Pj, and 8j have the same meaning as before. Along with body rotationsa.], they are prescribed. The unknown vectors Z and Ware always parallel, andthey are of infinite length. The unknown circle point k 1 is at infinity in the unknowndirection of Z and W in the first position. The effect of the infinite lengths of Wand Z is this: The connection between the unknown center point m and the movingbody 7T becomes a turn-slide. This enforces the line of the unknown vector D,which is embedded in 7T, to turn and slide through m. Hence m is a finite concurrencypoint. (See also Chap. 4, Path Curvature Theory, for infinitesimal concurrency point.)Therefore, we distinguish it with the superscript c, and do likewise for Z and W.Finally, the tip of the unknown D, fixed in the body 7T, is connected to P by the

k~ at 00

\// ..... ----" pl"i D

_::::.::------0. ' / ~-- / '

I \ 11Ti r I

I I 'E i" I\ I e J I

/~ai / /I O'j I\ Pj /

\ / /\ \ / ..... /~'

\ /\ / we (WC --+~)

z cei" i \ ~ \( z c --+ ~) \~

k~ at 00

\

\\

\

Figure 3.14 The unique tum-slid e dyad for four FSP of plane 1T forms one half of a fourlink turn-slide mechanism . The other half is one of a single infinity of pivoted dyads (seeFig. 3.15).


unknown embedded vector E. Observe that, while the rotations of both D and Eare the prescribed aj, E does not change in length, but D does, by the unknownstretch ratios Pj. With these, the equation of closure in standard form becomes

D(pjeiaj - I) + E(eiaj - I) = 8j

where aj and 8j are prescribed. The compatibility equation takes the form

p2eia2 - I eia2 - I 82p3eia3 - I eia3 - I 83 =0p4eia4- I eia4 - I 84

which expands to

(3.22)

(3.23)

(3.24)

(3.25)

(3.26)

where the Aj are the same as those defined in Eqs. (3.7) and (3.8). We now assumean arbitrary real value for P2, separate the real and imaginary parts of Eq. (3.24),and solve for P3 and P4, obtaining

IfYl(-Al - p2eia2A2) fYl(e ia4A4) If(-A 1 - p2eia2A2) f(e ia4A4)

P3 = I fYl(e ia3A3) fYl(e ia4A4) If(eia3A3) f(e ia4A4)

and

IfYl(eia3A3) fYl(-Al - p2eia2A2) If(e ia3A3) f(-A 1 - p2eia2A2)

P4=

IfYl(e ia3A3) fYl(e ia4A4) If(eia3A3) f(e ia4A4)

where fYl( • ) signifies the real part of ( • ) and f( • ) signifies the imaginary partof ( .). The resulting compatible set of Pj, j = 2, 3, 4, can now be substitutedback into any two of Eqs. (3.22) and these solved simultaneously for D and E, thuslocating m" and completing the solution.

It is to be noted that the concurrency point mC thus found is unique : It is theunique slider point or finite Ball's point for the inverted motion in which the fixedplane of reference and the moving plane 7T exchange roles. Thus it is seen thatregardless of the arbitrary choice for the value of P2, the end result is the same:There is only one turn-slide dyad that can guide 7T through the four prescribed finitepositions. To complete a single-degree-of-freedom four-link mechanism we find apivoted dyad (one of the infinite number available for four finite positions), thusobtaining the mechanism shown in Fig. 3.15, one of the single infinity of such fourposition motion-generator turn-slide four-link mechanisms. Furthermore, if designconditions make it desirable to change it to a tumbling block mechanism (Fig . 3.16),the same vector representation and derivations would apply.


Figure 3.15 One of Kaufman 's singular solutions yields the turn-slide dyad in this four-linkturn-slide mechanism [see Eqs. (3.22) to (3.26) and Fig. 3.14].

Figure 3.16 Equat ions (3.22) to (3.26) can also be used to find the tumbling-blockdyad in this four-FSP motion generator four-link mechanism.

Finding the Poles in Motion-Generation Synthesis

Any pole Pj k , j, k = 1, 2, 3, 4, j ¥- k, can be found as follows (see Fig. 3.16a):

(3.27)

Sec. 3.7

iy

Motion Generation : Five Positions

Qj

1ft -- __ Direct ion of(a,l

199

a,

Figure 3.16.a Derivation of Eqs. (3.27) to (3.29) for finding the poles in planar motion-generationsynthesis.

_ 8k -8j

Pj - ei(ak-a jl- 1

Pjk = rj - Pj

3.7 MOTION GENERATION: FIVE POSITIONS

(3.28)

(3.29)

In previous sections we observed that a four-bar linkage could be synthesized forfour prescribed positions as a motion generator. Thanks to Burmester and thosewho continued his work, we know that there are ideally an infinity of pivoted dyadsfor any four arbitrarily prescribed positions, and that any two of these can form afour-bar mechanism whose coupler will match these prescribed positions. But canwe synthesize four-bar motion generators for five arbitrary positions?

Our first hint toward the answer to this question came when the tabular formulation was developed (Table 2.1). Although the table shows that there are no freechoices for five prescribed positions, the number of real equations and the numberof real unknowns are equal, indicating that these equations can be solved.

The second hint will come from further examination of the four-position caseand the resulting center- and circle-point curves. Suppose, as in Fig. 3.4 that thecircle- and center-point curves are plotted for prescribed positions I, 2, 3, and 4.In addition, a new set of curves for the same first three positions plus a fifth position

200 Kinematic Synthesis of Linkages: Advanced Topics

PLOT OF " CURVES"1 - 1'12 _ ._ . 1'13 --- - 1'14 - -

Chap. 3

5.00

.--- -/' \

1 .00}, 6

53.00

2.00

l.00

0.00-2.00 -1.00 0.00 1.00 2.00

Figure 3.17 Four combinations of centerpoint curves for 02 = 1.5 + 0.8i. 03 = 1.6+ 1.5i. O. = 2.0 + 3.0i. and 05 = 2.3 +3.5i; and a2 = 10°, a3 = 20°. a. = 60°,and as = 90°. MI signifies the centerpointcurve for precision points I, 2, 3, and 4;M2 = I, 2, 3, and 5; M3 = I, 2, 4, and 5;and M4 = I, 3, 4, and 5. This examplewas programmed on the LINeAGES package by M. Richardson [218]. Two existingfive-point dyads are drawn in. Figure 3.18shows the resulting four-bar five-point motion-generation solution .

are superimposed over the first set. If these curves intersect, a common solutionexists and a Burmester pair (or dyad) has been found that will be able to guide aplane through all five prescribed positions. Figure 3.17 shows an overlay of allfour combinations of four-precision-point motion-generation cases whose intersectionslocate the center-point solutions for the combined five-position problem. Figure 3.18shows the four bar solution for the five precision points of Fig.3.17 in its first position.

The circle- and center-point curves can be shown to be cubics [107,108], sothere are a maximum of nine intersections. There are two imaginary intersectionsat infinity and , discounting the intersections marking the poles P t 2, P t 3, and P23,

there is a maximum number of four usable real intersections. [P t z • P t 3 , or PZ3

could also be used, but this would be tantamount to the "point-position reductionmethod" (see Sec. 2.9 and Ref. 169) with its frequent accompanying difficulties ofretrograde crank rotation.] Since usable real intersections will come in pairs, wecan expect either zero, two, or four solutions for any five arbitrarily prescribed precisionpoints. Let us see whether this geometric concept can be verified by mathematicalmethods.

Referring again to Fig. 3.2, the same standard form for the equations may bewritten (four equations in this case):

W(eiJlj - 1) + Z(eiaj - I) = 8j ,

The augmented matrix of this system is

j = 2,3,4,5 (3.30)

eiaz - 1eia3 - Ieia 4 - Ieia5 - 1

(3.31)

Sec. 3.7 Motion Generation: Five Positions 201

Li nkLength s:

InputLink L, :

Ao( X,Y)- 0.364

3.33 5A( X,Y)- 0.760

2.837

Bo( X,Y)- 0.484

2.515B(X, Y)- 0.931

1.936

OutputLin k L3 :

3.002.001.00

a,

0.00

Ls

L3

2.00 -H1-."-+--+-----t-----t------1

4.00 -,----r----,------,-------,

3.00 +-~+--t-----t----~~--___l

0.00 -+--,---.---"-,---I---,---+---,,----i

- 1.00

1.00 -t--\--\--t-----t------t--------j

C-R

LinkageTy pe:

L, = 0.64L2 = 2.94L3 = 0.73L4 = 2.15Ls = 0.92Ls = 0.83

This is th e Linkage in th e Fir st Positi on

Figure 3.18 For the five precision point s given in Fig. 3.17, two Burmester pairs exist and areshown in the figure. The resulting four-bar is shown in its first position. (For four -bar notation,see Fig. 2.59.) Other precision points are signified by P; and the prescribed angles by line Pja].

For system 3.30 to have simultaneous solutions for the dyad vectors Wand Z, Mmust be of rank 2. Thus there are two compatibility equations to be satisfied simultaneously for five prescribed positions:

ei fJ2 - I ei a2 - I 82

ei fJ3 - I ei a3 - 1 83 =0ei fJ4 - 1 ei a4 - 1 84

and

ei fJ2 - 1 ei a2 - 1 82

ei fJ3 - 1 ei a3 - 1 83 =0ei fJ5 - 1 e i a5 - 1 85

(3.32)

(3.33)

Since the second and third columns of determinants (3.32) and (3.33) contain prescribed data, the only unknown (unprescribed) reals are {3i' j = 2, 3, 4, and 5 inthe first column. Thus there are no free choices here (Table 2.1) to help solve thesetwo complex (or four real) equations, which are nonlinear and transcendental in {3i

(j = 2, 3, 4, 5). If the solutions for {3i are real, then with these values of {3i. Eqs.


(3.30) are compatible. Then there are up to four usable real intersections of theBurmester curves. This means that there will be up to four dyads that can be usedto construct motion generators for the five prescribed positions, for which Z andW can then be found from any two equations of the system of Eq. (3.30).

3.8 SOLUTION PROCEDURE FOR FIVE PRESCRIBED POSITIONS

The determinants [Eqs. (3.32) and (3.33)] will be expanded about their elements inthe first column (where the unknown {3j-S are):

li.2ei(32 + li.3ei(33 + li.4ei(34 - li.l = 0

li.;ei(32 + li.aei(33 + li.4ei(35 - Ii.~ = 0

(3.34)

(3.35)

where the li.j (j = I, 2, 3, 4) are the same as before [Eq. (3.8)] and the li.j (j = I,2, 3) are the same as the li.j except that each subscript 4 is replaced by a subscript5.

The complex conjugates" of these compatibility equations also hold true:

X2e- i(32 + X3e- i(33 + X4e- i(34 - Xl = 0 (3.36)

X;e -i(32 + Xae - i(33 + X4e- i(35 - X~ = 0 (3.37)

We can eliminate {34 and {35 from Eqs. (3.34) to (3.37) as follows. Equation (3.34)is multiplied by Eq. (3.36) and Eq. (3.35) by Eq. (3.37).t Equations (3.34) and(3.36) yield

li.4X4= li.1Xl - li.1X2e- i(32 - li.1X3e- i(33 - li.2Xlei(32

+ li.2X2+ li.2X3ei(32e - i(33 - li.3Xlei(33 + li.3X2ei(33e - i(32

+li.3X3

A more compact form of (3.38) is

Clei(33+ d l + Cle - i(33= 0

where

and

j = 1,2,3

Similarly, Eqs. (3.35) and (3.37) will yield

C2ei(33+ d 2+ C2e - i(33 = 0

• The superior bar indicates complex conjugates.

t After the Ii.. terms are put on the other side of the equals sign.

(3.38)

(3.39)

(3.40)

Sec. 3.8 Solution Procedure for Five Prescribed Positions 203

where C2 and d2 are the same as C1 and db but with primes on the 11j, j = 1, 2,3. Notice that (3.39) and (3.40) can be regarded as homogeneous nonlinear equationsin ei/3 2 and ei/33, containing their first, zeroth, and minus first powers , with knowncoefficients. Elimination of the powers containing {33 is accomplished through theuse of Sylvester's" dyalitic eliminant. This is begun by multiplying Eqs . (3.39) and(3.40) by ei/33, creating two additional valid equations:

C1e i2/33 + d 1ei/33 + C1 = 0

C2e i2/33 + d 2ei/33 + C2= 0

(3.41)

(3.42)

If ei2/33, eil /33, e i0/33 and e i(-1l/33 are considered as separate "unknowns," Eqs. (3.39)to (3.42) can be regarded as four homogeneous equations, linear in these four unknowns. Since these equations have zeros on the right-hand side, the determinantof the coefficients must be zero for the system to yield simultaneous solutions forthe four "unknowns." Therefore, the "eliminant" determinant is

0 C1 d l C1

E= 0 C2 d 2 C2 = 0C1 d l C1 0C2 d 2 C2 0

(3.43)

Note that we have successfully eliminated powers of ei/33 from the eliminant.Expanding the determinant, a polynomial in ei/3 2 is obtained:

(3.44)

where m = -3, -2, -1, 0, 1,2,3, and where all the coefficients am are deterministicfunctions of the prescribed quantities 8j and aj (j = 2, 3, 4, 5). Also note thata-k and ak (k = 1, 2, 3) are each other's complex conjugates, and that ao is real.Thus the expansion shows that E is real, so that its imaginary part vanishes identically.Therefore, only the real part of the eliminant is of interest. It has the form

L [Pm cos (m{32)+ qm sin (m{32)] = 0,m

m = 0,1,2,3 (3.45)

where pm and qm are known reals. By way of trigonometric identities, Eq . (3.45)can be transformed into a form containing powers of sines and cosines of {32 (up tothe third power), and then by further identities changed to a sixth-degree polynomialin a single variable, T = tan (/32/2), having the form

6

L AnT n =0O. I • . . .

(3.46)

We know that {32 = 0 is a trivial solution, which makes T = 0 a trivial root. ThusEq. (3.46) can be reduced to a fifth-degree polynomial. Also, from the determinantform of Eqs. (3.32) and (3.33), it is clear that the set of {3j = aj , j = 2, 3, 4, 5(here {32 = a2), is another trivial solution. Thus, after dividing the root factor

• Nineteenth-century English mathematician and kinemati cian.


[T - tan (a2/2)] out of the remaining fifth-degree polynomial equation, a quarticremains in T:

(3.47)

Equation (3.47) will have zero, two, or four real roots. Each real root yields avalue for /32, which can be substituted back into either Eq. (3.41) or (3.42) to find/33, and then into (3.34) and (3.35) to obtain /34 and /35' Then any two equationsof (3.30) can be solved for Z and W, yielding up to four Burmester point pairs forthis motion-generation case: m In), kIln), n = 1, 2, 3, 4. These, together with thestarting position PI of the prescribed path tracer point P, define the four dyadsW<n ), Z l n ). Z l n ) exhibit the same set of prescribed rotations aj, j = 2, 3, 4, 5,while the four different vectors W<n ) have four different sets of rotations /3/ n ), n =1, 2, 3, 4, j = 2, 3, 4, 5. The example in Figs. 3.17 and 3.18 has only two realroots and thus only two BPPs (n = 1, 2) which can be used to form one four-barlinkage which will guide its coupler plane through the five prescribed coplanar positions . In general , combining each of four BPPs with every other, we can obtainup to six different such four-bar motion-generator linkages . (See Appendix A3.2for computation schedule.)

3.9 EXTENSIONS OF BURMESTER POINT THEORY:PATH GENERATION WITH PRESCRIBED TIMINGAND FUNCTION GENERATION

Burmester theory was derived above for obtaining dyads suitable for a four-bar motiongenerator. What about path generation with prescribed timing and function generation with the four-bar? Also, can this theory be extended to other linkage types?Chapter 2 demonstrated that the dyadic standard form equation, Eq. (3.2), was usablein numerous cases. The Roberts-Chebyshev theorem will add more insight to thebroad applicability of the dyad form and of the Burmester theory.

Roberts-Chebyshev Theorem

An extremely useful property of planar four-bar linkages is revealed in the RobertsChebyshev theorem [125], which states that one point of each of three different butrelated planar four-bar linkages will trace identical coupler curves. This means thatthere will be two additional four-bar linkages associated with each "parent" fourbar linkage which will trace the same path as the parent (although the coupler rotationswill not be the same). These two additional linkages are called "Roberts-Chebyshevcognates" after their two independent English and Russian discoverers. We canform these cognates geometrically by building on the four-bar linkage shown in fulllines in Fig. 3.19 as follows:

1. Complete the parallelograms of ZI and WI and Z 2 and W2.

2. Find the third fixed point, C, of the Roberts configuration by making triangle

Sec. 3.9 Extensions of Burmester Point Theory 205

Figure 3.19 The Roberts-Chebishev Configuration consists of basic motion-generatorfour-bar mechanism consisting of two (out of the possible maximum of four) BurmesterPoint Pair (BPP) dyads, W'Z' and W2Z2. The dashed and dot-dashed four-barmechanisms are the cognates of the basic one.

m!Cm 2 similar to the coupler triangle k~Pkf such that C corresponds to P,m) to k~. and so on.

3. Find one cognate coupler triangle by making I:i.GPH similar to I:i.m 2m!C orI:i.kik~P. HC is the follower link of the dashed "right cognate" of Fig. 3.19.

4. Find the other cognate by making coupler triangle FIP similar to l:i.m !Cm 2

and/or I:i.k~Pki and connecting IC to form the follower link of the secondcognate, shown in dot-dashed lines. Note that ICHP is a parallelogram.Due to the three parallelograms that concur at P, Fig. 3.20 shows that thecoupler curves traced by P as a point of the initial four-bar or as a point ofeither one of its cognates are one and the same curve.

This property, that every four-bar linkage has two cognate linkages which tracethe same path as the parent four-bar, is extremely useful to designers. The cognatesare different linkages, even though they share one ground-pivot location with oneanother. A designer may find that although a particular linkage may trace a desired

206

k'1

I -----J,,\..k~

Parent Linkage

(b)

(c)

(d)

Figure 3.20 (a) three Roberts-Chebishev four-bar mechanisms: (b), (c),and (d); their common point P describes the same path in the fixed planeof reference; (e) the configuration contains seven similar triangles, demonstrated by the stretched-out configuration of Fig. 3.20(e).

"'---- --- -.---_.

I --,Jr--/

;'I

/I

F />S------/ Fp.'~....L.....f...L--~~~.J.....+-l....-.l-..l.o<:.()

I

//

I

/I

/

(e)

207


path, that linkage may not satisfy space requirements, or the transmission angle,mechanical advantage, velocity, and/or acceleration characteristics may not be acceptable. There are, however, two cognates available which , while they trace the samepath, in general will display different kinematic and dynamic characteristics.

It should be mentioned here that cognates are not equivalent linkages.Equivalent linkages are usually employed to duplicate instantaneously the position,velocity, and perhaps acceleration of a direct-contact (higher-pair) mechanism (suchas a cam or a noncircular gear) by a linkage (say, a four-bar) . The dimensions ofequivalent linkages are different at various positions of such higher-pair parent mechanisms, whereas the link lengths of cognates remain the same for any position of theparent linkage.

Other properties of cognates include the one developed by Cayley [36]: Thecommon coupler-path tracer point and the three instant centers of the three concurrentcouplers (with respect to ground) are collinear at all times and the line containingthese points is normal to the coupler curve in every position of the linkage system(see points IC!> lC2• and lCa in Fig. 3.20a). Another observation is: Each groundedlink of any of the three FBLs will exhibit the same angular rotations (and will rotateat the same angular velocity) as one of the grounded links of one of its cognateFBLs and the coupler link of its other cognate FBL, as shown in Fig. 3.19 (we willmake use of this property shortly). Still another noteworthy fact is that, if the parentlinkage and its two cognates were pinned together to form a movable lO-bar linkage,Gruebler's equation (Sec. 1.6) predicts that this linkage has -I degrees of freedom.This is an example of an overdosed linkage that has mobility due only to its specialgeometric properties. Yet another property of the Roberts-Chebyshev configurationis this: In addition to the four rigid similar triangles (the three coupler trianglesand the triangle formed by the three ground pivots) there are also three variablesize triangles, all of which remain always similar to the coupler triangles in thecourse of motion of the linkage. These are: Li.m1k~I, Li.ktm2H, and Li.FGC. Theproof may be started as shown in Fig. 3.20e; move m 2 away from m- along theextension of their connecting line until all three links between them are stretchedout in a straight line. Proceed similarly with C with respect to m 1. In the resultingstretched-out configuration, in which all link lengths have retained their originallengths, the above-mentioned seven triangles are all clearly similar. The rest of theproof is left to the reader as an exercise. [Hint: Move C' and (m 2)' toward m),keeping their triangle similar.] Then, by complex numbers and appropriate rotationoperators, show the similarity of the above-mentioned variable triangles with theother four.

Four-bar linkages are not the only linkages that have cognates . The slidercrank (a special case of a four-bar ; see Fig. 3.12), five-bar, six-bar, and in fact allplanar linkages have cognates. A complex-number proof of the existence of thefour-bar cognates, using complex numbers and appropriate rotational operators, canbe based on Fig. 3.20e. This is left to the reader as an exercise. For further development of the above-mentioned properties of cognates and a historical note, refer toRef. 125.

Sec. 3.9 Extensions of Burmester Point Theory 209

Four-Bar Path Generator Mechanisms (Four and FivePrecision Points)

In addit ion to the usefulness of cognates just mentioned, an important computationaladvantage may also be derived. By employing the Roberts-Chebyshev theorem, pathgenerators with prescribed timing may be obtained from motion-generator four bars.Let us look again at the geometric cognates of Fig. 3.19. Suppose that the parentfour-bar (mlktPkim2) is a motion generator that has been synthesized by eitherthe four- or five-precision-point technique. The rotations of the coupler link aj andthe displacements of tracer point P have been prescribed. Input link m1kt rotatesby f3} (beta rotations of the ground link in the first solution pair or dyad) whilethe output link m2ki rotates by f3~ (second dyad).

According to the Roberts-Chebyshev development, all three cognates trace theidentical coupler curves with their common tracer point. What do the individuallinks of the two other cognates rotate by? Noticing that m IF is always parallel tokiP and FP is always parallel to m1ki, it is clear that m-F rotates by aj whileFP rotates by f3}. Furthermore, m 2G rotates by aj, PGH and IC rotate by f3~,

and CH rotates by f3 }. Since the originally prescribed rotations aj have been transferred to the grounded links in the cognates, the cognates of a motion generator arepath generators with prescribed timing. For every four-bar motion generator therewill be two such four-bar path generators. This development may be utilized tosimplify both the four- and five-precision-point synthesis methods, so that the synthesisequations need only be solved once for both tasks : motion generation and path generation with prescribed timing. In the second case, the cognates may also be derivedvia the computer from the parent motion generator.

In the five-precision-point case, how many path generators with prescribed timingmight we expect? Since there are either 0, 2, or 4 real roots of the quartic [Eq,(3.47)], there will be 0, 2, or 12 path generators with prescribed timing for eachdata set.

Four-Bar Function Generator Mechanism (Four and FivePrecision Points)

In Sec. 2.16 it was demonstrated that the four-bar function generator could be synthesized in the "standard form" by treating it like a path generator with prescribedtiming where the path was specified along a circular arc (see Fig. 2.60). Furthermore,the tracer points along the arc were chosen so that as a link from the center of thearc to the tracer point rotated from one precision point to the next, this link wouldrotate by the prescribed output angles o/j. Correlation of this procedure with theRoberts-Chebyshev configuration can be observed by referring to points m -, m 2 , Fand P in both Figs. 3.20a and 3.21.

How many function generators might we expect from the five-precision-pointcase? One might initially guess that there will be a maximum of 12 function generatorsif all the roots of the quartic Eq, (3.47) were real, because we are treating the function


Figure 3.21 Four-bar function-generator synthesis. Given : a j. r l =1 + Z., and rj = 1 + Z. ei"' j , j = 2, 3, 4, 5, where Z. is arbitraryand aj and IjJj are analogs of the independent and dependent variable sof the function to be generated. With these, the standard form ofthe dyadic synthesis equation becomes : Z (e1aj - 1) + W (e1fJ j - I)= OJ where OJ = rj - rl.

Chap. 3

generator as a path generator with prescribed timing . This , however, is not thecase. First, only half of the four-bar path generator is required to form a functiongenerator, the dyad m 1FP, Figs. 3.20a and 3.21. Furthermore, since there are onlyfour different dyads that make up the 12 path generators, there are only four differentdyad solutions available. Also, there will always be one trivial solution since a circularare , centered at m 2, is being specified as the path of P. This solution will containzero length for the grounded link Z of the dyad and a coupler link W identicalwith the specified output link Z4 (with rotations ~j). Therefore, there is a maximumof only three different four-bar solutions; but at least one solution is guaranteedbecause complex roots [for T = tan (fi:zj2)] come in conjugate pairs and there shouldalways be a trivial solution. Table 3.2 summarizes the number of solutions thatcan be expected for zero, two, or four real roots to the quartic.

TABLE 3.2 NUMBER OF POSSIBLE SOLUTIONS FOR FOUR-BARSYNTHESIS WITH FIVE FINITELY SEPARATED PRECISION POINTSBY WAY OF THE STANDARD DYAD FORM [Ea. (3.30)).

Number of different four bar solutions expected

Number of Path generation Functionreal roots

I

Motion with generationof the quartic generation prescribed timing (See Fig. 3.21.)

0 0 0 02 1 2 14 6 12 3

Sec. 3.10 Further Extensions of Burmester Theory 211

3.10 FURTHER EXTENSIONS OF BURMESTER THEORY

Geared Five-Bar and Parallelogram Six-Bar Cognates

In the preceding section it was shown that the Burmester dyads can be arrangedtogether to yield four-bar mechani sms for motion, function, and path generationwith prescribed timing . Some other useful linkages, with more than four links, canbe synthesized from these same dyads using simple construction procedures. Supposethat one wishes to obtain a path generator with prescribed timing directly withoutcomputing the cognate of the motion generator. (Perhaps the motion generatorground-pivot locations are acceptable but the cognates exhibit an undesirable groundpivot location .) Then either the geared five-bar or parallelogram six-bar path generator(with prescribed timing) may be useful.

Referring to Fig. 3.22, complete the vector parallelograms of the original dyads(WI and ZI, W2 and Z2) of the four-bar motion generator, shown in dashed linesof Fig. 3.19. Disregarding the parent four-bar, connect the grounded links Z I andZ2 with each other by means of one-to-one gearing (using an idler to assure thatZI and Z2 perform identical rotations). Thus a single-degree-of-freedom geared fivebar linkage m I FPGm 2 is obtained which will trace the prescribed path of P withcorresponding prescribed input-crank rotations Uj . For each motion generator there

p

\I Z2 \

z' \I \irz: O:'j

"'j \

\

---J~

~o

Idl er Gear

Figure 3.22 Th e I: I gear rat io between ZI and Z2 of this single-degree-of-freedom gearedfive-bar assures that point P, the joint of WI and W2, will trace the same path as theparent linkage , the four-bar motion generator shown in dashed lines.

p

212

Figure 3.23 Same as the geared five-bar configuration of Fig. 3.22, but the gears havebeen replaced by chain and sprockets.

p

m'

T

Figure 3.24 Parallelogram linkage m 1m2'fS assures I : I velocity ratio betweenlinks Zl and Z' of the five-bar path generator. See Fig. 3.22 for the parentlinkage, a four-bar motion generator.

Sec. 3.10 Further Extensions of Burmester Theory

p

213

Figure 3.25 Same five-bar path generator as in Fig. 3.24 with an addedparallelogram linkage to avoid branching or binding at dead-center ofthe original parallelogram.

will be one geared five-bar, in which either grounded link can serve as the input.Another way to design this linkage is shown in Fig. 3.23, where the gearing is replacedby a chain or timing belt and two equal sprockets.

There is yet another way for converting the two-degree-of-freedom five-bar ofFig. 3.22 to a single degree of freedom besides using gears or sprockets with chainsor timing belts. The same objective can be accomplished by adding a parallelogramlinkage to the five-bar as shown in Fig. 3.24. Notice that the parallelogram connectedto Zl and Z2 is not unique . In fact, two parallelograms may be connected togetherto avoid the dead-center problem (Fig. 3.25). (This is another overdosed linkagewhose mobility is assured by its link proportions.)

The five-bar m 1FPGm 2 may be connected by gears of other than I: 1 ratio,but this would require combining two separate dyad solutions-both with the same8j but with different aj, where one aj would be proportional to the other.

Six-Bar Parallel Motion Generator

An extremely useful linkage is one that will trace a coupler curve while the couplerlink undergoes no rotation-a parallel motion (curvilinear translation) generator.One can easily observe that this is an inappropriate task for a four-bar linkage exceptin the trivial case of a circular coupler curve of a parallelogram linkage. The followingextension of the Roberts-Chebyshev construction yields a six-bar linkage with onelink performing curvilinear translation.

Begin by drawing the initial motion generator four-bar linkage mlk~Pklm2,

__-,..,H '

IIIIIIIIIIII

, I<, I

~G'-,

""-,-,

"-,-,~

m2 '

H

C -----jf~-- / I~ // I

/ I// I

/ I// I

/ IIIIII

, I

'-c\G-,

-,-,

""-,

":-...

Figure 3.26 Six-bar parallel-motion generato r (solid lines) derived from the parent linkage (four-bar motiongenerator mlkIPk~m2) and its right cognate (m 2GPHC). P and P' describe identical paths. If linkm 2' G' is added, a seven-bar (overdosed) parallel-motion generator with prescribed timing results becauselink m 2'G' performs the prescribed rotations aj .

214

Sec. 3.10 Further Extensions of Burmester Theory 215

whose point P traces the prescribed path (Fig. 3.19). Next , draw one of its cognates,say the right dashed cognate m 2GPHC. Now duplicate this cognate by moving itparallel to itself so that point C is coincident with m 1 (Fig . 3.26). This yields thefour-bar linkage C' H' P' G' and m2' . From past discussions we know that link CHrotates by f3}, as does m1kl . Since C'H' (same rotations as CH) and m1kl bothrotate identically, triangle m1klH' may be rigidly connected. Notice that both Pand P' trace the same path-thus PP' may be connected by a rigid link. This linkwill move parallel to itself while both P and P' trace out the prescribed path.Since we were able to form triangle m1klH', a single-degree-of-freedom linkage existswithout the need for links G' P'. G'H'. and m 2' G' (shown dashed in Fig. 3.26).The six-bar parallel motion generator is composed of the initial four-bar mlklPk~m2plus m1H'P'P.

Instead of using the right cognate as shown in Fig. 3.26, we could have usedthe left cognate . This would yield another six-bar parallel motion generator followinga similar construction. Thus, for every motion generator. there are two six-bar parallelmotion generators.

A closer look at these six-bars may yield some disappointment. The prescribedrotations aj are seemingly lost-neither the coupler link PP' nor the grounded linksrotate by the originally prescribed angles aj . However, by adding the dashed portionH' G' P' M2' to the solid links in Fig. 3.26, we obtain a seven-bar parallel motiongenerator with prescribed tim ing (an overclosed mechanism) where M2'G' rotatesby the prescribed a j rotations.

There are three additional useful observations to be made about Fig. 3.26:1. The parallel motion linkages derived from a four-bar motion generator have

a special quality: One can prescribe a combined "moving function generator" andparallel motion generator. This can be seen by observing that in Fig. 3.26 linkP P' does not rotate, while PKlk~ rotates by the prescribed angles ai. Thus therelative rotations between PP' and klPk~ are prescribed. One of the applicationsof this would be a flying shear, where the object to be cut moves along the prescribedpath and is supported by link PP' . Meanwhile a blade connected to klPk~ cutsthe object "on the fly."

2. Since the prescribed rotations aj are not a factor in the parallel-motiongenerating quality of the six-bar, one could find more six-bars by varying the choicesfor aj . Therefore, there are an infinite number of six-bar parallel motion generatorsthat will hit the prescribed precision points along the path of P.

3. Observe that the prescribed rotations aj are the rotations of one of thegrounded links in each cognate of the original motion generator. To make use ofthese prescribed rotations, the six-bar parallel motion construction technique shouldbe applied to the cognates rather than the parent motion generator. This will yieldtwo different six-bars per cognate for a total of four six-bar parallel motion generatorswith prescribed timing for every motion-generator four-bar.

Table 3.3 indicates the number of solutions expected for all the extensions ofthe Roberts-Chebyshev constructions described above. One can see that one motiongenerator four-bar breeds numerous useful offspring.


TABLE 3.3 MECHANISMS SYNTHESIZED BY WAY OF BURMESTER POINT PAIRSOBTAINED USING THE DYADIC STANDARD-FORM EQUATION AND EXTENDING THERESULTS ON THE BASIS OF THE ROBERT8-CHEBYSHEV THEOREM.

Four-barpath Geared

Four-bar generators five-barNumber of motion with prescribed Four-bar path generatorsreal roots generators timing function with prescribed

of the quart ic (Fig. 3.19) (Fig. 3.19) generators timing"equation (3.47) (The Parent Linkage) (Cognates) (Fig. 3.21) (Fig. 3.22)

0 0 0 - 02 1 2 1 14 6 12 3 6

a Several configurations possible (see Figs. 3.22 and 3.23).

3.11 SYNTHESIS OF MULTILOOP LINKAGE MECHANISMS

The dyad synthesis approach may be used to synthesize virtually all planar mechanisms. This was suggested in Chap. 2, where the standard-form equations werederived for the Stephenson III six-bar (see Fig. 2.66).

Why try to design a multiloop linkage such as the Stephenson III six-bar whenthe four-bar can do so much? If only three positions are required, the answer tothis question is that in most cases we do not need more than the four-bar chain.In the three-position case all coefficients in the dyad synthesis equations are eitherspecified or picked arbitrarily, so that even motion generation with prescribed timingis possible. Also, there are two infinities of solutions to inspect-usually more thanenough to find a "good" solution if the motion requirements are "well behaved. t'"

In the jour-position case, multiloop linkages become more attractive for severalreasons: (l) motion generation with prescribed timing is no longer possible with thefour-bar linkage; (2) even an infinite number of solutions which one can expect fromfour prescribed positions may not produce a suitable four-bar, especially if enoughrequirements are imposed on the final linkage (e.g., ground-pivot locations) ; and (3)multiloop linkages can exhibit more complex motion than four-bar linkages sincecoupler plane motions are no longer restricted by the requirement of two points tofollow either circular arcs or straight lines (except the Watt II six-bar, whichconsists of two four-bar chains). In the jive-prescribed-position case, multi looplinkages present a valuable alternative since at best there are only a finite numberof four-bar solutions.

Kinematic loops consisting of five, six, seven or more bars may be synthesizedfor more than five prescribed positions. Recall that in the Stephenson III sixbar of Fig. 2.66, the loop containing Zs. Z4. and Z3 has a loop closure equationZ s(e i1/Jj - 1) + Z4(e i/lj - 1) - Z3(ei'>'j - 1) = 8j [Eq. (2.31)]. This loop was

• No sudden changes in direction, velocity, and acceleration.

Sec. 3.11 Synthesis of Multiloop Linkage Mechanisms 217

GearedSix-bar five-bar

parallelogram combined path Six-bar parallel Seven-bar parallelpath generators and motion motion generatorwith prescribed constant-velocity- generators Six-bar parallel with prescribed

timingb ratio generation' with prescribed motion generationd timing(Fig. 3.24) (Fig. 3.22, but n '" 1) timing (Fig. 3.26) (Fig. 3.26)

0 0 0 0 01 4 4 2 26 16 24 12 12

b The basic five-bar of Fig. 3.24 (m 1FPGm2) remaining the same but any parallelogram m1STm2 may be added tocomplete the linkage. (See Fig. 3.25 for a seven-bar version which avoids the dead-center problem.)

'Requires two runs of the program, both prescribing the same path displacement vectors 8j but different Uj suchthat uj = n(uj), where n is any rational number. (The number of solutions in each row assumes that both quarticshave the same number of real roots.)

d By varying the prescribed Uj, an infinite number of sets of 2 or 12 new solutions may be generated.

analyzed in Table 2.2, which showed that these vectors could be synthesized for amaximum of seven positions. Since the rest of the mechanism could be designedfor only five positions, there is little reason to require seven from just one loop.In the great majority of design situations five precision points are sufficient. Ofmore importance perhaps is to make proper use for other purposes of the extrafree choices that occur in multiloop linkage synthesis.

In order to synthesize the triad Z3Z4ZS in the Stephenson III six-bar of Fig .2.66a for five precision points, Table 2.2 tells us that we must make free choices oftwo unknown reals. If we choose the vector Z3, the standard form will be achieved:

where

8} = 8j + Z 3(eiYj - I)

The other loops were described by (see Fig . 2.66b)

Zl(e i!f>j - 1) + Z2(e iYj - 1) = 8j

'4,(e iOj - 1) + Z7(e itlj - 1) = 8j

(3.48)

(3.49)

(3.50)

What tasks can we ask of this linkage? Two major tasks are evident afterinspection of the three equations above (recalling that the 8's and one set of therotations are prescribed for the standard form): (1) combined path andfunction generation (the path of point P and the rotations <pj and IjJj or OJ), and (2) motion generationwith prescribed timing (the path of point P and the rotations Y) and IjJj or OJ).

Besides the greater usefulness of this linkage for the designer, another "nice"by-product of this procedure for design is the free choice of Z3. This vector, whichforms the rest of the coupler plane once Zl and Z2 are synthesized, can be used


advantageously in three ways: (1) the shape of the coupler plane may be picked bythe linkage designer; (2) by choosing Z3 in a certain direction, the designer mayinfluence the form of the rest of the linkage: Zs, Z4, Z6' and Z7 (e.g., influence theresulting ground-pivot locations) ; or (3) by simply varying Z3' the designer may generate a larger number of solutions. With regard to the last observation, how manypossible solutions for five prescribed positions could we expect from either of thetwo combined tasks mentioned above? Equation (3.49), for j = 2, 3, 4, 5, yieldsup to four solutions ; Z3 may be varied between -00 and +00 in both x and y directions,and the four-bar left over (which includes Zs, Z6, Z4' and Z7) has up to 12 solutionsfor each value of Z3. Thus there are many infinities of solutions for five-precisionpoint synthesis of the Stephenson III six-bar mechanism of Fig. 2.66.

Example 3.2 [213]

The Stephenson III type of six-bar of Fig. 2.66a is to be synthesized as a combinedapproximate path and function generator for five prescribed positions . The path is anapproximate straight line while the function to be approximated is y = x 2• 1 ~ x ~

3, with the range in input ~</> = 40° and output ~I\I = 90°. The prescribed precisionpoints are

82= 0.7 - 0.5i.

83 = 1.5 - 1.1i.

</>2 = 10°,

</>3 =20°,

</>4 = 30° ,

</>s =40°,

84 = 2.55 - 1.8i

85= 3.6 - 2.6i

1\12= 14.06°

1\13 = 33.75°

1\14 = 59.06°

1\15 = 90.00°

Figure 3.27 shows one of the resulting linkage mechanisms in its first and fifth prescribedpositions, and Fig. 3.28 shows the second , third, and fourth positions of this same mechanism.

3.12 APPLICATIONS OF DUAL-PURPOSE MULTILOOPMECHANISMS

Multiloop mechanisms have numerous applications in assembly line operations.For example, in a soap-bar-wrapping process, where a piece of thin cardboard mustbe fed between rollers which initiate the wrapping operation, an eight-link mechanismis employed such as that shown in Fig. 3.29.* This mechanism is a combined functionand motion generator with prescribed timing. The motion of link Z3 is prescribedin order to pick up one card from a gravitation feeder (the suction cups mountedon the coupler must approach and depart from the card in the vertical direction)and insert the card between the rollers (the card is fed in a horizontal direction).The input timing is prescribed in such a fashion that the cups pick up the card

• This application was brought to the authors' attention by Delbert Tesar of the University ofFlorida.

Sec. 3.12 Applicat ions of Dual-Purpose Multiloop Mechanisms 219

iY

x Second Position

Figure 3.27 Synthesized six-bar simultaneous path and function generator of Ex. 3.2 in initial and finalprescribed positions.

Third Position Fourth Position

Figure 3.28 Intermediate positions of the mechanism of Fig. 3.27.


I Gravity Feedt for Card s

Chap. 3

8

Figure 3.29 Practical application of multiloop mechanism: wrapping-card feeder insoap packaging . Suction cups on link Z3take hold of the card and feed it betweenthe forwarding rollers at the right.

during a dwell period and release the card in a position and at a velocity that assuresthat it is fed into the rollers at approximately the same speed as the tangential velocityof the rollers.

In this example, there is a one-to-one functional relationship between the rotations of links Zl and Z4 since they are shown geared together with gears of equalpitch radii. The motion of link Z3 is prescribed by assigning the path of the tipsof Z2 and Zs (or Z6). Since Z3 is a rigid link, the distance between the tips of Z2and Zs must remain the same. In fact, we are free to choose the length and initialorientation of Z3. Thus the loop-closure equations are

Zl(eiq,j - 1) + Z2(e i'Yj - 1) = 8j

Z4(e iq,j - 1) + Zs(e i13j - 1) = 8j

Z7(e il/Jj - 1) + Z6(e i13j - 1) = 8j

(3.51)

(3.52)

(3.53)

This mechanism may be synthesized for five prescribed positions in two steps.First synthesize the right-side dyad by utilizing Eq. (3.51) (up to four possible solutions). Second, utilizing the four-bar generator option of the LINeAGES program,synthesize the left four-bar [Eqs. (3.52) and (3.53)] for path generation with prescribedtiming (up to 12 possible solutions). Thus, in general, we may expect up to 48solutions.

Sec. 3.12 Applications of Dual-Purpose Multiloop Mechanisms 221

Watt II Examples

One way to illustrate the tremendous number of multiloop linkage applications isto point out several instances in which Watt II six-bars are employed. The standardform equations for this linkage are easily found by recognizing that the Watt II isjust two four-bars connected together. Therefore, one simply synthesizes two fourbar function generators, making sure that the output link of the first and the inputlink of the second rotate by the same angles (see Ex. 2.5). For example, Rain [119]describes the need for a six-link mechanism for large angular output oscillation.He states: "It would be very difficult to solve this problem with one four-bar linkage,because it is difficult to design a four-bar linkage having such a large oscillation ofa crank without running into problems of poor transmission-angle characteristics; itmight be possible to use linkages in combinations with gears, but this would makethe mechanism more expensive, less efficient, and probably noisier." This statementof Rain's provides strong motivation for developing the kinematic synthesis of gearlessmultilink mechanisms. Figure 3.30 shows an agitator mechanism used in certainwashing machines . Certainly, Rain's advice would be seconded by the designer ofthis device. This Watt II six-bar has approximately 1500 of rotation on the outputlink.

Rain also cites another application which could be very nicely fulfilled by aWatt II six-link mechanism. A feeding mechanism (see Fig. 3.31) is required totransfer cylindrical parts from a hopper to a chute for further machining. A combinedpath and function generator will be an ideal solution to this problem. The Watt IIsix-link mechanism may be synthesized for this task. A schematic configuration ofthis linkage is shown in Fig. 3.31. Link 6 provides the rotating cupped platform

Input

Figure 3.30 Watt II six-link washing-m achine agitator mechani sm with crank AoA.coupler no. lAB, bellcrank BBoB' , coupler no. 2 B'C and rocker Coe. The latteroscillates 1500

•

In put

Figure 3.31 Schemat ic drawing of feeder mechani sm (a Wall II six-bar simultaneous path and functiongenerator with prescribed timing) .

(whose rotations are prescribed functions with respect to the input link) which transfersthe cylinder from the hopper to the chute while the prescribed path of the coupler(point P) positions the cylinder on the platform and then pushes the cylinder intothe hopper (see points P, and P2 in Fig. 3.31).

The following three examples were described by Kramer and Sandor [164].They suggested the design of an automobile throttle linkage whereby the followingangular positions must be coordinated:

Gas pedal movement Throttle opening

0° Closed5° 14°10° 28°15° 44°20° (to the floor) (Wide open) 60°

A six-bar is to be used instead of a four-bar, due to the space that the enginetakes up in the engine compartment. The required positions of the fixed pivots forthis Watt II linkage are shown in Fig. 3.32. The four-bar function generator sublinkages are first synthesized for prescribed input and output angles and then stretchrotated so that their ground pivots match these locations. Since the rotation of theintermediate bell crank is of minor importance, it is arbitrarily chosen to be alongthe function y = l.4x. For this choice, a final solution is shown in Fig. 3.32,while an analysis confirmed that the transmission angles vary from 560 to 900 and

222

Figure 3.32 In this example, the rotations of the accelerator pedal are to be coordinatedwith those of the carburetor throttle valve. The motion of the intermediate crank isnot of primary importance; its arbitrary choice can be used to influence the design asto proportions and transmission angles.

the maximum error of bell-crank rotations between precision points is 1.001°.If a more accurate solution is desired, another choice for the location of the fixedpivots and/or the rotation of the intermediate crank would be suggested.

In designing a linkage for the IBM Selectric typewriter, the printer elementneeds to be tilted a specified amount and the velocity and acceleration are requiredto be about the same in the vicinity of each precision point. The "tilt tape system"(Fig . 3.33) transforms a linear pull of the tilt latch to a rotation of the tilt bell

Figure 3.33 In the Watt II six-bar tilting mechanism for the type ball of the IBM Selectric typewriter,by pulling the tilt-2 latch, the tilt bell crank is made to rotate counterclockwise. The tilt arm is forcedto oscillate to the left, thus rotating the tilt pulley. (From Ref. 164.)

223


crank. The bell crank in turn rotates the tilt arm, which is connected to the printerelement by way of the tilt tape and tilt pulley. The tilt latch is to pull the tiltlever down from the starting position: 2.5, 5.0, 7.5, and 10.0°. The rotation of thebell crank may be arbitrarily chosen so that a solution results in acceptable velocityand acceleration values as well as linkage dimen sions. The tilt pulley is connectedto the tilt arm in such a way that the arm must rotate: 3.5, 7.1, 10.0, and 14.0°.After several unsuccessful passes with the computer-aided design and analysis programs , in a third run the function y = x 3 was prescribed for the bell crank. Thetransmission angles were excellent and the maximum error of bell-crank rotationsbetween precision points was 0.005°. A larger but similar mechanism is used torotate the printer element, and the two linkages are used in conjunction to give thetypehead over 80 print positions.

The control of the heating ducts in a compact car by way of a Watt II sixbar mechanism is shown in Fig. 3.34. The input crank Z'2 controlled by the drivermust rotate enough to open the valve Z2 on top while closing the flap Z/4 on thebottom (right), whereby the circulation of air is directed through the tubes leadinginto the cabin instead of through the exit passages at the rear of the vehicle. Forproper control of the air flow we must coordinate the following rotations:

Top valve Driver control Rear flap

(Closed) (Rest position) (Open)1P 12° 10°23° 24° 10°34° 36° 30°45° (open) 50° 40° (closed)

After synthesizing the two four-bar linkage mechanisms for function generation,they are stretched and rotated to match the fixed links: Zl = 2.800 - 9.oooi andZ'l = 9.000 - 1.5ooi. The resulting Watt II linkage is, for the first four-bar:

Z2 = +2.025 - 0.517i = top valve link

Z3 = +2.580 - 8.67li = coupler link

Z4 = -1.805 + 0.188i = driver-control link (lower branch of bell crank)

Zl = +2.800 - 9.oooi - fixed link

and for the second four-bar:

Z /2 = +0.368 + 1.420i = input link (upper branch of bell crank)

Z/3 = 9.432 - 1.409i = coupler link

Z'4 = 0.800 - 1.51li = output link

Z'l = +9.000 - 1.5ooi = fixed link

Sec. 3.12 Applications of Dual-Purpose Multiloop Mechanisms

(a)

225

z;

(b)

Figure 3.34 Watt II six-bar mechanism controls the airflow in the heating systemof an imported car : (a) original design, (b) proposed Watt II design. (FromRef. 164.)

The foregoing design examples illustrate a few of the many applications ofthe Watt II mechanism.

Case Study: Application of the Five-Precision-PointSynthesis in an Industrial Situation [91]

A linkage synthesis problem arose in building a machine for the assembly of a connector(which is used in the installation of telephones) shown in Fig. 3.35. Five metalclips are to be automatically inserted into the five slots in the plastic base of theconnector. The first attempt at building a production machine for this project used


• M"," C1 ;'~IIIIIIIIIIJII

Chap. 3

Figure 3.35 Telephone connector and metal clips. Five metal clips are to be inserted in thefive slots of the plastic base. (Courtesy of the 3M Company.)

five clip insertion heads which were fed by five separate feed bowls (vibratory feeders;see Sec. 5.6) positioned in the same configuration as the slots . The five heads insertedall clips simultaneously into the base, which was fixed to ground. Because of unreliableperformance of the insertion heads, considerable downtime resulted when anyoneof the insertion heads malfunctioned and the entire machine had to be shut downfor repair.

Rather than fixing the base to ground, a mechanism was sought to repositionthe base in each of the five desired positions under a single insertion head. Thetelephone connector would be indexed through the five positions necessary for onehead to insert all clips. Because of the simplicity of linkages as compared to othertypes of mechanisms for motion generation, a linkage was sought to move the telephoneconnector.

The first step in designing the motion generation linkage was to determinethe number of links needed to solve the problem. The four-bar was the first linkagethat was synthesized. All six solution linkages were obtained based on four real


roots to Eq. (3.47) . When these linkages were evaluated, however, it was foundthat none of them was the crank-and-rocker or double-rocker type which could easilybe built into a production machine.

Thus it became evident that a more complex linkage was needed to solve theproblem. Inasmuch as a multiloop mechanism would be sought, it was decided toplace two additional requirements on the linkage: (1) the input crank angles corresponding to the precision positions of the motion generator link would be equallyspaced and, (2) the crank should be capable of 360 0 rotation. These requirementswould simplify the indexing mechanism needed to drive this mechanism.

The eight-bar linkage type shown in Fig . 3.36 was chosen to solve this problem,although many six-bars or seven-bars could also have been considered.

The vectors describing the linkage in position 1, Zl through Zll' are shownin Fig. 3.37. Of these vectors Z3' Zs, and Zll are chosen arbitrarily by the designer.

A

Figure 3.36 Eight-bar linkage chosento successively move the plastic baseof Fig. 3.35 to the five positions, placingthe five slots in sequence under onemetal-clip-inserting head.

Figure 3.37 Vector representation ofthe linkage of Fig. 3.36 in position 1showing rotations to position j.Displacements OJ and rotations ()3j areprescribed (see Table 3.4).


When the crank rotates from position I to position j, the links rotate through angles()kj, k = 1 through 7. The standard-form equations for the eight-bar linkage are

(3.54)

(3.55)

(3.56)

(3.57)

where j = 2, 3, 4, 5. Equation systems (3.54) to (3.57) form the set of synthesisequations which are to be solved in order to obtain the desired linkage. The given(or known from having solved a previous equation), unknown, and designer-specifiedquantities for the synthesis equations are tabulated in Table 3.4.

TABLE 3.4 STRATEGY FOR SOLVING EQUATION SYSTEMS 3.54 TO 3.57 INSEQUENCE.

Given or knownEquation system from previously

number solved systems Unknown Designer-specified

(3.54) 8j , (J3). (Jlj Z" Z2, (J2j Z3

(3.55) 8» (J3j, (J2j Z., Z5' (J4j Z3

(3.56) 8), (J3j, (J6j or (J1j Zg, ZIO' (J1j or (J6j ZlI

(3.57) «; (J6j z, Z1' (JOj Z6

j= 2,3,4,5 j = 2,3,4,5

One slight problem exists with this procedure, however. When the last loops(Z6 and Z7) are synthesized, there is no guarantee that the ground pivots of the Z6'Z7' Zs loop will match up with the ground pivots already fixed by the base of Zsand Z9. Fortunately, since Z6' Z7' Zs is a function-generating loop, it can be stretchedand/or rotated without affecting the rotations ()4 j, ()Sj, ()6j . Therefore, the synthesizedvectors Z6, Z7 as well as Zs are simply stretched uniformly and rotated to fit thegap between the base of Zs and the base of ~.

After an eight-bar linkage has been synthesized by the procedure outlined above,it must be analyzed to determine if it has acceptable transmission angles throughout


its entire crank rotation. At the time when this project was accomplished an analysisprogram was written for this purpose which analytically stepped the crank throughincremental rotations and printed out the transmission angles at each step.Unfortunately, no interactive graphics hardware was available, so the synthesisanalysis process was time consuming.

A linkage thus designed is shown schematically in Fig. 3.38 together with thetelephone connector. Although this linkage has not been optimized, it is certainlyoperational. Its minimum transmission angle is 27° and its maximum link-lengthratio is 9.5. Once the necessary programming had been done, about 150 hours oftrial-and-error design effort were used to obtain this linkage. A model of the linkagewas built to demonstrate its motion. The final design is shown in Fig. 3.39. Interactivecomputer graphics displays of a synthesized linkage with appropriate analysis optionswould have significantly reduced the time of the trial-and-error design steps. Anonanalytical solution for this problem would have been very difficult.

F

Figure 3.38 Schematic of an eight-link mechan ism synthesized according to Fig.3.37, Eqs. (3.54) to (3.57), and Table 3.4. Bell crank EFD is input.


J

Figure 3.39 Scale draw ing of solution linkage synthesized according to Fig. 3.37, Eq. systems (3.54) to(3.57), and Table 3.4.

3.13 KINEMATIC SYNTHESIS OF GEARED LINKAGES

Planar geared linkages readily lend themselves to function, path, and motion generation. Function generation includes any problems in which rotations or sliding motionof input and output elements (either links, racks, or gears) must be correlated.

Sec. 3.13 Kinematic Synthesis of Geared Linkages 231

B

Figure 3.40 Geared five-bar dwellmechan ism.

For example, a frequently encountered industrial problem involves the generationof intermittent or nonuniform motions. Simple linkages, such as the four-bar draglink mechanisms, are usually the least complicated and most satisfactory devicesfor such tasks . However, when the desired motion is too complex to produce witha four-bar linkage or a slider-crank mechanism, a geared linkage can often be designedto fulfill the design requirements economically.

In a packaging machine, it may be necessary to connect an input shaft andan output shaft so that the output shaft oscillates with a prescribed dwell periodand timing while the input shaft rotates continuously. A simple geared five-barmechanism, such as that shown in Fig. 3.40, can be readily designed to producesuch a dwell period [45]'·

In the case shown, the gear ratios of the cycloidal crank AoA I A have beenchosen to produce a hypocycloid of four cusps (an astroid). The path of point Aon the pitch circle of the planet gear between cusps is approximately an arc of acircle, and the length of the coupler AB is equal to the radius of that arc . Thefollower link BoB has been arranged so that in its extreme right position its movingpin B coincides with the center of the arc . If desired, the coupler could be attachedto an output slider rather than to a rocker. Still different mot ion characteristicscould be obtained through the use of a three- or five-cusp hypocycloid.

Degrees of Freedom

Mobility of a planar geared linkage can be studied through the use of Eq. (1.3) (seeTable 1.1):

F = 3(n - I) - 2f1 - If2

where F is the number of degrees of freedom of the linkage, n is the number oflinks, f1 is the number of joints that constrain two degrees of freedom (revoluteand slider joints), and f2 is the number of joints that constrain one degree of freedom

• See also Prob. 3.49.


6,

Gear rat ios:

TA

TB2 Te,T=r2, T=r3 , T =r4

B, c , D

where TA is the number of teethin gear A, and so on.

Figure 3.41 Geared five-bar function generator linkage. Gear A is stationary, compound gears Band C are pivoted at linkjoints, and gear D is rigidly attached tothe output link. Gear ratios: TA ITa 1= rs,Ta2 I Tc l = rs, TC I l TD = r. , whereTA is the number of teeth in gear A, andso on.

(gear meshes in this case). If the gears are attached rigidly to p links, then , ingeneral, 12 = pl2. For instance, the mechanism of Fig. 3.41 has five bars and atrain of four gears. The gears are fixed at two points, to the frame and to theoutput crank. Thus n = 5, P = 2,11 = 5, and 12 = 1, so the mechanism has asingle degree of freedom. Here we tacitly recognized that gears Band C are idlersand therefore do not participate in the degrees-of-freedom computation when thequantity b is defined as given above. However, we could also regard the idlers asseparate links, in which case n = 7,11 = 7, and the number of gear meshes b =3. Since each 12 subtracts one freedom of rotation, the result is again a single degreeof freedom for the mechanism.

Synthesis Equations in Complex Numbers

The method of complex numbers is particularly well suited to the synthesis of gearedlinkages because links and gear ratios between links are readily represented and manipulated mathematically. When a limited number of precision conditions is imposedon a linkage, the method provides synthesis equations which are linear in the unknownlink vectors describing the mechanism in its starting configuration. In function generation with a single-loop mechanism, a linear solution can be obtained for one fewerprecision conditions than the number of bars in the loop. For example, in the caseof a geared five-bar (such as that shown in Fig. 3.41), a linear solution can be obtainedfor up to four first-order (finitely separated) precision points or for, say, two firstorder and one second-order precision point (the latter equivalent to two infinitesimallyclose precision points) (see Sec. 2.24). One can prescribe more than four precisionconditions for this mechanism, but the solution is made more difficult because someof the coefficients of the link vectors must then be treated as unknowns. Nonlinearcompatibility equations must then be solved.

For finite synthesis, vector loop equations and displacement equations writtenin terms of complex numbers form the system of synthesis equations. In higherorder synthesis involving prescribed derivatives, velocities, accelerations, and higher


accelerations, derivatives of the loop equations are taken with respect to a referencevariable or time . In both finite and infinitesimal synthesis, the system of synthesisequations can be made linear in the unknown link vectors . The coefficients wouldcontain the prescribed performance parameters and the gearing velocity ratios .These can be arbitrarily assigned convenient values by the mechanism designer.By varying these arbitrary choices the designer can obtain an infinite spectrum ofsolutions from which to select a suitable mechanism. All of these solutions willsatisfy the given precision conditions. Selection of the best available solution canbe based on such optimization criteria as most favorable transmission angles, bestgear ratios, or ratio between longest and shortest link lengths closest to unity .These criteria can be used either singly or in weighted combinations.

Geared Five-Bar Example [248]

Suppose that one wishes to synthesize a geared linkage of the type shown in Fig.3.41 to generate a function y = f(x) over some given range. Let the rotation ofthe input crank (<1» be the linear analog of x and the rotation of the output link(1jJ) be the linear analog of y.

One can represent this mechanism by a closed vector pentagon (Fig. 3.42).In the reference position of the mechanism, vectors Zh Z2' Za, Z4' and Zs definethe orientation and length of links 1, 2, 3, 4, and 5. In a general displaced positionof the mechanism, say the jth position, at which requirements on the motion havebeen prescribed, the mechanism is defined by these vectors multiplied by their appropriate rotation operators:

ia, =Zl

Z'2 = eN' i Z 2

(3.58)

Figure 3.42 Vector schematic ofgeared five-bar mechanism of Fig.3.41. Note that gear A is fixed tovector 1, which represents theframe, and gear 0 is fixed to vector5, which represents the output link.Compound gears Band C are freeto rotate.


(3.59)

A mathematical relationship must be developed to express the relative, andfinally the absolute, rotations of the various links. In other words, for a specifiedrotation of the input crank (link 2), can an expression for the rotation of some otherlink be found in terms of this input rotation and the gear ratios? Obviously, theremust be some such relationship since the mechanism has but one degree of freedom.

Determining the Effect of Gear Ratios on the Rotation ofthe Links

In reference to Fig. 3.43, the following general relationship (see Chap. 6 of Vol. 1)will prove very helpful in solving this rotation problem:

Tk - 1<Pj(k+ll = <Pjk + [<Pjk - <Pj(k-ll]~

~ k+l

where, for example, <Pjk is a finite rotation of the kth link from the first position ofthat link to its jth position. Tk refers to the number of teeth on the gear rigidlyattached to the kth link. For the linkage in Fig. 3.42 let <[.B and -j:C be the absoluterotations of gears Band C for any given input rotation <pj, measured from thestarting position of Z2. 'Y and fJ- are absolute rotations of links Za and Z4 fromtheir respective starting positions. Similarly, I\J is the rotation of link Zs, the output.Considering now each gear-pair separately, from Fig. 3.44, omitting the positionsubscript j, we have,

(3.60)

<!lj(k+1)

Figure 3.43 General geared pairshowing notation for the link rotations.

Figure 3.44 Input side of thegeared five-bar of Fig. 3.41.


Figure 3.45 Intermediate gears and floating link Z3 in the geared five-bar of Fig.3.41.

4 C

Figure 3.46 Output side of the geared fivebar of Fig. 3.41.

Continuing around the loop of the mechanism of Fig . 3.42, applying the generalrelationship to Fig. 3.45 yields

1 C = y + (y -1 B)'3

, where '3 = TB 2/Tc 1

Finally, applying the general relationship to Fig . 3.46, we obtain

, where '4 = Tc 2/Tn

Substituting Eq. (3.60) into Eq. (3.61) yields

1 C = y + [y - ('2 + 1)<{>]'3

1 C = y + y'3 - '2'3<{> - '3<{>

and substituting Eq. (3.63) in Eq. (3.62) gives

ljJ = p.+ [p.- (y + Y'3 - '2'3<{> - ' 3<{»]'4

=p.+p.~-y~-y~~+<{>~~~+<{>~~

and

(3.61)

(3.62)

(3.63)

(3.64)

So it can be seen that there is a direct relationship between ljJ, <{>. p., y and thegear ratios. As expected, the absolute rotations 1 Band 1 C of the idlers do notappear in this expression, but the gear ratios do.

Since <{> and ljJ are prescribed in accordance with the function to be generated,if values of yare assumed, then corresponding values of p. are defined by Eq. (3.64).

For convenience, let

Q = 1+ '4, (3.65)

Therefore,

ljJ = p.Q - yR + <{>S

1P. = Q (ljJ + Y R - <{>S)

(3.66)

(3.67)


Determining the Number of Precision Positions for Whichthe Mechanism of Fig. 3.42 Can be Synthesized

For the mechanism of Fig. 3.42 in the jth position , the vector equation of closurecan be written as follows:

(3.68)

Note that Zl is assigned the value I for convenience. Thi s is permissible becauseonly angular relationships are of interest (see Sec. 2.23).

Recall that the </>'s and ljJ's are prescribed and some of the y's may be chosenarbitrarily, in which case the JL's are known through Eq. (3.67).

Table 3.5 can now be constructed.

TABLE 3.5 NUMBER OF POSITIONS FOR WHICH THE GEARED FIVE-BAR MECHANISMOF FIG. 3.42 CAN BE SYNTHESIZED FOR PRESCRIBED INPUT AND OUTPUTROTAT~ONS, ONCE THE GEAR RATIOS ARE SPECIFIED

Number of Number ofindependent independent

Number of real equations unknown reals Arbitrary Numberprescribed [excluding [excluding ,... choices ofpositions Eq. (3.67)) owing to Eq. (3.67)) of reals solutions

1 2 Z2, Z3' Z., z, (8) 6, say Z2.•.• 0(00)62 4 Z2' Z3' Z., z, 'Y2 (9) 5, say Z2.3. 'Y2 O(oo)s3 6 Z2, Z., Z., z, 'Y2, 'Y3 (10) 4, say Z2' 'Y2.3 0(00)'4 8 Z2' Z3' Z., z, 'Y2, 'Y3, 'Y. (11) 3, say 'Y2.3.• 0(00)35 10 Z2' Z3' Z., z; 'Y2, 'Y3 , 'Y., 'Ys (12) 2, say, 'Y2.3 0(00)26 12 Z2, Z3' Z., z; 'Y2 to 'Y6 (13) 1, say, 'Y2 0(00)17 14 Z2' Z3' Z., z, 'Y2 to 'Y7 (14) 0 Finite

Up to four precision positions, the designer can pick all the y 's arbitrarily. Forfive, six, or seven positions , only some or none of the y's can be picked, and nonlinearcompatibility relationships must be solved for the remaining unknown y's. Thus itcan be seen that the limiting number of precision positions , beyond which nonlinearcompatibility equations become necessary in the solution, is four. For predeterminedgear ratios and scale factors , the mechanism may be synthesized for up to sevenpositions . However, if the gear ratios and scale factors are also regarded as unknowns,the number of attainable precision points can be further increased.

For four finitely separated first-order (finitely separated) precision points thesynthesis equations for this mechanism, written in matrix form, are

(3.69)


As all the quantities in the coefficient matrix are either prescribed or arbitrarilyassumed, one can easily solve this linear system of complex equations for the fourcomplex unknowns, Zz, Za, Z4' and Zs. Varying the arbitrary values 'Yz, 'Ya, and'Y4 and varying the choice of gear ratios and scale factors allows one to obtain aninfinite spectrum of solutions.

Example 3.3

Suppose that the function to be generated is y = tan (x), 0 ~ x ~ 45° . For fouraccuracy points with Chebyshev spacing, values of x and y can be found (see Sec.2.2):

X2= 13.89°,

x3=31.ll o,

Yl = 0.03

Y2 = 0.25

Y3 = 0.60

Y4 = 0.94

Let A<p = range of input crank = 90° and AIjJ = range of output = 90°. With these

<P2 = 24.36°,

<P3 = 58.80°,

<P4 = 83.16°,

1jJ2 = 19.80°

1jJ3 = 51.30°

1jJ4 = 81.90°

Figures 3.47,3.48, and 3.49 and Table 3.6 show some typical computer-synthesizedlinkages for generating the function y = tan (x). Note that the gears are not shown,but their effect is clearly in evidence: the gears transfer rotary motion directly , transmissionangles are of no interest between geared links.

Figure 3.47 Synthesized geared five-bar generating the tangent function . Example A ofTable 3.6 is shown in its first (solid), second(short dashed), third (uneven dashed), andfourth (long dashed) precision positions .Gears are not shown.

Figure 3.48 Example B of Table 3.6 is shown in its fourprecision positions (same notation as Fig. 3.47). Gearsare not shown .


Figure 3.49 Example C of Table 3.6 is shownin its four precision positions (same notat ion asFig. 3.47). Gears are not shown.

TABLE 3.6 THREE DIFFERENT GEARED FIVE-BAR DESIGNS SYNTHESIZED FORFOUR·PRECISION-POINT FUNCTION GENERATOR (See Fig. 3.42 and Table 3.5.)

Example A Example B Example C

Figure 3.47 3.48 3.49

Function y =tan x y =tan x y=tan xRange o ~ x s 45 ° o ~ x ~ 45 ° o ~ x s 45°

Scale factors t.cf> = 90 ° t.cf> = 90° t.cf> = 90°

t.1\1 = 90° t.1\1 = 90° t.1\1 = 90°

Gear ranos r, 3 3 3

' 3 0.5 0.5 0.5r, 0.5 0.5 0.5

Link vectorsZt 1.000, +O.OOOi 1.000, +O.OOOi 1.000, +O.OOOiZ. 0.402, -1.115i 1.335, +0 .027i 0.333 , - 1.126iZ3 -Q.709 , +O.475i -0.886, +0.846i -1.225, -0.482iZ. 1.714, -0.4686i 1.919, -Q.611i 2.029, +O.254iZ. -0.407, +1 .109i -1.366, -0.262i - 0. 137, +1 .354iz,

Arbitrary linkrotations 1'. = 20° 1'. = 0° "y. = 0°

1'3= 0° "Y3 = 0° 1'3= 20°1'. = 0° 1'. = 60° "y. = 40 °

Geared-Linkage Compatibility Equations

The system of equations for five-point function-generation synthesis of the gearedfive-bar of Fig. 3.42 is obtained by adding another equation in system (3.69). Thiswill yield the compatibility equation

ei</>2 eiY2 eill2 eio/J2

ei</>a eiYa eilla ei o/Ja

ei</>4 eiY4 eill4 eio/J4

e i</>s eiyS eillS eio/Js

With 'Y2 and 'Ys assumed arbitrarily, this expands to

=0 (3.70)

Sec. 3.14 Discussion of Multiply Separated Position Synthesis 239

(3.71)

where the A's are known. Equation (3.71) can therefore be solved geometricallyfor 'Y3 and 'Y4, as shown in Fig. 3.3 and Table 3.1 for /33 and /34' Then, with compatiblesets of 'Yio j = 2, 3, 4, 5, any four of the five equations such as Eq. (3.69) can besolved simultaneously for Zk, k = 2, 3, 4, 5.

In case of six-point synthesis, the five-column augmented matrix will have sixrows and therefore yield two compatibility equations, say, one consisting of the firstfour and the sixth rows, and another of the first three plus the last two rows.After assuming an arbitrary value for 'Y6, these will expand to

(3.72)

and

(3.73)

where all the A's are known. These are of the same form as Eqs. (3.34) and (3.35),and can be solved the same way for 'Yio j = 2, 3, 4, 5. Thus, with the assumedvalue of 'Y6, we will have compatible sets of values for all the 'Yj, j = 2, 3, 4, 5, 6.Any such set can be substituted back into our set of synthesis equations of the formof Eq. (3.68) with j = 1, 2, . . . , 6 (note that all angles for j = 1 are zero), andsolve any four of these simultaneously for Zk, k = 2, 3, 4, 5, thus yielding a solutionfor a six-precision point function generator geared five-bar.

3.14 DISCUSSION OF MULTIPLY SEPARATED POSITIONSYNTHESIS

Section 2.24 introduced the concept of prescribing precision points where the generatedfunction or path is to have higher-order contact with an ideal curve . By takingderivatives of displacement equations, contacts through two, three, and so on, infinitesimally separated positions can be obtained. When such infinitesimally separated positions (contained in a higher-order precis ion point) are specified in addition to finitelyseparated first-order (single-point match) precision points, we have "multiply separatedposition synthesis." Depending on the total number of prescribed positions andderivatives and the number of unknowns in the synthesis equations (see Table 2.6),the solution procedure may involve either linear or nonlinear methods. Two examplesof a nonlinear method follow.

Synthesis of a Fifth-Order Path Generator

In industrial practice, problems are frequently encountered that require the designof mechanisms that will generate a prescribed path in a plane. The problem isfurther complicated if the velocity, acceleration, and higher accelerations of the motionare critical, as might be the case where possible damage to the mechanism or tothe objects handled may result from large accelerations or rates of change of acceleration and the resultant shock. This example [250] presents an analytical closed-form


iY

TracerPoint

PathY = F(X)

x

Figure 3.50 Four-bar path generator for higher-order path generation.

solution for the synthesis of a four-bar linkage that will give fifth-order path approximation in the vicinity of a single precision point. Recall from Table 2.6 that this isthe maximum order that can be prescribed for a four-bar. At the single precisionpoint, derivatives of the path-point position vector up to the fourth are specified,which, when taken with respect to time, can be interpreted as the velocity, acceleration,shock, and third acceleration of the path-tracer coupler point. The closed-form solution to be derived in this section will yield a maximum of 12 different linkages foreach data set.

The problem is to synthesize a four-bar linkage with the notation shown inFig. 3.50 that will generate a path given by y = f(x) with prescribed timing, i.e.,with prescribed input-crank motion. First, as before, the input side (dyad) of thefour-bar shown in Fig. 3.51 will be synthesized. Then, for each possible solutionfor the input side, the corresponding output side solutions will be found .

In Fig. 3.51, the vector R locates the precision point on the prescribed idealpath; vector Z7 locates the unknown fixed pivot; Zl represents the unknown inputlink, and vector Z 2 represents one side of the unknown floating or coupler link.The following loop equation can now be written for the input side at the precisionpoint:

(3.74)

where <f> and yare rotations measured from some reference position shown in dashedlines.

In order to achieve fifth-order path generation, Eq. (3.74) must be successivelydifferentiated up to the fourth derivative with respect to time:


iY

We

"'e

/.fre(Xc

x

Figure 3.51 Input side of the four-bar path generator of Fig.3.50. Observe the M,); "fifth-order Burmester Point Pair ."

1

iweiq,Zl+ iWeeiYZ2 = R2

(ia - ( 2)Zle iq, + (ia; - WnZ2eiy = R3

(hi - 3aw - i(3)Zle iq, + (io.e - 3aewe - iw~)Z2eiy = R

[i«i - 6a(2)+ w4 - 4o.w - 3a2]Zle iq, 4

+ [i«ie) - 6aewn + w~ - 4o.ewe - 3a~]Z2eiy = R

1 234where R, R, R, R are the successive derivatives of R with respect to t, and

(3.75)

(3.75a)

(3.76)

(3.77)

d<f>w=-,

dt

dyWe=-'

dt

dwa=-,

dt

dco;a =-- ,

c dt

. daa= -

dt

. daeae=-

dt

dO.(i=-

dt

.. do.ea c =-;j(

Simplifications of Eqs . (3.74) to (3.77) are possible since a single precis ion pointis being considered, and P coincides with Pj in Fig. 3.51. Therefore,

<f>=y=O

242

and

Kinematic Synthesis of Linkages: Advanced Topics Chap. 3

With these Eqs. (3.74) to (3.77) become

R =Z7+Z1 +Z2

1

R = iWZl + iWeZ2

2R = (-W2+ ia)Zl + (-W~ + iae)Z2

3

R = [-3aw + i(o. - W3)]Zl + [-3ae We + i(o.e - W~)]Z2

(3.78)

(3.79)

(3.80)

(3.81)

(3.82)

4

R = [W4 - 4o.w - 3a2+ i (a - 6aw2)]Zl

+ [W~ - 4o.ewe - 3a~ + i(ae - 6aeW~)]Z2

The prescribed quantities in Eqs. (3.78) to (3.82) are the position vector R1 2 3 4

with its time derivatives R, R, R, and R, plus w, a , 0., a, which are the angular

velocity, angular acceleration, angular shock, and angular third acceleration of theinput link. The unknown quantities are the complex vectors defining the input sideof the mechanism in its starting position Z7' Zl, and Z2 plus the unprescribed quantitiesWe, ae, o.e, and a e, which are the angular velocity, angular acceleration, angularshock, and angular third acceleration of the coupler link.

The path function y = f(x) is introduced into the equations by the positionvector R and its derivatives. The first derivative of R is defined as

1 dR dR dSR=-=--

dt dS dt(3.83)

(3.84)

where S represents the scalar arc length along the path, measured from some referencev

point on the path. The term dR/dS is a unit vector tangent to the path at theprecision point and the term dS/dt is the speed of the tracer point along the path,which is a scalar. The second through fourth derivatives of Rare

R= d2R

(dS)2 + dR d2S

dS2 dt dS dt2

R= d3R

(dS)3 + 3 (d2R)

dS d2S + dR d

3S

dS3 dt dS2 dt dt 2 dS dt3(3.85)

(3.86)

4 _ d4R

(dS)4 (d3R)

(dS)2 d2S

R-- - +6 - - -dS4 dt dS3 dt dt2

+ 4 (d2R)

dS d3S

+ 3 (d2R)

(d2S)2

+ dR d4S

dS2 dt dt3 dS2 dt2 dS dt4

The solution of the synthesis equation (3.78) to (3.82) will be accomplished


by first solving Eqs. (3.79) to (3.82) for the unknown link vectors Z l and Z2 andthen returning to Eq. (3.78) for solving it for the unknown vector Z7.

The solution of Zl and Z2 requires the simultaneous solution of four equationswith complex coefficients linear in two complex unknowns. In order for simultaneoussolutions to exist for Eqs. (3.79) to (3.82), the augmented matrix of the coefficientsmust be of rank 2. The augmented matrix is

M=

iw

-w2 + ai

-3aw + (a - w3)i

iWe

-3aeWe + (ae - w~)i

1

R2

R3R4

W4 - 4aw - 3a2 + (a - 6aw2)i W~ - 4aeWe - 3a~ + (ae - 6aew~)i R(3.87)

This will be assured by the vanishing of the following two determinants:

1

iw iWe R2

0 1 = -W2+ ai -W~ + aei R =03

-3aw + (a - w3)i -3aeWe + (ae - w~)i R(3.88)

1

iw iWe R2

O2= -w2+ia -w~ + ia; R = 04

W4 - 4aw - 3a2 + (a - 6aw2)i W~ - 4aeWe - 3a~ + (ae - 6aew~)i R(3.89)

The two complex "compatibility equations" (3.88) and (3.89) may be solved for thefour unknown reals, We, ae, ae and a e. Expanding the determinants 0 1 and D2

according to the elements of the second column and their cofactors, separating realand imaginary parts and employing Sylvester 's dyalitic eliminant results in a sixth degree polynomial in We with real coefficients and with no constant term , whichcan be written as follows:

H w~ + Jw~ + K w~ + Lw~ + M w~ + N We = 0 (3.90)

where the coefficients H to N are deterministic functions of the prescribed quantitiesin the first and third columns of the determinants in Eqs. (3.88) and (3.89).

Factoring out the zero root results in

Hw~ + Jw~ + Kw~ + Lw~ + MWe + N=O (3.91)

The solution of (3.91) gives five roots for We. An examination of Eqs. (3.88) and(3.89) shows that We = W is also a trivial root. Dividing out the We W rootresults in

(3.92)


where Aj , j = 0, 1, . . . 4, are known real coefficients. This leaves four possibleroots remaining as solutions. These four remaining roots are real roots and/or complex pairs. Only the real roots are possible solutions for We ' Each real value ofWe is substituted back into the real and imaginary parts of Eqs. (3.88) and (3.89)to solve any three of the resulting four real equations simultaneously for the corresponding values of a e, ae, and ae associated with each of the four We values.Anyone of these sets of We , a e, ae , and a e values are then substituted into anytwo of the original synthesis equations (3.79) to (3.82), from which Zl and Z2 aredetermined. Then Z7 may be obtained by solving Eq. (3.78). Since four is themaximum number of possible real roots of the polynomial (3.92), there may existfour possible sets of input-side vectors Z7' Zh and Z2. The output-side dyad, Zaand Z4, with Zs locating its ground pivot, is synthesized by the same method asoutlined above, using the same We , a e, ae and ae set as the prescribed rotation oflink Z4, solving the compatibility equations for the rotation of Za, say Wa, aa, a a,and aa , and then going back to solve for the dyad and ground-pivot vectors. Itcan be shown that this procedure will yield the same set of values for these vectorsas those found for the input-side dyad. Thus, since each of the four such dyadscan be combined with any of the other three to form the path generator FBL, therewill be 12 such FBLs : up to 12 possible solutions for this higher-order path generationsynthesis with prescribed timing, just as for the five-finitely-separated-precision-pointcase described in Sec. 3.9.

Example 3.4

The solution of the synthesis equation s and the analysis of the synthesized linkageswere carried out on the IBM 360 digital computer by programs based on the foregoingequations. An example of a solution is given in Table 3.7 and Fig. 3.52.

Figure 3.52 Example of Table 3.7.A four-bar higher-order path generatorsynthesized for the path y = XI?".

The ideal path is shown dashed and thegenerated path is shown solid.

Sec. 3.14 Discussion of Multiply Separated Position Synthesis

TABLE 3.7 THE MECHANISM SHOWN IN FIG. 3.528 ,

SYNTHESIZED FOR PATH GENERATION WITH FIVEINFINITESIMALLY CLOSE POSITIONS (FIFTH-ORDERAPPROXIMATION WITH PRESCRIBED TIMING)

Path: y=xex

Precision point: R = (0.000, 0.000)

w = 1, a = a = ii = 0 (constant-velocity input crank)

245

dsdt = 1,

cflsdt. = 1,

d's d'sdt' = dt' = 0

Link vectors :

Z, = 0.81691Z. = -0.23055Z, = -1.34553Z,= 2.43881

-1.33976;1.32842;1.15162;

-2.27172;

a In Fig. 3.52 the generated path appears to depart from theideal path on the same side at both ends of the curve.However, in reality, the generated curve does depart the idealcurve on opposite sides, which would indicate an even numberof (infinitesimally close) precision points. This, however, isnot the case, because in the positive x and positive y quadrantthe departure on the positive y side is so slight that it is detectable only in the numerical values of the computer output.

m

If the motion We, ae , Ue, and ae of link Z2 is prescribed together with R,m = 0, 1, 2, 3, 4, the preceding synthesis would be motion generation synthesis ofthe Zh Z2 dyad with five infinitesimally close prescribed positions . Figure 3.51 showsone of the up-to-four M, k 1 Fifth-Order Burmester Point Pairs associated with sucha dyad . Using one other of the up-to-four such dyads together with the first, afour-bar higher-order motion generator is obtained. There are up to six such fourbar mechanisms, with a total of 12 cognates. The cognates are higher-order pathgenerators with prescribed timing . The geared, parallelogram-connected and tapeor chain-connected five-bar path generators, as well as the parallel-motion generatordiscussed for five finitely separated (discrete) prescribed positions, can all be adaptedfor higher-order synthesis with the method of this section, as can the many multilooplinkage mechanisms presented in the preceding sections.

Position-Velocity Synthesis of a Geared Five-Bar Linkage

The geared-five bar linkage of Fig. 3.41 was synthesized for four finitely separatedpositions offunction generation. According to Table 3.5, this linkage can be designedfor seven total positions but the five-, six-, and seven-position cases will involve compatibility equations.

Following the logic laid out in Sec. 2.24, the number of prescribed positionsin column 1 of Table 3.5 may be either finitely or infinitesimally separated. Note,however, that an acceleration involving three infinitesimally close positions may notbe prescribed without the position and velocity (two infinitesimally close positions)

r-,I .....I .....I .....I .....IIIIII

' ---_ ... -

(b)

Figure 3.53 (a) geared five-bar function generator is shown in the first precision position (solid). Thejth position (dashed) correspond s to a rotation of the input link Z\ by <l>J ; (b) scale drawing of Ex. I,Table 3.8 in its three Chebyshev-spaced second-order precision positions.

246


at that location also being prescribed. Figure 3.53a shows another form of a gearedfive-bar with F = I, since there is a geared constraint between link 1 and link 2,forming a cycloidal crank. Let us write the equations and synthesize this gearedfive-bar for six mixedly separated prescribed positions of function generation-threepairs of prescribed corresponding positions and three prescribed corresponding velocitypairs for the input and output links [85].

We will use the following notation for the rotational operators:

{

Aj=ei4>j

Given: Ej = -':~(TS/ T2)4> jp.j = e'o/JJ

Unknown: Vj = eh'j

The equation of closure for this mechanism may be written as follows:

AjZ1 + EjZ2 + VjZa +P.jZ4 = -1,

The first derivative loop equation

).jZ1 + EjZ2 +VjZa + jJ.jZ4= 0,

j = 1,2,3

j = 1,2, 3

(3.93)

(3.94)

where the superior dot represents differentiation with respect to the input-crank rotation. Here

. .(dl/l)p.j = I d</> j p.j

are known from prescribed data for j = 1, 2, 3, and

. .(d"l)Vj = I d</> j Vj

are unknown. Note that for j = I, </>1 = 1/11 = "11 = 0, and therefore Al = E1 = VI

= P.1 = 1. The augmented matrix of systems (3.93) and (3.94) in full array is asfollows:

I -1A2 E2 V2 P.2 -1

M= Aa Ea Va P.a -1(3.95)

i E1 i'YI jJ.1 0iA2 E2 iY2V 2 jJ.2 0iAa Ea iYava jJ.a 0


This matrix must be of rank 4 to ensure that system (3.93)-(3.94) yields simultaneoussolutions for Zk. Thus if we say that

D I = det(M)1 .2.3.4.s = 0

D 2 = det(M)1,2.3.4.6 = 0

(3.96)

(3.97)

where the subscripts designate row numbers, we can regard system (3.96)-(3.97) asthe compatibility system. This system, containing four real equations, can be solvedfor four real unknowns. Elements in the first, second, and fourth columns of Eq.(3.95) are known from prescribed data. Column 3, however contains fiveunprescribedreals. If we assume 11 arbitrarily, this leaves the four unknown reals 12, 13, 12,and 13. Expanding D I and D 2, each according to the elements of the third columnand their cofactors, we get

D I = Ii. I + 1i.2V2 + 1i.3V3 + 1i.4i11 + ili.s12V2 = 0

D2 = 1i.'1 + 1i.'2V2 + 1i.'3V3 + 1i.'4i11 + ili.s13V3 = 0

(3.98)

(3.99)

(3.100)

where the Ii.'s are the appropriate cofactors, known from prescribed data. DivideEq. (3.98) by ili.sV2 and Eq. (3.99) by ili.sV3' obtaining

.!!.. -I ~ ~ - I 1i.4i11 -I '-'A v 2 +'A +'A V2 "3+ 'A v 2 +12- 0I Ll.s I Ll.s I Ll.s I Ll.s

Ii.' Ii.' Ii.' Ii. ' i ._ I V- I +_2 -1+_3+~V-I+1' = 0.A 3 .A V2V 3 .A .A 3 3I Ll.s I Ll.s I Ll.s I Ll.s

Simplifying the notation of known quantities, we rewrite Eq. (3.100):

al + a2v2"1 + a3v2"lv3 + 12 = 0

The complex conjugate of Eq. (3.102) is

iii + ii2v2 + ii3V2V3"1 + 12 = 0

Subtracting Eq. (3.102) from Eq. (3.103) and dividing by 2i we obtain"

al y + a2y cos 12 - a2x sin 12 + a3y cos (13 - 12) - a3X sin (13 - 12) = 0

(3.101)

(3.102)

(3.103)

(3.104)

Similarly, combining Eq. (3.101) with its complex conjugate and using primed symbolsfor the known factors yields

(3.105)

In Eqs. (3.104) and (3.105) all a and a' values are real deterministic functions ofthe known coefficients in columns 1, 2, and 4 of the matrix M. To simplify Eqs.(3.104) and (3.105) we use the following identities:

cos (13 - 12) = cos 12 cos 13 + sin 12 sin 13

• Altern atively, we can say that, since 'Y2is real, the imaginary part of Eq. (3.102) is zero, whichalso leads to Eq. (3.104).

Sec. 3.14 Discussion of Multiply Separated Position Synthesis

sin ('Ya - 'Y2) = sin 'Ya cos 'Y2 - cos 'Ya sin 'Y2

249

where

1- T~cos 'Yj =__J ,

1+TJ. 2Tj

sm y > 1+T~J

'YjTj = tan 2

With these we rewrite Eq. (3.104) :

1 - T~ 2T2 [1 - T~ 1 - T~ 4T2Ta]al

y + a2y 1 + T~ - a2X 1 + T~ + aay 1 + T~ 1 + T~ + (l + T~)(l + T~)

2Ta(l - T~) - 2T2(1 - T5)-a =0

ax (1 + T~)(l + T~)

Multiply by (1 + T~) (l + T~):*

aly( l + T~)(l + T~) + a2y(1- T~)(l + T~) - a2X2T2(l + T5)

+ a ay[(l - T~)(l - T5)+ 4T2Ta] - 2a ax[Ta(l - T~) - T2 (1 - TnJ = 0

Expanding, we get

aly (l + T~ + T~ + T~T~) + a2y (1 - T~ + T~ - T~T5) - 2a2X(T2 + T2T~)

+ aay [(1 - T~ - T~ + T~T~) + 4T2Ta] - 2a ax (Ta - T~Ta - T2 + T2T5) = 0

Arrange in descending powers of Ta:

T~[alY (l + T~) + a2y (1 - T~) - 2a2X T2 - aay (1 - T~) - 2a ax T2]

+ Ta[4aayT2 - 2a ax (1 - T~)] + Tg[aty (l + TD + a2y (1 - T~)

- 2a2X T2 + aay (l - T~) + 2a ax T2] = 0

This is in the form

(3.106)

(3.107)

(3.108)

(3.109)

(3.110)

where Pj (j = 0, 1, 2) denote second-degree polynomials in T2 with real coefficients .Similarly, from Eq. (3.105) we obtain

(3.111)

where 7Tj (j = 0, 1,2) also denote second-degree polynomials in T2 with real coefficients.We eliminate Ta by writing Sylvester's dyalitic eliminant:']

• Multiplica tion through by (I + T~) (I + T~) introduces the extraneous roots of ±i for T2 and

t Multiply Eqs. (3.110) and (3.111) by T3, yielding two more equations, which are third-degreepolynomials in T3' The resulting four equations will have simultaneous solutions for T~n ), n = 0, I, 2,3, if the augmented matrix of the coefficients is of rank 4, which leads to Eq. (3.112).


0 P2 PI Po

8 1=0 7r2 7rl 7rO =0P2 PI Po 07r2 7rl 7rO 0

Chap. 3

(3.112)

Expanding, we obtain an eighth-degree polynomial in 72 (since every element of 8 1

is a quadratic in 72) with two trivial roots (±i).Similarly, by eliminating 72 from the system of Eqs. (3.104) and (3.105) we

obtain an octic in 73, whose solutions also include four trivial roots: ±i and

73 = tan (~3) and 73 = tan (~3)

Corresponding simultaneous values of the nontrivial roots of 72 and 73 can beidentified by direct substitution in systems (3.110) and (3.111) of one 72 value at atime. This will yield two roots for 73 from Eq. (3.110) and two roots for 73 fromequation (3.111), one of which will be a common root satisfying the system (3.110)(3. II I ) and identical with one of the nontrivial roots of the octic in 73. Indeed,the procedure of the foregoing paragraph need not be performed to find these roots.Instead, any two different nontrivial real roots of Eq. (3.112) can be used as 72 and73. The maximum number of simultaneous nontrivial 72, 73 pairs is six.

Thus having found up to six pairs of values for 12 and 13, each such pair canbe substituted back in Eqs. (3.108) and (3.109) to find corresponding values for 'band 1'3'

Anyone of such sets of solutions for 12, 13, 1'2, and 1'3 can then be substitutedback into the original system of Eqs. (3.93) and (3.94), any four of which can thenbe solved simultaneously for the mechanism dimensions in the starting position, namelyZk, k = I, 2, 3, 4, yielding up to six different designs.

Example 3.5

Table 3.8 lists several results of the geared five-bar function generator synthesized forthree second-order precision points . Although respacing for optimal error has not beenemployed, the maximum error in several examples (2, 4, 5, 6) is considerably smallerthan the optimum four-bar synthesized by Freudenstein [104] for the identical set ofparameters (i.e., the function, range , and scale factors). In some cases (1, 3) the outputprovides a "near fit" to the ideal function for a sector of the range. An extra firstorder precision point has contributed to the accuracy of the function generator in examples(1, 4).

Specifying a second-order precision point and receiving a third-order precisionpoint does not seem to be unlikely. Example 3 has one second-order and two thirdorder precision points.

Since a second-order precision point is actually two precision points infinitesimallyclose to one another, the actual curve approaches and departs from the ideal curvewithout crossing it. Second-order precision points have applications wherever the firstderivative of a function (tangent to a curve) must be reproduced exactly. As suggestedby McLaman [181], if all the precision points are second order, the maximum errorcan be halved by shifting the ideal curve by one-half the maximum error. Examples2a, 5a, and 6a each have only second-order precision points which are shifted in parts

Appendix: A3.1 The Lineages Package 251

b to halve the maximum error. The negative gear ratios, examples 3 and 4 denotethat the gears lie on the same side of their common tangent (a hypocycloidal-crankmechanism) (see Fig. 3.40).

Example I is shown in Fig. 3.53b in its three Chebyshev-spaced second-orderprecision positions. The output generates x 2, corresponding to an input of x for 0 ~

x ~ 1. The range of both the input and output is 90° . An arbitrary gear ratio of 2and )'1 of 1.5 gave an extra precision point at x = 0.2299. This extra precision pointcauses the error for nearly 60% of the range to be less than 0.0176°.

Example 4 is a special case where Z2 is for all practical purposes zero . Thuswe have a four-bar mechanism which , because of unprescribed precision points at x =0.0666 and x = 0.2157, has eight precision conditions (one third-order, one first-order,and two second-order precision points) . Note that the maximum error over the entirerange is 0.0584°, while over 95% of the range the error is less than one-tenth of themaximum error of the optimal four-bar linkage of Ref. 104.

APPENDIX: A3.1 The LINeAGES Package*

It is not the purpose here to fully explain all the options of the interactive subroutinesof the LINeAGES package [78,83,84,218,270]. Some of the subroutines will beillustrated, however, by way of an example. Since there are two solutions for eachchoice of 132 (Fig. 3.2) (for which the compatibility linkage closes (Fig. 3.3), eachsynthesized dyad is designated by a 132 value (0° to 360°) and a set number (lor2) indicating whether it is the first or second of the two available solutions for 133and 134 for the particular 132.

Example 3.6

The assembly of a filter product begins by forming the filtration material into what isknown as a filter blank. Next the filter blank is placed by hand onto a mandrel.This mandrel is part of a machine that completes the assembly of the filter. The objectiveof this problem is to design a four-bar linkage mechanism for removing the filter blanksfrom the hopper and transferring them to the mandrel.

Figure 3.54 diagrams the design objective. A gravity-feed hopper holds the semicylindrical filter blanks with the diametral plane surface initially at a 27° angle fromvertical. The blank must be rotated until this diametral plane is horizontal on themandrel. The position of the hopper can be located within the sector indicated, althoughthe angle must remain at 27° .

At the beginning of the "pick and place" cycle, it is desirable to pull the blankin a direction approximately perpendicular to the face of the hopper. To prevent foldingthe filter blank on the mandrel, it is necessary to have the rotation of the blank completedat a position of approximately 2 em above the mandrel and then translate without rotationonto the mandrel. The motion of the linkage should then reverse to remove the completedfilter from the mandrel and eject it onto a conveyor belt . After this , the linkage shouldreturn to the hopper and pick another blank. Because of the requirement of both forwardand reverse motions over the same path, a crank-rocker would have no real advantageover a double-rocker linkage.

An acceptable linkage solution (a four-bar chain is desirable here) must have

• Available from the second author.

I\)U1I\)

TABLE 3.8 GEARED FIVE-BAR FUNCTION GENERATOR' OF FIG. 3.53, EX. 3.5.

Z vectors Precisionin initial points 01 Geared Four-bar Remarks Shape 01

Mech· Scale Angular position' geared Iive-bar maximum on geared errorcurveanism Function Range lactors' Gear velocity" 01 geared Iive-bar maximum error Iive-bar 01 geared

number Y" Xo S; x S x. Aef>,AljI ratio ;0, Iive-bar atx= error (Freudenstein) performance Iive-bar

1 x' O:S;x:S;l 90,90 2 1.5 0.2759 + 2.291 i 0.0666-, 0.2299" , 0.0802 " 0.0673" Extra precision~J-o.1451-0.3883i 0.5000", 0.9333-, poinl at x = 0.2299;

0.2270 - O.638i lor 0.03 :s; x :s; 6, Ir'-'-1.358 - l.264i error < 0.0176 "

2a x' O:S;x :S;1 90,90 2 1.5 -3.34O +2.131i 0.0666-, 0.5000" , 0.0708 " 0.0673" All precision-0.2015 +0.1525i 0.9333- points are

2.638 - 1.671 i second-order ,-,",-/,

-0.0968 - 0.6122i

2b x' O:S;x:S;l 90,90 2 1.5 Same 0.0169',0.1517', 0.0354 " 0.0673 " Shifting precision0.3611', 0.6409', points 01 example 2a0.8403', 0.9788' yields 1/2 1he - -maximum error; --- "'" ...

Y.. = Y..+ 1/2 19_1(2a)

3 x' O:S;x :S;1 90,90 -2 1.0 0.2946 + 1.396i 0.0666e. r 0.3816 " 0.0673 " For 0.33 < x :s; 1.0,-o.0148 -0.1921i O.!iOOO<- r error < .011 ; lor .-.-0.0918 - 0.7607i 0.9333- 0.40 < x < 0.56 ............-1 .188 - O.4433i error < 0.001

4 XU O:S;x :S;1 90,90 -2 1.5 - 3.80 3 +4.852i O.0666c.t 0.0584 " 0.146 " Extra precision0.0071 - 0.0052i 0.2157', 0.5000" point at 0.2157 ; .-

2.424 - 2.659i 0.9333" lor 0.05 :S; x :s; 1.0, .r '-J'",-,"

0.3712 -2.187i error < 0.0138 "

Sa xt-' O :S; x :S;1 90,90 2 1.5 -3.511 + 1.435i 0.0666- 0.0951 " 0.412 " All precision-0.3032 +0.2282i 0.5000" points are

2.951 - 1.405i 0.9333< second-order ~ --0.1368 - 0.2573i

5b ",.1 O ~x ';l 90,90 2 1.5 Same 0.0142',0.1540", 0.0476 · 0.412· Shilling (Sa)

0.3763·,0.6152', yields VI .- .-. -0.8582', 0.9829- maximum error -- .....

68 x' O';x ';l 90,90 2 1.5 - 3.615 + 1.251 i 0.0666<, 0.5OQ()<, 0.1281· 0.566· All precision-0.3235 + 0.2889 i 0.9333< points are '- /""'\.../"\. ~3.059 - 1.411i second-order-0.1193 - 0.1292i

6b x' O';x ,;1 90,90 2 1.5 Same 0.6338" 0.0641 · 0.566· Shilling (68)0.8359" yields VI r-''-'" ......

maxilTMJm error

• The units 01 !he scale lactors are in degrees per unit 01 x. The velocity 1. is in radians per second . The Z vectors are (from !he top down) Z" Zo. z.. and z.. where Z. = 1.

I x' function generator also good for x".,e Precision point derived from Chebyshev spacing (S8COlld-«der precision points).

" There is an extra , unprescribed first-order precision point present

• These are precision points obtained by shifting !he error curve 01 example 2a upward by VI 01 the maximumerror 01 that example .

' These turned out to be gratuitous third-order precision points.

I\)U1W


HOPPER

~~-BLANKS

\-ACCEP TABLE AREA

FOR HOP PER ANOLINKAGEY

MANDREL

(not shown)CONVEYOR BELT

Figure 3.54 The prescribed task of motion generation for Ex. 3.6: Filterblanks are to be taken from the hopper and placed on the mandrel.

ground pivots and linkage motions within areas that do not interfere with the hopperand the filter assembler. Also, since the resulting linkage may be driven by an addeddyad (to provide a fully rotating input), the total angular travel of the input link ofthe four-bar synthesized here should be minimized so as to obtain acceptable transmissionangles for the entire mechanism, including the driving crank and connecting link, formedby the added dyad, which actuates the input of the motion-generating four-bar linkage.

This example is a typical challenging problem that often faces linkage designersin practice. Some of the constraints are firm, whereas others can vary within somespecified range. This means (mathematically) a number of infinities of solution possibilities. The computer graphics screen is an ideal tool to help survey a large number ofpossible solutions.

Method of Solution

The problem clearly requires motion-generation synthesis (or rigid-body guidance),in which the position and angle of the filter blank is specified at different precisionpoints. Four points along a specified path and four corresponding angular positionswere chosen.

The first set of precision points chosen are shown in Table 3.9. The mandrel

TABLE 3.9 FIRST ATIEMPT AT SYNTHESIZING A FOUR-BARMOTION GENERATOR WITH FOUR PRECISION POINTS(EX. 3.6, FIG. 3.54).

Position X coordinate (cm) Y coordinate (cm) Rotation (deg)

1 0 0 02 1 7 03 17 18 604 38 21 117


was designated position 1, while the second position was picked above the mandrelwith no rotation (to prevent folding the filter blank). The third position was chosento be about halfway between the second and fourth positions with approximatelyhalf the required rotation. The fourth position corresponded to the angle and positionof the hopper.

A portion of the M-K curves (center- and circle-point curves) is shown inFig. 3.55. The solid and short-dashed lines represent the portions of the centerpoint curve from sets I and 2 of the {3j (j = 2, 3, 4), respectively, while the longdashed and dash-dotted lines are the circle-point curves from these sets. Figure3.56 shows an M-K curve for set I solutions only, where the {32 values are correlatedto their m-k; positions on the curves by letters corresponding to the table alongthe left-hand side of the figure. For example , letter B represents m and k 1 pointsfor {32 = 30°. The results of interactively locating ground pivots and moving pivots(by using the crosshairs on the graphics screen which can be positioned by the operatorto indicate his choice of a point on the M or K curve), corresponding to {32 =330° and {32 = 30°, are also shown in Fig. 3.56. As the designer locates a groundpivot with the crosshairs, the computer finds the moving pivot of the dyad and drawslines on the screen to represent the dyad. These two dyads formed what looked tobe an acceptable four-bar solution to be further analyzed. Coupler curves of fourbars picked are generated by another subroutine. Figure 3.57 shows that the couplercurve shifts to the left between points I and 2, approaches precision point 4 vertically,and has a cusp at point 3. These are all unacceptabe characteristics. Althoughthis linkage is not useful, what about the others which also satisfy the same set ofprescribed precision points?

PLOT OF "-K CURVES ( BOTH SETS )"1 - K1-- " 2 - - - K2---

/v----II/ /~~

---I'- .....

,

1<, ,

---- _ ........ _ 01

<, o\ 0, \,

c\ ' .' \

\ I)" -,-" . ---1--- ._ /

-10.0-20.0 -10 .0 0 .0 10.0 20.0 30 .0 40.0

50.0

0.0

10 .0

20 .0

30.0

40 .0

Figure 3.SS Plot of M-K curves (both sets), Ex.3.6, Fig. 3.54, for the four precision point s specifiedin Table 3.9.


PLOT OF M-K CURVE S (SET 1)M -K--

Chap. 3

--

~I

1\/

~ v~~ ----~---~

-<,<, /<, 0,

0

/1

o A50 . 0

30 B

60 C

90 D~ 0.0

120 E

150 F30. 0

180 G

210 H

2~0 120 .0

270 J

300 K

330 L10•0

360 M

0.0

-10.0 0 .0 10 .0 20 . 0 30 .0 ~0.0

33030

Figure 3.56 Plot of M-K curves (setI), Ex. 3.6, Fig. 3.54, for precisionpoints of Table 3.9. One of an infinitenumber of four-bar linkage mechan ismsformed from two M-K dyads: the fixedlink is LB (nearly vertical, line notshown). BB and LL are groundpivoted links and LPtB is the coupler

50.0 triangle, with coupler LB not shown.

30 .00

25.00

20.00

15.00

10.00

5.00

0 .00

- 5 . 00

COUPLER CURVE

0

,q /, ,

//

,, /

/ - -:0- '

I

<,

-10.00

-5 .00 5.00 15 .00 25 .00 35.00

0.00 10 .00 20.00 30. 00 40.00

L e LABEL UITH THETA 1 CODE

Figure 3.57 Coupler curve of the four-barmechani sm of Fig. 3.56 (Table 3.9).

Figure 3.58 shows a copy of one of the other options available (the so-called"BETAS") in the LINeAGES package, which is helpful in surveying possible dyads.Grounded link (W) rotations /33 and /34 from sets I and 2 (BTA31, BTA21, BTA41,and BTA42) are plotted here against /32, Notice that there are no values of /33, /34between 300 < /32 < 3300. This shows that the "compatibility linkage" does notclose for this range of /32, Another useful design characteristic of this plot is theability now to pick dyads in which link W exhibits constant directional rotationbetween precision points 1 to 4. For example, the solution corresponding to /32 =


PLOT OF BETAS US BETA2BTA31 - BTMI - - BTA32 - - - BTM2 - --

Figure 3.58 Plot of betas versus12e.e 188 .8 248.8 388.8 368 .8 beta two.68 .e

!IJ

I! ! II I I 'I i . !

/ i I iI I ;i i/ ~

/ J \ ./

' I

" I V'r- . i\:

8.8

8.8

68 .8

128.8

388.8

188 .8

368 .8

10° from set 1 has constant directional rotations with {J3 = 110° and {J4 = 180°(see Fig. 3.58).

After investigating a representative number of other four-bar solutions withoutsuccessfully improving the coupler path trajectories, another set of precision pointswas picked. An attempt was made to position the hopper above and closer to themandrel (Table 3.10).

The "TABLE" subroutine of the "LINeAGES" package helps to survey alarge number of four-bar combinations (in sort of a "shotgun" manner) as to whetherthey branch between precision points and what the resulting maximum link-lengthratio is. Figure 3.59 shows a sample TABLE from the second choice of precisionpoints. Twelve dyads corresponding to user specified {J2'S from set 1 or set 2 formthe vertical and horizontal axes of the table (both dyad sets are from set 2 of the{J2'S in Fig. 3.59). Each of the 36 boxes of the matrix lists a measure of the mobility(see Sec. 3.1 of Vol. 1) and the link-length ratio of the resulting four-bar mechanismmade up of these dyads . Table 3.11 lists the mobility abbreviations generated bythis table. Regions that showed promise were expanded, with an eye toward the

TABLE 3.10 SECOND SET OF PRECISION POINTS FOR THEFOUR-BAR MOTION GENERATOR OF EX. 3.6, SELECTED AFTER THEHOPPER (FIG. 3.54) WAS MOVED CLOSER TO AND DIRECTLY ABOVETHE MANDREL.

Position X coordinate (em) Y coordinate (em) Rotation (deg)

1 0 0 02 1 7 03 9 17 604 17 22 117


TABLE OF LINKAGE PARAMETERS

MINIMUM TRANSMISSION ANGLE~~MAXIMUM LINK LENGTH RATIO

Chap. 3

SET

2

60 T-RR TOG TOG TOG TOG T-RP2 .1 2 1 3 .0 10 .9 1'1 .0 ****

50 R-C T- RR T-RR TOG TOG T-RR1 .9 1 .6 3 .0 11 0 1'1 .1 ****

'10 T-RR T-RR T-RR T-RR TOG T-RR1 .8 1 5 3 .1 11 .3 1'1 .'1 ****

30 TOG T-RR T-RR T-RR TOG T-RR1 .9 1 7 3 . 3 11. 9 15 .0 ****

20 TOG T-RR T-RR T-RR T-RR T-RR2 .9 2 .'1 '1 .1 13 .8 16 .'1 ****

10 TOG TOG BRAN T-RR T-RR T-RR5 7 'I 7 7 1 21 .2 21 .8 ****

62 -300 312 32'1 336 3'18 360

SET 2

Figure 3.59 Table oflinkage parameters of four-bar mechanism sformed from two dyads. For example. the first box in the toprow refers to the mechanism formed out of a dyad obtainedwith f32= 60°. and f33.f3. taken from set 2. plus the dyad obtainedwith f32 = 300°. also with f33.f3. from set 2.

TABLE 3.11 ABBREVIATIONS FOR FIG. 3.59.

R·C

ROC

BRAN

T-RR

RR-T

TOG

Rocker-crank mechanism (input side is rocker)

Double-rocker does not toggle between precision points

Linkage branches

Linkage passes through toggle position between precision points when input is drivenbut does not toggle if follower is driven

Double-rocker will toggle if follower is driven

Double-rocker will toggle when either side is driven

Link ratio greater than 99 .9 :1

No solution for one of the dyads

(Blank) no linkage-essentially identical dyads

BETAS output to ensure that the total required input angle rotation (/34) was nottoo large. For example, the four-bar formed by the /32 = 50° and /32 = 300° solutions(both from set 2 in this case) looks promising, with the maximum link-length ratio = 1.9. When driven from the /32 = 50° side, it would be a rocker-rocker linkage;while driven from the /32 = 300° side, it would be a crank-rocker. Unfortunately,no acceptable solutions were found from this search.

One further attempt was made at specifying the precision points. Precision

Appendix: A3.1 The Lineages Package

TABLE 3.12 THIRD SET OF PRECISION POINTS FOR THEFOUR-BAR MOTION GENERATOR OF EX. 3.6.

Position X coordinate (em) Y coordinate (em) Rotation (deg)

1 0 0 02 3 5 53 27 22 904 35 24 117

259

point 2 was given at a slight angle (see Table 3.12); also, it was moved downward,closer to, and over to the right of precision point 1.

After a search through several possible solutions from this set of precision points,a final solution was found. The dyads of this linkage are shown in Fig. 3.60 andthe coupler curve in Fig. 3.61. The total angular travel of the input link is only113°, which is small enough to drive with another dyad. The link-length ratio (seeTable 3.13) for the entire linkage is 2.51 and the transmission angles are satisfactoryover the range of motion. The "choose" option of Table 3.13 allows the user toput two dyads together to form a four-bar mechanism. The "side 1" and "side 2"columns give specific numerical data of interest for the four-bar; the "minimum transmission angles" row indicates that with either side driven the linkage would be arocker-rocker linkage. The coupler angles refer to the angles PAB and PBA (seeFig. 3.60). Finally, the linkage fits within the physical constraints required (Fig .3.62 displays the mechanism in its four prescribed positions).

This example represents a typical design situation in which there are numerousconstraints that would be difficult to make part of the mathematical model. With

PLOT OF "-K CURVES (SET a)"-K --

J 0......<, 0

/ <,

~\/

/

~K)

\. /

~"- /

"-,,- ir'/

e ~e.e

3e 8

6e C

9. ~e.ela. E

15. F

18. G

21. Ae .•

24. I

27e J

3.. U ••

33. L

36. "-Ie.'

-I'.' e.e 18.' a••' 31.' 4'.'

a.34.

Figure 3.60 Plot of M-K curves (set 2). Final solution of thefour-bar motion-generator of Ex. 3.6 with precision points listedin Table 3.12.

260

A 0 25.00B 30.0C 60.0D 90.0 20.00E 120 .0F 150.0G 180 .0 15.00H 210 .0I 2<40.0J 270.0 10.00K 300.0L 330.0

5 .00

0.00

-5.00

-10.00

Kinematic Synthesis of Linkages: Advanced Topics

- _-t '~/

/'

/

/ '£

./

.....v,r

~<,

r-

Chap. 3

30.00 48. ee Figure 3.61 Coupler curve of the final3S. ee solution, a four-bar motion-generator of

Ex. 3.6.

10.0e 20.005.00 15.00 as.eeL • LABEL UITH THETA 1 CODE

-15.00e.e0

TABLE 3.13 " CHOOSE" SUBROUTINE-FOUR-BAR MOTION GENERATOR

SYNTHESIZED WITH THE INTERACTIVE " L1NCAGES" PACKAGE (EX. 3.6).

SIDE 1 SIDE 2

SOLUTION SET 2 2

BETA 2 340 .00 18.00

BETA 3 281 .35 60.93

BETA 4 247.13 40.06

MX 16.16 -4.66

MY 7.17 23.63

KX .85 13.98

KY 10.54 15.51

INPUT LENGTHS (LINKS 1 AND 3) 15.68 20.34

COUPLER SIDE LENGTHS (LINKS 2 AND 4) 10.58 20.88

.......... TOGGLES IF SIDE TWO DRIVEN ••••••••• *

MINIMUM TRANSMISSION ANGLES ROCKER ROCKER

COUPLER ANGLES (PAB AND PBA) 115.31 27.25

COUPLER LENGTH (LINK 5)

GROUND LENGTH (LINK 6)

14.0426.54

MAXIMUM LINK LENGTH RATIOS

TOTAL (LINKS 1-6)

FOUR BAR (LINKS 1,5,3,6)

COUPLER (LINKS 2,4,5)

2.511.891.97

Appendix : A3.2 261

Figure 3.62 Linkage of Fig. 3.60 in the precisionpoint positions.

, , \' ...,, ,, ,

,.r - ::\ .::: ..~,

"":':.' . ' --., ..' - " -.~

, ,

<, -- :;;; 'j-- -.....-- -e ..-. ~ .

, r-:»<r; ,,

~ ',I

)--><;;V ,,,,

1</ r--::::.~...

V5."

4'."35.••

3'."as."

a. ...

15."

18."

....-5 ... 5... IS... as.ee 35."

.... I. ... a.... 3....THIS IS THE LI"KAGE I" THE PRECISIO" POI"T POSlTIO"S

the designer "in the loop," nonspecified constraints may be considered as well, especially when there is a manageable total number of solutions and a visual method ofsurveying many solutions at once, as is available with the LINCAGES package.

APPENDIX A3.2

Computation schedule for standard form synthesis of motion-generator dyad for fivefinitely separated positions of the moving plane-Burmester Point Pairs." See Fig.3.2 for notation.Prescribed quantities:

Rj, j = 1,2,3,4,5

a], j =2,3,4,5

Compute:

Compute ak ,

j = 2,3,4,5

k = 2, 3, 4 according to Eq. (3.8)

Compute:

a' = I(e iaa - l )2 (e ias - 1)

(eia 2 - 1)1(e ias - 1)

k = 1,2,3,4; k = 1,2,3

• Enables the reader to write a program for five precision positions.


Using the above-obtained (now known) quantities, compute the real coefficients

of a fifth-degree polynomial in the unknown r = tan ~2 according to the following

sequence (superior bar signifies complex conjugate):

a=a'a""

b' =~~ K;,3

n = L a~ -a~,k =l

3

n' = -a~+ L (aic)2k=l

b=b'a" - b"a'" + b'" n' - b"" n

e = b"e' - b'e" + e'" n' - e"" n

d' = K; A;;, d" = e' - e", d = d'd"

u = ~3"K;, f' = ~ ~2' f= f'u

h' = ~~ K;, h = h'u, k = f - ii, k = [k ]

g'=~~~, g"=~2K;, g'" =g'+g"

gy = uxg~' + uyg;' , v = igy(4k)

m =-4g~ -2k 2, p=ad

q = ae + bd + k2, S = aii + be + ed + v

It = 2: (a 2+ b2+ c2 + d 2+ m), where a = [a], etc.

A 1 =-6py -4qy -2Sy

A 2 = - 15Px - 5qx + Sx + 3t

A 3 = 20py - 4Sy, A 4 = 15px - 5qx - Sx + 3t

As= - 6py + 4qy - 2Sy, A 6 = - Px + qx - Sx + t

Check: A o = px + qx + Sx + t = 0

aj=Aj+tlA 6 , j=O,1,2,3,4

Solve the following fifth-degree polynomial equation having real coefficientsby means of any polynomial-solver routine for all five roots, both real and complex,for the unknown r:

Check: one of the roots should be:

Chap. 3 Problems

70=1 - cos U2

263

This is a trivial root. Discard it and all complex roots . Keep the remainingreal roots:

(If no real roots remain, no solution exists. Go back to use different prescribedquantities.)

Using the real roots, compute {32 as follows:

(up-to-four different values)

where

u = 1 or 2 when there are 2 real roots (71 and 72)

and

u = 1 or 2 or 3 or 4 when there are 4 real roots (71) 72, 73 and 74)

Take one of the {32 values and use Table 3.1 to solve for {33 and {34' Thenuse any two equations of the system Eq. (3.3) to compute Wand Z. Repeat thisfor all (up-to-four) (32 values.

This completes the computation of the (up-to-four) Burmester Point Pairs .

PROBLEMS

3.1. Several different four-bar linkages are shown in Fig. P3.1.(a) Find the two four-bar cognates for the specified four-bar(s).(b) Compare the coupler curves of the cognates with the parent four-bar(s).(c) Construct the single degree of freedom geared five-bar path generator mechanism

(with the same path).

3.2. Figure P3.2 shows four slider-crank mechanisms. Find a cognate and compare couplercurves for the slider crank in:(a) Figure P3.2a(b) Figure P3.2b(c) Figure P3.2c(d) Figure P3.2d

3.3. Figure P3.3 shows several four-bar linkages that should form the base of a six-bar linkagethat generates the same path as point P of the four-bar, but does so without rotatingthe coupler link of the six-bar . Construct those six-bars and compare paths of thefour-bar and six-bar.

3.4. We wish to synthesize a six-bar motion generator to move stereo equipment from ashelf to closed storage when not in use. Rotation of the coupler link is not permitted,to avoid tipping the turntable or having the equipment slide off the moving platform.Figure P3.4 shows a four-bar that was synthesized such that its ground pivots are con-

(o) (b)

a

(e)

a

az

a

Figure P3.1

264

(d)

(I)

(9)

(e)

(h)

a

p

265

266

p

(a)

Range of Rot at ionsfor Link m2 - k2

(d)

p

p

m1

Figure P3.3

Chap. 3

iy

Problems 267

Shelf

P,

Figure P3 .4

strained with the area from 1.5 to 4.0 in the x direction and from - 5.0 to - 7.0 in they direction. The precision points prescribed were:

Precision points

Number x y Coupler angle

1 16 - 4 0 0

2 12 0 20 0

3 7 2 50 0

4 0 4 95 0

Design a six-bar linkage based on this acceptable four-bar such that the coupler of thesix-bar does not rotate the turntable.

3.5. A six-bar linkage is to be designed to raise a portable bench grinder from an initialposition on the bench to a final position resting against the garage wall. Figure P3.5

Wall

••••••

Figure P3.S


shows a synthesized four-bar in an intermediate position along the required path .Based on the cognate development in this chapter, design a six-bar to carry the grinderalong the path without rotating the grinder .

3.6. Design a six-bar parallel motion generator to lift a small boat off the top of a car to arack attached to the rafters of a garage. Figure P3.6 shows an acceptable four-barthat was synthesized for the following prescribed positions of motion generation :

Precision points


1 0 0 0 0

2 1.8 2.9 36 0

3 3.7 3.3 480

4 6.0 3.0 60 0

Use this four-bar in the development of your solution.

x

Figure P3.6

3.7. Design a six-link, approximate double-dwell mechanism (to replace a cam linkage) usingall revolute pairs based on the synthesized four-bar of Fig. P3.7a. Notice that thefour-bar traces a symmetric coupler curve [125] which has two circular arc sections.The output link should have a change of angle of 15° (see Fig. P3.7b) while the minimumtransmission angle at the output is 65°. .

Chap. 3

Figure P3.7

Problems

(a)

0 AOs =2.44AOA = 1.00BOs = 2.00AB = 2.00BC = 2.00AC = 3.00

OutputRotati on

!6

3 4Dwell

Input Rotat ion

(b)

269

21T

3.8. Design a six-bar linkage with revolute joints to replace a cam double-dwell mechanism .The output should oscillate 20° while the two approximate dwell periods should be100° and 35° of input link rotation. Figure P3 .8 shows the prescribed path of theprimary four-bar linkage (path generation with prescribed timing) together with thefive precision points and the input timing:

Precision points

Number x y Input crank rotation

1 0 0 0°2 -26.4 - 5.0 100°3 -21 .0 - 7.2 220 0

4 -7.0 -4.0 240 0

5 0.0 -;-4.25 255 0

(a) Synthesize a four-bar path generator with prescribed timing for the precision points.(If you wish to synthesize for only four precision points, leave out the fourth point.)

(b) Design the rest of the dwell mechanism such that the output link will swing only20°.


P,

Prescribed Path

Chap. 3

Input Timing Figure P3.8

3.9. A small autoclave is to be used to sterilize medical instruments. The door must bestored on the inside of the autoclave when it is open. The door must be closed by amechanism from the inside to form a seal with a gasket that allows the steam pressureto reach 15 psi on the inside of the vessel, forcing the door to stay closed. FigureP3.9 shows the space limitations of the door during its movement as well as its initialand final positions. Synthesize a four-bar mechanism that will open and close the auto-

pJ~~ (9.6, 11.25)

_ FinalPosition

Figure P3.9 Side view of autoclave . Door is 11.25 units; opening is 10.5 units .Ignore the thickness of the door.

Chap. 3 Problems 271

clave door. The suggested precision points are:

Precision points


1 0 8.7 0°2 2.8 11.125 -40.62°3 7.5 11.6 -79.24°4 9.6 11.25 -90.00°

3.10. Pick a fifth precision point for the linkage in Prob. 3.9 and compare the best linkagesynthesized with all five points to the linkage synthesized with four points. Suggestedadditional precision point: x = 0.8135, y = 9.5640, angle = - 16.8842°.

3.11. Synthesize a four-bar linkage that will move a small door from a vertical position infront of an automobile headlamp to a horizontal position above the headlamp. FigureP3.10 shows the door in five precision positions during its travel. The linkage yousynthesize must fit into the space available (within the rectangle). Although the precisionpoints are along a straight-line path, a slider is to be avoided if possible due to thewear that will occur in long-term use of this mechanism. Use the following precisionpoints (skip precision points 3 or 4 if only prescribing four points) :

I Precision points


1 2.0 8.0 0°2 4.0 8.0 30°3 6.0 8.0 50°-4 8.0 8.0 70°

. 5 10.0 8.0 90°

o

9

8

7

6

5

4

3

2

o ~~~~~~~~~::-0.~2 3 4 5 6 7 8 9 10 11

Un its

Figure P3.10 Rotations are 0°,30 °,50°,70°,90°cw.


3.12. Figure P3.11 shows a small bucket that is to be dumped into a large container and adesired path for the center of the bucket. Synthesize a four-bar mechanism that willlift the bucket and dump its contents into the container. Ground pivots are attachedto the container. Synthesize for either four precision point s (leaving out precision point3) or for all five precision points.

Precision points


1 0 0 0°2 -0.5 4 5°3 -1 .5 5 5°4 -2.0 5.5 60 °5 -2.5 5 120°

.4

x

Figure P3.11

iy

.2T5

1tl=14 ====:zz==!J.1_2

iy

x Figure P3.12


3.13. Synthesize a linkage to pick an object off the ground at (2, 0) and smoothly translateit, rotate it, and put it down at (0, 2) (see Fig . P3.12). Choose (2, I) and (1, 2) asadditional precision points and rotate the coupler 0°, 30°, 60°, and 90°, respectively.Restrict the ground pivots to within the triangle formed by the origin (0, 0) and theinitial (2, 0) and final points (0, 2).

3.14. Figure P3.13 shows a square 20 x 20 unit s with the origin at the center. Inside thesquare is a circle of radius 5 units centered at the origin. Design a four-bar motiongenerator that will contain the arrow within the coupler triangle and have it point atthe lower right comer. Contain the ground pivots within the square. The four prescribedprecision points are:

Precision points


1 5 0 0°2 0 5 7.12°3 -5 0 29.74°4 0 -5 36.86°

(-10,10)

( - 10, - 10)

(10,10)

(10 ,- 10) Figure P3.13

3.15. Choose a fifth precision point for Prob. 3.14 and find a linkage that gives similar results[we suggest (-2.38, 4.02) and a coupler angle of 13.44°]'

An obstacle is blocking the path of an object as shown in Fig. P3.14. Synthesize afour-bar linkage using the given precision points to move the object over the obstacle.

274

y

4 -

P33 - •

P22- •

Kinematic Synthesis of Linkages: Advanced Topics Chap. 3

1 1-

0,0

I

Obstacle

I

2

I

3

I

4

I

5 X Figure P3 .14

Precision points


1 0 0 0°2 1 2 45 °3 2 3 0°4 3 2 315 °

3.17. Add a fifth preCISIOn point to those used in Prob. 3.16 and find a four-bar linkagesolution. The suggested point is (0.799, 1.42) and a coupler angle of 59°.

3.18. Synthesize a four-bar linkage that can be used to open the garage door in Fig. P3.15.The following precision points and door angular positions are given:

Precision points

Number x y Angle of door

1 0 6 90°2 0.5 6.5 60°3 1 7 30°4 1.5 7.5 0°

(Notice that these points lie on a straight line.)

3.19. Figure P3.16 [(a) front view and (b) side view] is taken from U.S. patent 4,084,411(A. B. Mayfield). This device is a radial misalignment coupling that transmits constantangular velocity between shafts . The two shafts (12 and 13)are shiftable during operation,and the linkage system remains dynamically balanced.


Garage

~

8

•7 •

•6

5

4

3

Door_

2

32

a ~------ J'r- .......J

a

Figure P3.1S

dO

28 26

/3

(a) Determine the degrees of freedom of this device.(b) Describe briefly how this mechanism works.(c) Why is it designed the way it appears? Could you make some improvements?


3.20. Figure P3.17 shows a hydraulically driven industrial lift-table mechanism. The mechanismlifts a table (4 ft X 8 ft) on which a weight of 2500 lbf (max.) can be placed to aheight of 4 ft. The mechanism and table will collapse into a box-shaped region 4 ft X8ft Xlft.(8) Determine the number of degrees of freedom of this mechani sm.(b) Describe briefly how and why it works.

-- - - -- ----

Figure P3.17

3.21. Figure P3.18 shows a version of a lazy-tongs linkage .(8) Determine the number of degrees of freedom of this mechanism.(b) Describe briefly how and why it works.(c) Can you design a different mechanism for this task?

• Figure P3.18

3.22. A hinge is to be designed to be entirely inside a container when the lid is closed. FigureP3.19 shows a proposed six-bar design in the form of parallelograms.·(8) What type of six-bar is this?(b) Write the standard-form synthesis equations for a motion-generator task of moving

the lid with respect to the container.

• Suggested by T. Carlson.


t

th

t

-t-h

-{-

Side View: Closed

Front View

----- Side Walls - - -

_ - - Mechanism~--

Lid

Side Wall

Side View: Open

Figure P3.19

(c) For a set of four precision positions of your choice, design a six-bar to go throughyour specific set of positions.

3.23. Every golfer realizes the necessity of a good drive (or first shot) and also the importantrole that consistency plays in a good game. With these needs in mind, an automatictee-reset mechanism for use at a driving range was conceived.* This machine wouldhelp in practicing driving by automatically replacing another golf ball on the tee, andit would aid consistency because the golfer would not need to change his or her stance.The machine's task is to take one ball from a ball reservoir, gently place it on a tee,and retract out of the way without knocking the ball off (see Fig. P3.20). A crank-

BallReservoir

Figure P3 .20

4

--------~.... .......... "-

~CIUb /<;" ...._--------_' "\"'\crv / ......~ ------.;./...._--

j;;T:e3* This problem was originated by J. Peters, S. Yassin, and J. Arnold .


rocker motion-generator four-bar is desired. Measure the precision points from thefigure and synthesize a four-bar for this task. Ground pivots are allowed to be belowground since a permanent placement for this mechanism is assumed. The entire mechanism should be well out of the way of the golfer's swing in the rest position (position4).

3.24. An interesting mechanism is used by cabinetmakers-the Soss [254] concealed hinge(see Fig. P3.21). The entire mechanism is very compact and is embedded into thewood wall and door of the cabinet . Figure P3.21b shows that it will open 180°.(a) What type of linkage is the Soss hinge (see Chap . 1)1(b) Write the standard-form equation for the synthesis of this mechanism.(c) How does this design compare with the type shown in Fig. 3.191

(a)

Closed

Figure P3.21

Open 90°

(b)

Open 1800


3.25. Figure P3 .22 shows a type of mechanism used as an automobile window guidance linkage[275). To enter the door cavity properly, the window should have minimal rotationwhile the path of p follows a prescribed trajectory.(a) Verify that this mechanism has a single degree of freedom.(b) What type of linkage is this?(c) Write the synthesis equations for the linkage in the standard form.(d) Synthesize a mechanism of this type that satisfies a path and space constraints of

your choice.

Figure P3.22

3.26. A lock mechanism for a window must be designed such that the key will tum theinput crank of a slider-crank mechanism while the slider (the bolt) will travel a totalof 0.5 in. (see Fig. P3 .23) . The restricted space for the mechanism is such that themaximum distances are H = 0.65 in. at <P = <Pmax and L = 1.75 in. at <P = O. Themechanical advantage in the first position of the mechanism must be maximized whilethe deviation angle 8, is minimized at <P = </>max. Design the slider-crank (it. h L)for this objective.

Inputby Key

Bolt

IBait Movement~-----L-----'"

Figure P3.23 First position schematic .


3.27. It is proposed to design a Fowler wing-flap mechanism of the type shown in Fig. P3.24.·The objective is to avoid sliding contacts which are employed in present designs.The motion specified is a linear translation along the mean chamber for 15% of thewing width, followed by a 40° rotation downward. Further constraints are that thelinkage fits in all its positions inside a 10° angle between the top and bottom of thewing profile, and approximate the motion specified above between the precision points.(0) Determine the number of degrees of freedom of the linkage in the figure.(b) Check graphically to see that the mechanism shown accomplishes the design objec

tives.(c) Write the synthesis equations for this linkage for four prescribed positions in standard

form.(d) Describe how the synthesis will proceed. How many possible solutions are there

for this objective?(e) Design a mechanism of the type shown in the figure.

o--~--6

- --

--------------~------// 0 0 B

IfI\\ I' --- -

Figure P3.24

3.28. A linkage is required'[ to duplicate the motion of the human finger from the knuckleto the tip of the finger (Fig. P3.25). After careful study, a Watt I six-bar linkage (seeFig. 1.9) was chosen as the most likely to match four prescribed positions and to benarrow enough to match the size of a finger. The positions of P and the rotations oflink 6 are:

First position: finger fully extended, parallel with the back of the hand :

81 = 0, Oi;

Second position: finger slightly bent, as if one were holding a medium-sized glass:

82 = 1.475- 5.650i;

Third position: finger and thumb touching as if one were holding a piece of paper :

83 = 5.350 - 8.100i;

Fourth position: fingers almost forming a closed fist, such as when grasping asteering wheel:

84 = 10.350 - 6.650i;

• This problem was suggested by J. Boomgaarden [22].

t This problem was suggested by Kevin J. Olson.

Chap. 3 Problems

t1.25 em t1.5em

(a)

281

(b)

p

(2)

(e)

f/J

(d)

Figure P3.25

The four-bar labeled 1 (Fig. P3.25d) must be synthesized first. But to do this, onemust relate the 8j vectors to the 8j vectors . This can be done by choosing the vectorZ and solving for the following vector equation.

j = 1,2,3,4

choosing Z = -1.90, 1.00i.The calculated positions for point P' and rotations for coupler 4 become:


x y aO

8', 0 0 0

8'2 1.214 -3.263 44.5

8'3 3.217 -4.966 80

8', 6.348 -5.212 133

Chap. 3

Synthesize a Watt I six-bar for this task. The most challenging aspect of synthesizingthis linkage is designing the mechanism to fit the constraints of a human finger. Themechanism should be approximately 10 em long, 2.5 em from the tip to the first joint,2.5 em between the first and second joints, and 5.0 ern between the second joint andthe knuckle. The height of the first joint should not exceed 1.25 em, the second shouldbe no more than 1.5 em, and the knuckle should not be greater than 2.0 cm.

3.29. Figure P3.26 shows four different schematics of bucket loaders seen on work sites.(a) For each bucket loader:

(a)

Figure P3.26(b)

Chap. 3 Problems

(e)

(d)

Figure P3.26 (cont.)

283

(1) Draw an unsealed kinematic diagram.(2) What kind of mechanism is this? What task does it perform?(3) Determine the degrees of freedom of the mechanism.(4) How would you synthesize this mechanism in the standard form?

(b) Compare the performance of each design based on intuition and the knowledgegained in part (a).

3.30. Figure P3.27 shows schematics of two alternative designs for desk lamp mechanisms.(a) For each desk lamp mechanism:

(1) Draw an unsealed kinematic diagram.(2) What kind of mechanism is this? What task does it perform?(3) Determine the degrees of freedom of the mechanism.(4) How would you synthesize this mechanism in the standard form?

(b) Compare the performance of each design based on intuition and the knowledgegained in part (a).


(a)

(b)

Figure P3.27

Chap. 3

3.31. Figure P3.28 shows a schematic of a fill valve mechanism for a toilet tank .(a) Draw an unsealed kinematic diagram of this device.(b) What kind of mechanism is this? What task does it perform?(c) Determine the degrees of freedom.(d) How would you synthesize this mechanism in the standard form?

Chap. 3 Problems

FloatRod

285

t tt

WaterEntersHere

FillTube

;;/&N/$ //'?I I1 I Figure P3.28

3.32. Figure P3.29 shows an automobile hood mechanism [different from the two in Chap.I (Figs. 1.2c and P1.24»).(a) Draw an unsealed kinematic diagram of this device.(b) What kind of mechanism is this? What task does it perform?(e) Determine the degrees of freedom of this mechani sm.(d) How would you synthesize the mechanism in the standard form?(e) Compare this linkage's performance to that of the other two hood linkages based

on intuition and knowledge gained in parts (a) to (d).

Pins toFrame

Figure P3.29


3.33. An industrial robot manipulator designed for extracting formed articles (such as castings)is shown in Fig. P3.30 (designed by C. A. Burton, U.S. patent 3,765, 474, courtesy ofRimrock Corporation, Columbus, Ohio) . This machine was designed so that the linkagecould move out of the way of the die-casting process and have a nearly straight line

In it ial Posit ion

(a)

(b)

One MainSupport

ExtractingPosit ion

+

Figure P3.30


motion for up to a 70-in. stroke. The mechanism that closes the extractor is not shownhere.(a) Determine the degrees of freedom of this linkage.(b) Determine the type of this mechanism .(c) Write the standard form equations for the synthesis of this device.(d) Determine the length and accuracy of the straight-line path .(e) Try to design a linkage with better straight-line characteristics.

3.34. Figure P3.31 shows one of the early designs for typewriters. A multiloop linkage transfersthe finger movement of the typist to the magnified movement of the type bar.(a) Determine the degrees of freedom of this linkage.(b) What type of mechanism is this?(c) Write the equations for this mechanism in the standard form.(d) Design a typewriter mechanism with your own set of four or five precision points .

oFram ePivot s

Figure P3.31

3.35. Figure P3.32 shows the Garrard Zero 100 "zero-tracking" error mechanism for "elimination of distortion" in playback ." The articulating arm is designed to constantly decrease

Figure P3.32

/ "\/ 1/1/ ,

/ / , II, II

, I I

J I

I I »>.>Articulating ·Arm Pivot

PIckup HeadRemains Tangent

--- to Groove AcrossEnure Record--------• Popular Science. November 1971, pp. 94-96.


the pickup head/tone ann angle so the pickup head forms a tangent with the groovebeing played.(a) What task does this four-bar satisfy?(b) Write the standard-form synthesis equations for this task .(c) Design your own tone ann mechanism using your choice of four or five precision

points .

3.36. Interactive computer graphics were used to design a wing-flap mechanism [201) (seeFig. P3.33). Some desired positions were digitized and displayed on the computer screen(Fig. P3.33a). Four such positions yielded the m-k curves shown in Fig. P3.33b.A final linkage was picked interactively as shown in Fig. P3.33c and d.(a) Pick four or five of your own design positions and synthesize your own linkage

for this task.(b) Include in this synthesis the trailing flap that is shown in Fig. P3.33d.

.../ . , -0-, '. "...- - r--I---,,

II t,, ,, , ,, I~7

, I,

J' ,I ,.

., -,

'. "/,I •-, ,"

"~

'.

(b)

Figure P3,33

Chap. 3 Problems

--...,-II~

~~ ...........~l~ ~

,~~..l.-'r

~i'~~~~~

i:~ <, \

-,

~(c)

289

Figure P3.33 (cont.)(d)

3.37. Figure P3.34 shows a proposed design of an aircraft spoiler assist device [201].(A spoiler is a device that "spoils" the airflow around a wing to decrease lift. Spoilersare used in landing and for roll control.) The "q pot" is to act as a balancing deviceto the spoiler. With the assistance of the q pot, no power, other than that exerted bythe pilot, is required to actuate the spoilers. A linkage is desired to balance the q potand the spoiler throughout 60° rotation of the spoiler . After the entire dynamic systemwas modeled, a governing equation was derived. Using Chebyshev spacing for fourinput positions within a 90° range results in the following values of 0:

01 = 138.4254°,

O2 = 162.7794°,

03 = 197.2215°

0. = 221.5746°

For an approximate 60° spoiler rotation, the following values of </> result:

</>1 = 10.7342°,

</>2 = 33.0073°,

</>3 = 51.5953°

</>. = 60.3387°

Four coordinated positions of each crank and the locations of the fixed pivots werespecified at the interactive graphics console, after which the computer displayed theM-K curves shown in Fig. P3.34b. After selecting several linkages from the curves,

AEROD YN AMICPRESSUREINl EI

' q POI '

------/,

P

(a)

f-1\

,f'J/ SPOILER

, , ~~':" '-

~t._. ..:. :.: --"

1\-- --- ~~ ,

--- --" ---

, :, ..'....,.~

','

:... j b:.I

.,.-~ V, ,

I ...........( , vz...-,

(b )

/;V~!1{W~

~V ~

tl r>:

~~ ~E:/ ~

/

~

-

(c)

Figure P3.34 (d) Final design; pilot has only to break linkage past dead-center positio n and q pot willthen assist mot ion.

290

j I11Jll111 j I j I

I

(d)


the linkage shown in Fig. P3.34 c and d was chosen because it best fits within thespace requirements.(a) Using the precision points above, see if you can find the same or a better mechanism

for this task .(b) How sensitive is the final choice to small variations in input data (i.e., if you truncate

the decimals on the angles, what happens?)

3.38. A helicopter skid is to be retracted to clear a large rotating antenna (Fig . P3.35). Amechanism was designed for this task as shown in Fig. P3.35 [201).(a) Determine the degrees of freedom of this mechanism.(b) Write the synthesis equation for this mechanism in the standard form.(c) Pick four or five positions from the figure and design your own retracting linkage .

(a)RETRACTIONLINKAGE

Figure P3.35 Helicopter skid retraction mechanism.

291

2

3

4

5

OOWN

ACTUATOR

(c)

6

292

(b)


•

UP


3.39. Figure P3.36 shows several suggested mechanisms that have been designed [208,209]to replace bulky, noncosmetic, steel slide joints with linkages located entirely belowthe stump for the through-knee amputee. These mechanisms exhibit instant centersand fixed centrode (Chap. 4) that pass through the femoral condyle (upper portion ofthe knee) for stability reasons . These mechanisms are all different than the one shownin Fig. 1.16. For one or more of these designs:(a) Draw the kinematic diagram of the mechanism, and check the degrees of freedom.(b) What type of linkage is this?(e) How would you synthesize such a mechanism for this task?(d) Pick four or five prescribed positions and design your own through-knee prosthesis.

(0)

Figure P3.36

20'

( b) (c)

50 '60'70'eo'90'

L , - (. - - 1.540, 0.000)

.... = (0.750, 2.720)

" =(- 0.665,5.090)

~ , - (0- .445.6 '00.- )

Figure P33. 6 (cont.)

(d)

(e)


3.40. A six-link mechanism of Fig. 1.13 was designed for generating the five given path precisionpositions of the coupler point of link 5:

P1(X I> Y1) = (5.0,6.0)

P2(X2, Y 2) = (3.9,5.71)

P3(X3• Y3) = (3.06,5.202)

P4(X4 • Y4 ) = (2.716,4.429)

Ps(Xs• Y s) = (3.386,4.936)

Figure P3.37 shows the designed mechanism. The calculated values of the coordinatesof Co and C 1 are

Co = (X, Y) = (2.347846,2.916081)

C I = (X, Y) = (0.045052, 6.231479)

See if you can duplicate these results .

y

7.0

6.0

5.0

4.0

3.0

2.0

1.0

- 1.0 o

- 1.0

1.0 2.0 3.0 4.0 5.0 6.0 7.0 x

Figure P3.37


3.41. Prove that Eq. (3.38) contains only real terms.

3.42. Design a four-bar path generator with prescribed timing such that the path is an approximate straight line traveling through the following precision points along a straight line:

Precision pointsInput angle

Number X Y 8°

1 0.0 -0.25 0.02 1.0 -0.25 -16.0933 1.4 -0.25 -21.4234 1.79 - 0.25 -26.4045 2.16 -0.25 -31.006

3.43. Could the approximate straight-line four-bar mechanism of Prob. 3.42 be used to forma dwell mechanism? How?

3.44. A Stephenson III six-bar linkage (Figs. 1.13 and 2.66a) is to be designed so that thecoupler bodies rotate in opposite directions so as to be used as a "flying shear" or acrimping tool." The following precision points are to be used for the initial dyad (links5 and 6):

PrecisionpointsCoupler rotation

Number X Y a O

1 0.0 0.0 02 -0.625 -0.22 103 -1 .0 -0.625 204 - 1.25 - 1.25 375 - 1.0625 - 1.50 58

Once an acceptable dyad is chosen, design the rest of the linkage (we suggest specifyingZ3 = -1.007 - 1.092i) .

PrecisionpointsCoupler rotation

Number X Y a O

1 0.0 0.0 02 -0.4172 -0.3809 - 103 -0.5658 -0.9036 -204 -0.3900 -1.636 -325 -0.3369 -1 .8407 - 45

3.45. Dry powder ingredients for forming ceramic tile are contained in the hopper.At the proper time of the press cycle, the gate pivots open to dispense the "dust" intoa transfer slide which transports the dust to the die cavity on the next stroke.'] Thehopper and gate are existing. It is desired to use a 2-in.-stroke air cylinder to openand close the gate. It is further desired to have an adjustable gate opening to meter

• This problem contributed by A. S. Adams.

t Suggested by M. Nelson.


the amount of dust. The gate opening is to be variable by 2° increments from 10° to18° using the constant 2-in. air cylinder stroke. Design a mechanism for this task.

3.46. A four-bar path generator with prescribed timing is to be synthesized to generate asausage-like curve to be used to make a double-dwell mechanism [210]. The five precisionpoints are:

Precision points(polar form)

Input angleNumber R 8° po

1 1.0 0.0 0.02 t .740 - 29.50 117 .03 1.740 -10.70 150 .04 1.740 10.30 191 .05 1.740 25.90 228.0

(a) Design an acceptable four-bar for this task.(b) Complete the design for a double-dwell mechanism with good transmission-angle

characteristics.

3.47. Design a four-bar motion generator for five prescribed positions [210]:

Precision points(polar form)

Coupler angleNumber R 8° aO

1 1.5 0.0 0.02 1.275 33.7 12.03 1.0 90.0 24.04 1.275 146.3 36 .05 1.5 180.0 48.0

3.48. The table lists three examples of four-bar function generation [210]. The first exampleis identical with Freudenstein's optimum four-bar function generator based on Chebyshevspacing [104].

(A) (B) (C)

Function X' X X+sinX-a(X+2)

Interval of X 0:5:x :5:1 0 :5:X :5:6 0 :5: X :5: 1Range of .p (deg) 90.0 100 .0 85 .0Range of ljJ(deg) 90.0 60.0 60.0Precision points:

Xl 0.033689272 0.1468304511 0.02447174185X. 0.24917564 1.1236644243 0.2061073739X. 0.54280174 3.00000000 0.50000000X. 0.81636273 4.763355757 0.7938926261X. 0.9786319 5.8316954 0.9755282581

Design one or more of these five-point function generators.


3.49.· In the design of mach inery, it is often necessary to use a mechanism to convert uniforminput rotational motion into nonuniform output rotation or reciprocation. Mechani smsdesigned for such purposes are almost invariably based on four-bar linkages. Such linkages produce a sinusoidal output that can be modified to yield a variety of motions.

Four-bar linkages cannot produce dwells of useful duration. A further limitationof four-bar linkages is that only a few types have efficient force-transmission capabilities.Nevertheless, the designer may not choose to use a cam when a dwell is desired andaccept the inherent speed restrictions and vibration associated with cams. Therefore,he/she goes back to linkages.

One way to increase the variety of output motions of a four-bar linkage, andobtain longer dwells and better force tran smissions, is to add a link and a gear set.

Figure P3.38a shows a practical geared five-bar configuration including paired

Di splacerPiston

Cross Bar

PowerDisplacer-

Piston RodPiston

BufferGas-Tight SpaceStu ffing

Power-Box esPiston

Inpu tRod

Crank

Cont rolRods

(a)

Gears

(b)

Planet Gear

Stationary -----Sun Gear

(c)

Figure P3.38 (a) fixed-crank external gear system; (b) stirling engine system; (c) external planetary gearsystem.

• This problem adapted from Ref. 28.

Chap . 3 Problems 299

external gears pinned rotatably to ground. The coupler link (cross bar) is pinned to aslider. The system has been successful in high-speed machines, where it transformsrotary motion into high-impact linear motion . A similar system (Fig. P3.38b) is usedin a Stirling engine.(a) Verify the degrees of freedom of geared linkages in Fig. P3.38a and b.(b) Draw all the inversions of the geared five-bar in Fig. P3.38a.(c) Figure P3.38c shows a different type of geared dwell mechanism using a slotted

output crank.Verify the degrees of freedom of this mechanism.

3.50. A multi loop dwell linkage has been designed" as a combination punching and indexingdevice. The principle used is based on synthesizing a nearly circular portion of a fourbar coupler curve. While the four-bar traces that portion of the curve, a dyad pinnedto the tracer point at one end and to ground at the other (the intermediate point beinglocated at the center of curvature of the path) will exhibit a near dwell in the dyadsegment that is pivoted to ground. Figure P3.39a and b show photographs of the drivein two positions. A computer-generated animation of the motion of the dwell mechanismat 20° increments of the input crank angle is displayed in Fig. P3.39c. The complexnumber method was used to design a portion of this linkage.(a) Determine the degrees of freedom of this mechanism.(b) Describe the function of each loop of the dwell mechanism.(c) Describe how this mechanism could be synthesized using the standard-form approach.

(a)

(b)

Figure P3.39

• By W. Farrell, D. Johnson, and M. Popjoy under the direction of A. Midha at PennsylvaniaState University, September 1980 and described in Ref. [185].

Index ingLi nk

Figure P3.39 (cont. )

300

(c )

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Sandor, Arthur G. Erdman-Advanced Mechanism Design_ Analysis and Synthesis Vol. II (1984)

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