MIT Radiaton Lab Series V27 Computing Mechanisms and Linkages

8/8/2019 MIT Radiaton Lab Series V27 Computing Mechanisms and Linkages

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COMPUTING MECHANISMSAND LINKAGES


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iiMASSACHUSETTS INS TITUTE OF TECHNOLOGYRADIATION LABORATORY SERIES

Board of EditorsLOUISN.RIDENOUR,ddor-in-Ch@

GEOR~EB. COLLINS, Deput!/ Edztor-in-ChiefBRITTON CHANCE , S. A. GOUDSMIT, R. G. HER~, HUBERT M. JAMWS,JULIAN K. KN IP P,JAMES L. LAWSON , LEON B. LI~FORD, CAROL G. MONTGOMERY, C. NEWTON , ALEERTM. STOISE, LouIs A. TURNER, GEORC~ E. VALLEY, J R., H ERBERT H. WHEATON

1.2.3.4.5.6.7.8.9.10.11.12.13.14.15.16.17.18.19.20.21.22.23.24.25.26.27.28.

RADAIi SYSTEM l;KCXNE~R1~~ --l~zdenou~R.4DAR .\IDS TO N.4v1GAmo-~a/LR.ADAR BEAcoNs lioherlsLoRA~Pzmce,T1cKen zfe, a nd JVoodwardPULSE GEN ER.iroRs


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*

CO MPIJTINGMECHANISMS

AND LINKAGES

By AN TONIN SVOBODAEciitedby HIJB13RT M. JAMES

OFF ICE OF SCIEA-TIF IC RESEARCH AND D13VELOPM.ENr

NATION.4L DEFENSE RESEARCH COMMITTEE

FIRST EDITION

NEW YORK AND LONDONMcGRAW-HILL BOOK COMPANY, INC.

1948*


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.4 -=L. c

COMPUTING MECHANISMS AND LINKAGES

COPYRIGHT, 1S48 , EIYTHEMCGRAW-HILL Bo o < COMPANY, INC.

PRINTED 1>-THE l?N-ITEE ST LTES OF .iMERI~A

All rights reserved. This book, orparts thereof, ntay z,oi be reproducedin icn~j form dho,d permission of

the pubttshers.

THE M.4PLE PRESS (X)?I~PAh-Y,ORK, PA.


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Forezuord

T HE tremendous research and developmen t effor t tha t wen t in to thedevelopmen t of radar and rela ted techniques dur ing Wor ld War IIresu lted not only in hundreds of radar set s for milit ary (and some forpossible peacet ime) use but also in a grea t body of informat ion and newtechniques in the elect ronics and high-frequency fields. Because thisbasic mater ial may be of gr ea t va lu e t o scien ce and en gin eer in g, it seem edmost important to publish it as soon as secur ity permit ted.

The Radiat ion Labora tory of MIT, which opera ted under the super -vision of t he Na tiona l Defen se Resea rch Comm it t ee, u nder took t he gr ea tt ask of pr epa rin g t h ese volumes. Th e wor k descr ibed h er ein , h owever , isthe collect ive result of work done at many labora tor ies, Army, Navy,university, and industr ia l, both in this cou nt ry and in England, Canada,and other Domin ions.

The Radia t ion Labora tory, once it s proposa ls were approved andiin an ces pr ovided by t he Office of Scien tific Resea rch a nd Developmen t,chose Louis N. Ridenour as Editor-in-Chief to lead and direct the ent ireproject . An editor ia l sta ff was then selected of those best qualified forthis type of task. Finally the authors for the var ious volumes or chaptersor sect ions wer e ch osen from among th ose exper ts who were int imatelyfamiliar with the var ious fields, and who were able and willing to wr it ethe summaries of them. This ent ire staff agreed to remain a t work a tMIT for six months or more after the work of the Radiat ion Labora torywas complete. These volumes stand as a monument to this group.

These volumes serve as a memorial to the unnamed hundreds andth ou sa nds of ot her scien tists, en gin eer s, and ot her s wh o act ua lly ca rr iedon t he r esea rch , developmen t, a nd en gin eer in g wor k t he r esu lt s of wh ichare herein descr ibed. There were so many involved in this work and theywor ked so closely t oget her even t hou gh oft en in widely sepa ra ted la bora -t or ies th at it is impossible t o n ame or even t o kn ow t hose wh o con tr ibu tedt o a pa r ticu la r idea or developmen t . On ly cer ta in on es who wr ot e r epor tsor ar t icles have even been ment ioned. But to all those who cont r ibutedin any way to th is grea t coopera t ive development enterpr ise, both in th iscou nt ry an d in E nglan d, t hese volumes a re dedica ted.

m L. A. DUBRIDGE..$#,. 4 v


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Preface

THE work on linkage computers descr ibed in this volume was car r iedout under the pressure of war . War gives lit t le oppor tun ity for theadvancement of abst ract knowledge; all effor ts must be concent ra t ed onmeet ing immedia te needs. In developing techniques for the design oflin ka ge compu ter s, t he a ut hor has t her efor e been for ced t o con cen tr at e ont inding pract ica l methods for the design of computers ra ther than ondeveloping a unified and systemat ic analysis of th e subject . The war hast hu s given t o t his wor k a specia l ch ar act er t ha t it migh t n ot ot her wise h avehad.

The impulse to the development of the methods presen t ed in thisvolume for the mathemat ica l design of linkage computers grew out of acollabora t ion of the au thor with his fr iend, Dr . Vladimir Vand. That col-labora t ion was begun in France in 1940, and was brough t to a prematureend by the progress of the war . Though these ideas and methods havela rgely been developed by the author since that t ime, he wishes toemphasize tha t credit for the in it ia t ion of the work is shared by Dr .Vand. It must be ment ioned also that the techniques descr ibed in thisbook were for the most par t developed before the au thor became asso-cia ted wit h t he Ra dia tion La bor at or y.

The author wishes t o express sincere gra t itude t o Dr . H. M. J ames, theeditor of this volume, who gave the book its presen t form, cont r ibu t ingma ny examples an d ma ny im pr ovem en ts t o t he m et hods. (Sees.: 6.7,6.8,6.15, 8.6.)

The book would never have been completed in such a shor t t ime with-ou t t he assist an ce of Miss Con st an ce D. Boyd, wh o r ea d t he m an uscr ipt s,and Miss Elizabeth J . Campbell, Mrs. Kathryn G. Fowler , Miss VirginiaDr iscoll, and Miss Pat r ica J . Boland, who ca lcu la ted th e tables and dr ewnomograms. The author also wishes to thank Dr . I. Maddaus, J r., forbibliographical research.

The publishers have agreed that ten year s aft er the da te on whicheach volume in this ser ies is issued, the copyr igh t thereon shall berelinquished, and the work shall become par t of the public domain.

A. SVOBODA.., PSARA,CZECHOSLOVAmA,

J u ne, 1946.:. vii


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Contents

FOREWORD BYL. A. DUBBIDGE,.. vPREFACE.&AP . 1. COMPUTING

INTRODUCTION

,,. viiMECHANISMS AND LINKAGES. 1

11.1. Types of Comput ingMechanisms . I1.2. Su rveyof the Problem of Computer Design 21.3. Orgrmizat ionof the Presen tVolume 5

ELEMENTARYOMPUTINGECHANISMS 61.4. Addit ive Cells. . . . . . . . . . . . . . . . . . . . ...61.5. Mult ipliers . . . . . . . . . . . . . . . . . . . . . . . . 121.6. Resolvers, . . . . . . . . . . . . 151.7. Cams. . . . . . . . . . . . . . . . . . . . . . . . ...19I.S. In tegra tors . . . . . . . ,.23

CHAP.2. BAR-LINKAGE COMPUTERS 272.1.2.2.2.3.2.4.2.5.2.6.2.7.

In t roduct ion. . . . . . . . . . . . . . . . . . . . . ...27Hietor iea lNotes. . . . . . . . . . . . . . . . . . ...28The P roblem of Bar -linkage-computerDesign 3 ICharacter ist icsof Bar -liikage Computers. 32Bar Linkageswith One Degreeof Freedom 34Bar Linkageswith Two Degreesof Freedom. 37ComplexBar -linkageCompu ters. 40

CHAP. 3. BASIC CONCEPTS AND TERMINOLOGY. 433.1. De6nit ions . . . . . . . . . . . . . . 433.2. HomogeneousParametersand Variables 473.3. An Opera torFormalism. 493.4. Graphica lRepresen ta t ionof Opera tors 513.5. The Squareand Square-rootOpera tors 54

CHAP.4. HARMONIC TRANSFORMER LINKAGES. 58THE HAB~ONXCRANSFORMED.. . . . . . . 58

4-1. Defin it ionand Geomet ryof the HarmonicTransformer . 584.2. Mechaniza t ionof a Funct ion by a Harmonic Transformer . 61ix


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x4.3.4.4.45.4.6.4.7.48.

CONTENTSThe Idea l Harmonic Trans former in HomogeneousParameter s , 62Tables of HarmonicTransformerFunct ions 63Tot al St ru ct ur alE rr or of a Non idea lH armon icTr an sformer 67Calcula t ion of the Structura lError Funct ion ~H~of a NonidealHarmonicTransformer . 6SA Study of the Structu ra lE r ror Funct ion 6H~. 71A Method for t heDes ignof Non idea lHa rmon icTransformer s. 75

HARMONIC TRANSFORMERS IN SERIES.49. Two Ideal HarmonicTransformersin Series. .410. Mechanizat ionof a GivenFun ct ionby an IdealDouble HarmonicTransformer. . . . . . . . . . . . . . . .4 .11. Pre liminaryFit to a MonotonicFunct ion.412. Pre liminaryFit to a NonmonotonicFunct ion4.13. Impr ovemen t of t h e F it by a Met hod of Successive Approxirna -t ions. . . . . .4.14. Nonideal Dou ble H ar mon ic Tr an sfor mer s.4.15. Alterna t ive Method for Double-harmonic-t ransformer Design

777?798289!)1m

][)1

CHAP.5. THE THREE-BAR LINKAGE . . . . . 1075.1. Fundamenta lEqua t ions for the Three-ba rLinkage. 1075.2. Classifica t ionof Three-ba rLinkages 1085.3. Singu la rCasesof Three-bar Linkages, 1 ~25.4. The Problem of Design ing Three-bar Linkages. . 117

THE NOMOGRAPHIC kfETHOD . . . . . . . . . . . . . . . . . . . . . 11855.5.6.5.7.5.8.5.95.10.5,11.512.513.5.14.

Analyt ic Basis of the homographic Method. 118The Nomographic Char t . . . . . . . . . . . . . . . ...120Calculat ion of the Funct ion Genera ted by a Given Three-barLinkage .,.,.......,,, . . . . . . . . ...122

Complete Representa t ion of Three-bar-linkage Funct ions by theNomogram, ,., . . . . . . . . . . . . . . . . . ..125

Restatement of the Design Problem for the NomographicMethod, . . . . . . . . . . . . . . . . . . . . . ...127

Survey of the Nomographic Method . 128Adjustment of ba and a, for Fixed AX,, AX,, b, . 132Alterna t ive Methods for Over lay Const ruct ion . . . . 136Choice of Best Value of b, for Given AX,, AX,. . . . 137An Example of t he Nomographic Method. . 139

THE GEOMETRICETHonFORTHR~E -BLRLINKAGEDESIGN. . . . 1455.15. Sta tementof the Problem for the Geomet r icMethod. . 1465.16. Solu t ion of a SimplifiedProblem. . . . . . 1475.17. Solut ion of the Basic Problem. . 1515.18. Improvemen tof t he Solu t ionby Succes siveApproxima t ion s. 1545.19. An Applica t ion of the Geometr ic Method: Mechaniza t ion of the

Imgar ithmic Funct ion . . . . . . . . . . . . . 156


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CONTENTS xiCHAP. 6. LINKAGE COMBINATIONS WITH ONE DEGREE OF FRf3E-DOM . . . . . . . . . . . . . . . . 166

COMBINATION OF Two HARMONIC TRANSFORMERS WITH A THREE-BAR I .lNK-AGE . . . . . . . . . . . . ., .,,., 1666.1. Sta tement of the Problem. . 1666.2. Factor iza t ion of the Given Funct ion . 1686.3. Example: Factor ing the Given Funct ion 1716.4. Example: Design of the Three-bar -linkage Component 1746.5. Redesign of the Terminal Harmonic Transformers 1866.6. Example: Redesign of the Terminal Harmonic Transformers 1876.7. Example: Assembly of the Linkage Combinat ion, 193

THREE-BAR LINKAGES IN SERIES. 1956.8. The Double Three-bar Linkage 195

CHAP. 7. FINAL ADJUSTMENT OF LINKAGE CONSTANTS 1997.1. Roles of Graphica l and Numer ica l Methods in Linkage Design. 19972.GaugingP arameters . . . . . . . . . . . . . . . . . . ..2o273. Use of the Gauging Parameter in Adjust ing Linkage Constant s. 2017.4. Small Varia t ions of Dimensional Constants 2057.5. Large Varia t ions of Dimensional Constants 2057.6. Method of Least Squares 2067.7. Applica t ion of the Gauging-parameter Method to the Three-ba r

Linkage . . . . . . . . . . . . . . . . . . . . . ...2077.8. Applica t ion of the Gauging-parameter Method to the Three-bar

Linkage. An Example,...,.. . .,........2097.9. The Eccen t r ic Linkage as a Correct ive Device. 217

CHAP. 8. LINKAGES WITH TWO DEGREES OF FREEDOM 2238.1. Analysis of the Design Problem 22382. Possible Grid Genera tors for a Given Funct ion . 2268.3. The Concept of Gr id St ructu re 2288,4. Topologica l Transformat ion of Gr id Structu res. 2328.5. The Significance of Ideal Gr id Structure . 2338.6. Choice of a Nonideal Gr id Genera tor . . . 2388.7. Uw of Grid St ructu res in Linkage Design. . . . 243

Cu . 9. BARLINKAGE MULTIPLIERS . . . .2509.1.9.2.9.3.9.4.9.5.9.6.9.7.9.8.

The Star Grid Genera tor ,, .250A Method for the Design of Star Gr id Genera tors with AlmostIdeal Gr id St ructu re.....,.. . . . . . . . . ...251

Gr id Genera tors for Mult iplica t ion . 256A Topologica l Transforma t ion of the Gr id St ructu re of a Divider . 258Improvement of the Star Gr id Genera tor for Mult iplica t ion. 264Design of Transformer Linkages. 271Analyt ic Adjustmen t of Linkage Mult iplier Constan ts 277Alternat ive Method for Gauging the Er ror of a Gr id Genera tor 281


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]I xii CONTENTS

~H,4P. 10. BAR-LINKAGE FUNCTION GENERATORS WITH TWO DE-1 GREES OF FREEDOM . . . . . . . . . . . . . . . . . . ...284

10.1. Summaryof the Design Procedure. . . . . . . 284,I 10.2. Example: First Approximate Mechanizat ion of the Ballist ici Funct ion in Vacuum . . . . . . . . . . . . . . . . ...286I 10.3. Example: Improving the Mechanizat ion of the Ballist ic Funct ionin Vacuum .,,..... . .292

10.4. Curve Tracing and Transformer Linkages for Noncircula r Scales 29541 APPENDIX A. TABLESOF HARMONICRANSFORMERUNCTIONS>APPENDIX B. PROPERTIESF THETHREE-BAR-LINKAGEOMOGRAMINDEX . . . . . . . . . . . . . . . . . . . . . . . .

. . . 301333

. 353


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CHAPTER 1INTRODUCTION

1.1. Types of Comput ing Mechanisms.Comput ing mechanismsmay be divided in to two dist in ct t ypes: ar it hm et ica l compu tin g machines,familia r to the layman through their common use in business offices, andcont inuously act ing comput ing mechanisms and linkages tha t range incomplexity fr om simple cams and lever s t o enormously complex devicesfor t he dir ect ion of na va l an d a nt iair cr aft gu nfir e.

The ar ithmet ica l comput ing machines accept inputs in numericalform, usually on a keyboard, and with these numbers per form the simplear ithmet ica l oper at ions of addit ion , subt ract ion , mult iplica t ion , anddivision-usually by t he iter at ion of addit ion and su bt ract ion in cou nt ingdevices. Th e r esu lt s a re fina lly pr esen ted t o t he oper at or , a ga in in n umer -ical form. In their simplest forms these machines have the vir tue ofa pplica bilit y in a wide va riet y of compu ta tion s, in clu din g t hose r equ ir in gvery high accuracy. By elabora t ion of these devices, as by the in t roduc-t ion of pu nch ed-t ape con tr ol, t heir possibilit ies for au tomat ic oper at ioncan be grea t ly increased. Character ist ic of their opera t ion , however , istheir product ion of numer ica l results by calculat ions in discrete steps,involving delays which are a lways appreciable and may be very largeif t he r equ ir ed calcu la tion is of complex form.

Con tin uou sly a ct in g compu tin g mecha nisms a re less flexible an d h aveless pot ent ia l a ccur acy, but th eir applicability t o t he instantaneous or t ot he con tin uou s solu tion of specific pr oblem s-even qu it e complex on esmakes them of grea t pract ica l importance. They may serve as mereindica tors of the solut ions of a problem, and require fur ther act ion byhuman agency for the complet ion of their funct ion (speedometer , slideru le); or t hey may t hemselves pr odu ce a mechanica l act ion funct ionallyrela ted to other mechanica l act ions (mechanica l governors, automat icgunsight).

Cont inuously act ing computers fall in to two main classes: funct iongen er at or s and different ia l-equat ion solvers. Fun ct ion gen er ator s pr o-du ce mechan ica l a ct ion susually displa cemen ts or sh aft r ot a tion stha tare det it e funct ions of many independent var iables, themselves in t ro-duced in to the mechanism as mechanica l act ions. Simple examples ofsu ch mech an ism s a re gea r differ en tia ls, two- a nd t hr ee-dimen sion al cams,slide mult iplier s a nd divider s, lin ka ge compu ter s, a nd mech an ized n omo-gr am s. Compu ter s of t he secon d cla ss gen er at e solut ion s of some defin it e1


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2 INTRODUCTION [SEC.1.2different ia l or in tegrodifferen t ia l equa t ionoften an equat ion tha tinvolves funct ions cont inuously determined by var iable externa l cir -cumstances. Elementary devices of th is type are the in tegra tor s, com-ponen t solver s, speedomet er s, a ndpla nimet er s.From these elementary devices one can build up complicated mecha-nisms tha t per form elabora te calculat ions. We may ment ion theirapplicat ion in gunsights, bombsights, automat ic pilot s (for airplanes,submarines, ships, and torpedoes), compensa tors for gyroscopic com-passes, t ide predictor s, and other robots of var ied types.

The present volume will dea l only with the problem of design ing con-t in uou sly a ct in g compu tin g mech an ism s.

1.2. Survey of the Problem of Computer Design .-There is no setrule or law for the guidance of a designer of complex mechanica l com-puter s. He must weigh against each other many diverse factor s in theproblem: th e accu racy r equ ir ed; th e cost , weight , volume, and shape of t hecomputer ; it s iner t ia and delay in act ion; the forces requ ired to opera te it ;it s resistance to shock, wea r , and changes in weather condit ions. He mustconsider how long it will t ake to design the computer , how easily it canbe built , how easily it can be opera ted by a crew, whether suitable sourcesof power will be available, and so on . Th e complexit y of t he t heor et ica land pract ica l problems is so grea t tha t two designers working on a givenpr oblem will n ever a rr ive a t pr ecisely t he same solu tion .

For pract ical reasons, a designer should be asked to find a computertha t meets cer ta in specified tolerances, ra ther than the best possiblecomputer for a given use. He should know what will be the maximumtolera t ed er ror of the computer , the maximum cost , weigh t , and volumeoccupied, the maximum number of opera tors in the crew, the maximumnumber of servomechanisms allowed, and so on . Tolerances provide acon ven ien t means for cont rolling th e developmen t of th e compu ter , andif established in a pract ica l waythey permit some freedom of choice bythe designer .

Ch oice of Appr oa ch t o t heDesign P roblem .The t ype of compu ter t o bebuilt is somet imes indica ted in the specificat ions. If not , the first t askof the designer is to decide whether the computer is to be mechanical,elect r ical, opt ica l, or a combina t ion of these. At the same t ime tha t th isimpor tant decision is made, the designer must weigh in his mind the pa ththa t his th inking will follow. There are two pr incipal methods for design-ing a computer : the const ruct ive method and the analyt ic.

The const ruct ive method makes use of a small-sca le model of the realsystem with which the computer is to deal. For example, a con st r uct iveant ia ircraft fir e-con t rol computer might determine the elements of thelead t r iangle by mainta in ing within itself and measur ing the elements of asmall model of th is t r ian gle.


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SEC.1.2] SURVEY OF THE PROBLEM OF COMPUTER DES IGN 3In using t he a nalyt ic met hod, t he designer con cen tr at es on t he ana lyt ic

r ela tion s between t h e va ria bles in volved. A r ela tion between va ria bles,such as

(1)can be given mechanical expression in terms of displacements or shaftrota t ions, withou t regard to the na tu re of the quant ity represen t ed by thevar iables x, y, and z. For example, one may possess two devices tha tgenera te ou tpu t displacements xy and x/y, respect ively, given input dis-placements z and y. Combining these with a th ird device for adding theirou tpu t displacements, one can then pr oduce a computer tha t , given inputdisplacements x and y, genera tes a final ou tput displacement z havingcon t inuously the va lue specified by Eq. (1). The computer is then amechaniza t ion of Eq. (1), ra ther than a model of any specia l systeminvolving var iables x, y, and z thus rela ted.

Computers designed by analyt ic methods consist of units (cells)tha t mechanize fa ir ly simple relat ions, so connected as to provide amechaniza t ion of a more complex equat ion or system of equat ions. Forany given problem a grea t va r iety of designs is possible. This var ietyar ises in pa rt fr om t he possible ch oice among mecha nical cells mech an iz-ing a given elementary rela t ion , and in par t from the var iety of ways inwhich the rela t ion between a given set of var iables can be given analyt icexpression. Thus, each of t he equat ions

Z=; (yj+ l),( )=X y+~, YZ7J= Z(yz + 1),

(2a)(2b)(2C)[all equiva len t to Eq. (l)] suggest s a differen t method of connect ing

mechanical cells in t o a complet e compu ter . Th is flexibilit y in a na lyt icdesign methods makes it possible to a r r ive at designs that a re in genera lmore sa t isfactory mechanically than those obta ined by const ruct ivemethods.

In the pr esen t volume we shall be con cerned ent ir ely with mechanica lcompu ter s design ed by t he a na lyt ic m et hod.

Block Dia gr am of t he CompuLer .To ea ch formu la tion of t he pr oblemin analyt ic terms there cor responds a block diagram of the computer .in thk diagram each analyt ic rela t ion between variables is r epresen tedby a square or similar symbol, from which emerge lines represen t ing theva ria bles in volved; a lin e r epr esen tin g a va ria ble common t o two r ela tion swill connect the cor responding squares in the diagram. In mechan ica lterms, each square then represen ts an elementa ry computer tha t estab-


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4 INTRODUCTION [SEC.1.2lishes a specified r ela t ion between th e var iables, and t he con nect ing linesrepresen t the necessary connect ions between these elementa ry com-puter s. By examinat ion of block diagrams the designer will be able tosee the pr incipal vir tues of each comput ing scheme: the complexity of thesyst em , t he wor kh g r an ge of var ia bles, t he a ccu ra cy r equ ir ed of in dividualcomponents, and so on . On this basis he can make a t least a ten ta t iveselect ion of t he block diagram t o be u sed.

Select ion oj Compon en ts jor t he Compu ter .-Kn owin g t he a ccu ra cy an dmechanical proper ties r equ ired of each comput ing elem ent , t he design ercan select the elementa ry computers from which the complete device is tobe bu ilt .

As an example of the diverse factor s to be borne in mind, let us supposethat it is required to provide a mechanical mot ion propor t ional to theproduct of two var iables, X1 and X2. A slide mult iplier of average sizewill a llow an er ror of from 0.1 per cent to 0.5 per cen t of the whole rangeof the var iable; th is er ror will depend on the quality of the const ruct ion-on the backlash and the elast icity of the system. A linkage mu lt iplierwill have an er ror of some 0.3 per cen t due to it s st ructure, pract ically noer ror from backlash , and a slight er ror due to elast icity of the system if theunit is well designed; the space required by a linkage mult iplier is small,but it s er ror cannot be reduced by increasing its size. If these devices don ot pr omise sufficient accur acy, t he designer must use mult ipliers basedon other pr inciples. It is possible to per form mult iplicat ion by use oft wo of th e precision squar ing devices illust r ated in Fig. 1.23, by con nect -ing these in the way suggested by the equat ion

X1X2 = +(X1 + x2)* *(X1 X2)2. (3)The er ror of such a mult iplier may be as low as 0.01 per cen t , but thesystem has an appreciable iner t ia . About the same accuracy is at ta in-able by a mult iplier based on th e different ia l formula for mult iplica t ion ,

d(xlxz) = xl dxz + x2 dxl; (4)this employs two integra tor s, and is commonly used when two quant it iesa re t o be mult iplied in a differ en tia l an alyzer . Th is sch eme is u sefu l on lywhen it is possible to a llow a slow change in a constant added to theproduct XIX~a change which will result from slippage in the in tegra-t or s, n egligible for a single mult iplica tion bu t a ccumu la tin g wit h r epet it ionof t he oper at ion .

From this discussion it should be evident that there is no bestmult iplier . Simila r ly, other components of a computer must be select edwith due regard for their specia l character ist ics and the demands to bemade upon them.

Mathematical Design oj the [email protected] the block diagram oneshould proceed to the mechanical design of a system through an in ter -


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SEC.1.3] ORGAN IZAT ION OF THE PRESE>T VOLUME 5media te step-tha t of establishing the mathemat ica l design of thesystem. The mathemat ical design ignores the dimensions not essent ia lto the nature of the computa t ion to be ca r r ied out -diameters of shafts,dimensions of ball bear ings, dimensions of the fram~but specifies thedimensions of lever s measur ed bet ween pivots and joints, t he size of fr ic-t ion wheels, t enta t ive gear diameters and gear ra t ios. The proper t ies ofthis design should be studied carefully, because this usually leads to achange in some deta il of the design , and somet imes even to choice of a newblock diagram.

Final Steps in the Design .-F rom the mathemat ical design of thesystem one can pr oceed to the design of a working model. The elemen t sof this model should be accessible ra th er than massed t oget her , in expen -sive, and quick to manufacture. If the performance of the working modelis found to be sat isfactog, the fir st model can be designed. Here theingenuity of the designer must be used to the maximum. The par ts ofthe mechanism must be a rranged compact ly to decrease space require-ments, weight , and the effect s of elast icity and thermal expansion, butthey should not be massed in such a way that assembly is difficu lt , orrepair or servicing impossible. Somet imes division of t h e whole compu t erinto severa l independent par ts is advisable. F inally, the computer can bebu ilt a nd t es ted aga in st specifica tion s.

1.3. Organiza t ion of the P resen t Volume.It is not possible to dis-cuss in one volume all elements of the problem of computer design . Thisbook will dea l pr in cipa lly wit h ba r-lin ka ge compu ter sspecifica lly, wit hthe mathemat ica l design of elements for such computers. Bar linkagesa re mechanica lly very sa t isfactory, and computers built from them havemany impor tant vir tues, but the mathemat ical design of these systemsis rela t ively difficult and is not widely understood. There are few stand-ar d bar -linkage element s for compu ter s; it is usually n ecessar y t o designthe components of the computer , and not merely to organize standardelemen ts in to a complex a ssembly. It is hoped that the design methodsto be descr ibed here will lead to their more genera l use.

Bar linkages can be used in combin ation with t he standar d comput ingmechanisms. For this reason , and for the contrast with the bar link-a ges wh ich a re t o be discu ssed la ter , t his volume begin s wit h a br ief su rveyof some mor e or less st an da rd elem en ts of mech an ica l compu ter s. Chap-ter 2 is devoted to a genera l discussion of bar linkages. Chapter 3establishes terminology and descr ibes graphica l procedures of whichextensive use will be made. Chapter s 4, 5, and 6 discuss, in order of theirin cr ea sin g complexit y, ba r lin ka ges wit h on e degr ee of fr eedomgen er -a tor s of funct ion s of one independen t va ria ble, Ch apt er 7 in dica tes somemathemat ica l methods of importance in bar -linkage design. Finally,Chaps. 8, 9, and 10 develop methods for the design of bar-linkage gener -


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6 INTRODUCTION [SEC.14a tors of funct ions of two independent variablesa field in which barlin ka ges h ave ver y st rikin g a dva nt ages.

ELEMENTARY COMPUTING MECHANISMSThe remainder of th is chapter will gi~e a br ief su rvey of elementary

comput ing mechanisms, or cells, of more or less standard type. Dis-cussion of bar -linkage cells will be defer red to Chap. 2.

1.4. Addit ive Cells.- Addit ive or linear cells establish linearr ela t ions bet ween mecha nical mot ion s of t he cell, usually shaft r ot at ionsor slide displacements. If these a re descr ibed by parameters X1, X2, X3,t he cell will compu te

X3= Q. X,+ Q. X,+C. (5)Here Q, Q, and C are constants depending on the design of the cell and thechoice of the zero posit ions from which Xl, XZ, and X3 are measured. By

FIG. 1 1.Bevel-gear clifferential.proper choice of the zero posit ions, C can always be made to vanish; inwhat follows it will be assumed tha t th is has been done.

The bevel-gear deferent ia l (Fig. 1.1) is a well-known linear CCI1f[J rwhich all t hr ee pa ramet er s a re r ot at ions. The parameter Xl is the rot :~-t ion of the shaft S1 from a predetermined zero posit ion , X1 = O; the posi-t ive direct ion of rota t ion is indicated by symbols represen t ing the hewland tail of an a r row with this dir ect ion . The parameter X2 is the rota t ionof the shaft SJ from a similar zero posit ion; Xj is the rota t ion from its zeroposit ion of the cage C car rying the planeta ry bevel gears G. The zeroposit ions ar e not indica ted in the figure.

The equat ion of the bevel-gear different ia l isX, = 0.5X, + 0.5XZ. (6)

To der ive this it is convenien t t o consider the va lue of X2 corr esponding t ogiven values of X, and Xs. Let us consider the different ia l to be or igina llyin the posit ion X1 = X2 = X3 = O. The parameters Xl and X3 can thenbe given their assigned values in two steps, the fir st a rota t ion of both the


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SEC,14] ADDITIVE CELLS 7shaft S1 and the cage C th rough the angle X3, and the second a rota t ion ofthe shaft S1 through an addit iona l angle Xl Xi. In the first step thedifferen t ia l moves as a unit ; the shaft Sz is rota ted through the angle X3.In the second step, the cage is sta t ionary and the movement of the shaftS1 is t ransmit ted to the shaft SZ with it s sense of rota t ion reversed; therota t ion through angle Xl X3 of the shaft SI causes rota t ion throughXt Xl of the shaft SZ. The tota l rota t ion of the shaft SZ is thenX2 = Xt + (Xt X,), from which Eq. (2) follows immediately. It is,of course, essent ia l tha t all rot a t ions be taken as posit ive in the samesense.

It is remarkable tha t Eq. (6) is independent of the ra t io of the bevelgear ing of the clifferen t ia l; the essent ia l character ist ic of this type of

X3

FIG.1.2.Cylindrica1-gearlifferen t ial.

different ia l is t ha t t he gear in g of t he cage t ransmits th e rela t ive mot ion ofthe shaft S1 to the shaft SZ in the ra t io 1 to 1, but with r eversed sense. Itis not necessary to use bevel gears in the cage to obta in this resu lt ;cylindr ica l gears can accomplish the same purpose. A cylindr ica l-geardifferent ia l is shown in Fig. 1.2. This differen t ia l is equivalen t to thecommon bevel-gea r differ en tia l, except in it s m ech an ica l fea tu res. It isfla t ter , and easier to const ruct in la rge numbers, bu t there is one moregea r mesh than in the common type; there may be more backlash andmore fr ict ion . It should be noted, however , tha t bevel gears a re subjectt o axia l as well as radia l forces in their bear ings, and that these may alsoincrease friction.

The spur -gear differen t ia l shown in Fig. 1.3 has only t wo gea r meshes,and is quite fla t . The planeta ry gears G in their cage C do not inver t themot ion of the shaft SI when t ransmit t ing it to the shaft S3, but can bemade to t ransmit it a t a ra t io differen t from 1. The eauat ion of th is


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8 INTRODUCTION [SEC.1.4differen t ia l is

X, = QX, + (1 Q)X,. (7)To prove this rela t ion we can use the same method as before. Let usbegin by con sider in g t he differ en tia l in t he zer o posit ion ,

X1=X,=X3=0.We wish to find the value of X3 cor responding to given X, and X2. Wein t roduce the angles Xl and X2 in two steps, fir st tu rning both the shaft

FIG. 1.3.Spur -gear clifferent ial.

FIG.1.4.Differentialwith axia lly displaced spira l gear .S, and the cage C through the angle X2, and then the shaft S1 through anaddit ional Xl - X,. In the fir st st ep the different ia l is tu rned as a r igidbody; the shaft S, is a lso turned through the angle X2. ln the secondst ep the shaft Ss is tu rned through Q(xl X2); its total motion isX, = X, + Q(X, X2), in agreement with Eq. (7).

If we make Q = Q = 0.5 by proper choice of the gear ra t ios, we canobt ain a differ en tia l equ iva len t t o t he bevel-gea r differ en tia l. Th e fa cttha t the fr ee ch oice of Q gives to th is differen t ia l a la rger field of applica-bility does n ot n ecessar ily mean t hat th is differ ent ia l sh ould be pr efer red


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SEC.1.4] ADDITIVE CELLS 9tothosewith Q = 0.5; it inconvenien t tousedifferent ia ls with Q =0.5a s pr efa br ica ted st an da rd elemen ts.

A deferen t ia l mth axially displaced spiral gea r is shown in Fig. 1.4.The parameter ,, which measures the axial displacement of the spiralgea r a ndt hepin PZ, invar iable on ly ~\ -ithinfinit e limits. Th e mecha nicalst ructure of this different ia l is, however , much simpler than that of thediffer en tia ls a lr ea dy men tion ed, for wh ich a ll pa ramet er s ca n ch an ge wit h-out limita t ion. The equat ion of this differen t ia l is

X3=X1*9X2, (8)

where nis the number of threads per inch along the axis of the spiral gearon the shaft Sz and m is the number of teeth on the gear with which it

FIG.1.5.Differential worm gearing.meshes. Thehelica l mgleofthe gears should beat least 450 for smootha ct ion a nd small ba ckla sh .

Thediferent ia l worm gear ing shown in Fig. 1.5 is used for the samepurpose asthe preceding differen t ia l, especia lly if the range of values ofX, cor responds to a la rge fract ion of a revolu t ion of the shaft S, or evento severa l revolu t ions of this shaft . The equat ion of this d%-ent ialis

x,= ++ X1+ +X2 (radians) (9)where t is the number of teeth of the worm gear , m is the mult iplicity ofthe threads of the worm, and R is the radius of the worm gear .

The sign in Eqs. (8) and (9) depends on the sense of the threads of thespira l or worm gea r.

Th e scr ew differen tia l sh own in Fig. 1.6 combin es an axial t ra nsla tionXl of a screw with a t ransla t ion X2 of the nut N with respect to the screw;

x3 = xl + x,. (lo)


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I

10 INTRODUCTION [SEC.1.4To obta in the fir st t ransla t ion , the pin P on which the screw tu rns isdisplaced by X1. Therota t ion of thescrew comes fromthegear G,wh ichmeshes with a cylindr ical rack C and slides a long it . The real inpu t

IWILI

x,i *

-+E=Eaii c-kFIG.1.6.Screwdifferential.parameter of the differen t ia l is not X,, bu t the angle X, th rough which therack is tu rned. The equa t ion of the differen t ia l is then

X3 = x, * kx4. (11)The sign depends on th e sense of the screw; k is a constan t determined bythe gear ra t io, the number of th reads per inch on the screw, and their

.++FIG.1.7.Belt different ial.

mult ipli~ity. All t hr ee pa rameters of th isd iffer en t ia l h ave con st r uct ive lim it s.

The belt deferen t ia l (F ig. 1.7) makesuse of the inextensibility of a belt ing onsevera l pu lleys. In pract ice, chains,st rin gs, a nd specia l ca bles a re u sed a s belt s.Th e equa tion of th e belt differen tia l is

X3 = c 0.5XI 0.5X2, (12)where C is a constan t depending on thech oice of zer o poin ts of t he pa rameter s.

The tension in the belt must not fa llbelow zero a t any t ime; if it does, the beltwill sa g a nd t he equ at ion of t he differ en tia l

will not hold. To obta in posit ive act ion in the direct ion of increasing X3,it is necessa ry to preload the belt by put t ing a load on the ou tpu t pu lleyior instance, by a spr ing tha t can exer t a force la rge enough to produce th edesir ed act ion . The maximum driving force requ ired for thk dMer -en t ia l will t hen be about twice the force necessa ry to opera t e it withou tpreloadlng.

Th e loop-belt di~er en tia l (F ig. 1.8) has t he belt in g in t he form of a loopwith length independen t of the posit ion of the pu lleys, The belt can then


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SEC-1.4] MULTIPLIERS 11be preloaded (turnbuckle B) withou t adding to the dr iving force of thediffer en tia l, except by t he in cr ea sed fr ict ion in t he bea rin gs.

Belt differen t ia ls a re some-t imes used to add a la rge numberof parameters; t hey a re easilycombined in ba t t er ies , a s indica t edschemat ica lly in Fig. 1.9. In such -an ar rangement the parameter XTmay have so la rge a range that itis impract ica l to use a slide as theou tpu t t ermin al. It is bet t er FIG.1S.-Loop-belt cl ifferen t ia l.pract ice to use a drum (dashedline in Fig. 1.9) on which the belt is wound on. and a t the same t imewound off. To preven t slippage, the belt should make many turns on the

FIG. 1.9 .Loop-bel t d ifferen t ia l forthe evaluat ion ofX1= CX12XZ+2X, 2X,+ 2X, 2X6.

drum and be fastened to it ; a chainon chain sprockets may also be usedas the belt .

The above enumera t ion does notexhaust the possibilit ies for linearmechanica l cells; t here are manyvar ian ts the use of which may bedict a ted by specia l cir cumst ances.

As a ru le, when a differen t ia l isused in a comput ing mechanism, t woof its members (the input terminals)a re moved by externa l forces; th isr esu lt s in movement of a th ird mem-ber (the output terminal) which is inturn requ ired to furn ish an appreci-able for ce. If differen t ia ls wer e fr ic-t ionless, any two of their threeterminals cou ld be used as inputterminals. In rea lity, only a few oft he differ en tia ls descr ibed h er e h avecomplete interchangeability of theterminals. For instance, with thescr ew clifferen t ia l [Fig. 1.6) it is im-possible to have Xd as the outpu t

pa ramet er if t he helical a ngle of t he scr ew is so low tha t self-locking of t henut on the screw occurs; it is possible to use Xl as an outpu t parameter ,and, of course, a lso X,. With the chfferen t ia l worm gear ing of Fig. 15,X, is an impract icable out put pa ramet er .


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12 INTRODUCTION [SEC.1.5106. Mult iplier s.-Mult iplier s a re computers tha t establish between

t hr ee pa ramet er s a r ela tionRX3 = X, .X,, (13)

( where R is a constant tha t depends on the type of mult iplier and on it sdimensions.The act ion of the slide mult iplier shown in Fig. 110 is based on the

proport ion ality of the sides of t wo similar t r iangles. These are t r iangleswith hor izonta l bases, and ver tices a t th e cent ra l pin shown in the figure:

Iv;,: .-

FIG.1.10.Slidemultiplier.the first has a base of length R and alt itude Xl, the second a base of lengthX2 and alt itude X8. Thus

R x,T, = X (14a)or

RX3 = X,X2. (14b)The figure gives a schemat ic ra ther than a pract ica l design; t he lengths oft he sliding su rfa ces as sh own a re n ot gr ea t en ou gh t o pr even t self-lockin gin a ll possible posit ion s of t he mech an ism . Th ese len gt hs det ermin e t hespace requirements for mult iplier s of this type; they must be rela t ivelyla rge in two direct ions. It is difficu lt to make this t ype of mult iplierprecise. The pins in slots, as shown in the figu re, a re mechanica llyinadequate, and roller slides on rails must be used. One can not ach ievethe same end by increasing the dimensions of the mult iplier because the


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SEC.1.5] MULTIPLIERS 13elast icity of par ts comes in to play, not only when the par t s a re opera t ingin a comput er , but also when t hey are being machined.

The slide mult iplier shown in Fig. 1.11 saves space in one direct ion .There are fewer sliding contacts, and the slides are easier to const ruct .

Fm. 1.11.Slidemultiplierwith inputsXI, XI - Xz.

.

I ?m . 1.12.Intmwct ion nomogr am for mu lt ip lica t ion z; = z j . z*.This device cannot mult iply Xl and X2 direct ly t o compute RX3 = X1X2;the input terminals must be given transla t ions of Xl and X, X2. Thedifference is easy to obta in if the parameters a re genera ted as shaftr evolu tion s befor e en ter in g t he mult iplier ; scr ews ca n t hen be u sed in st ea d


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14 INTRODUCTION [SEC.15of t he slides sh own in t he figu re, a nd t he r equ ir ed differ en ce ca n be formedby a gea r differ en tia l.

Nomogr aph ic Mu lt iplier s.A mu lt iplier t ha t is st ruct ur ally r ela ted t oa n omogr am for mu lt iplica tion will be ca lled a n omogr aph ic mu lt iplier . Su ch mult iplier s can be der ived fr om in t er sect ion or a lignmen t nomogr ams;t he examples t o be given h er e a re r ela ted t o in ter sect ion nomograms.

FIG.1.13 .An in ter sect ion nomogram for mult ip lica t ion , ob tained from the nomogram inFig. 1.12 by a p roject ive t r ansformat ion .Figure 1.12 shows an in ter sect ion nomogram for mult iplicat ion in an

unusual form , t he full sign ificance of which will be ma de clea r in t he la tt erpar t of this book. Thk represent s the formula

Xi = XiXk. (15)It con sist s of t hr ee families of lin es, of con st an t xi, xi, a nd z~, r espect ively;through each poin t of the nomogram passes a line of each family, cor -responding to values of xi, ~i, and x~ which sat isfy Eq. (15). (The linesin thk par t icula r figure are drat i for va lues of the zs that a re powers of1.25; th is is not of immediate impor tance for our discussion .) The mult i-


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SEC.1.6] RESOLVERS 15plier of Fig. l.lOisst ructurally rela ted to th is nomogram. .The rota t ingslide can be brough t to posit ions cor responding to the radia l lines in thenomogram; the hor izonta l and ver t ica l slot s cor respond st ructu ra lly tothe hor izon ta l and ver t ica l lines on the nomogram, and the pin that con-nect s all slides mechanica lly assures a t r iple in ter sect ion of these lines.The va lues of xi, xi, and xk cor responding to the posit ions of the th r eeslides must then sat isfy Eq. (15); to complet e the mult iplier it is on lynecessary to provide scales from which these va lues can be read, or , as isdone in Fig. 1.10, to provide mechanical connect ions such tha t termina ldispla cemen ts a re pr opor tion al t o t hese qu an tit ies.

By a project ive t ransformat ion of the nomogram in Fig. 1.12 one canobtain the nomogram in Fig. 113, where lines of constant va lues of the

FIG.1.14.Nomograph ic mu ltiplier.var iables Zi, ~j, and xk form three families of radia l lines in t er sect ing inth ree cen ter s. The obvious mechanica l ana logue of th is nomogram formu lt iplica t ion is shown in .F ig. 1.14. It consist s of th ree slides th at r ot at eabout cen ter s cor responding to the cen ter s of the radia l lines in Fig. 1.13;these slides a re bound together by a pin , which establishes the t r iplein tersect ions found in the nomogram, and the cor responding va lues ofZi, Zi, and zk are read on circular scales. It will be noted that the scaledivisions ar e not uniform. Such nonuniform sca les a re of more genera luse than one might expect . Often one will have to dea l with var iablesgenera t ed with nonuniform sca les by some other computer ; by properchoice of the project ive t ransformat ion one can then hope to produce amult iplier of th is t ype wit h similar ly deformed scales.

1.6. Resolvers.The resolver is a specia l type of mult iplier . Itgenera tes a parameter X3, and usually a lso another pa rameter X4, as aproduct of a parameter Xl and a t r igonometr ic funct ionthe sine or


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16 INTRODUCTION [SEC. 1.6cosin~f a parameter X2. The equa t ions are

X3 = Xl sin Xz, (IOU)X4 = x, Cos X.2. (16b)

The name of this device is der ived from its act ion as a resolver of a vect ordispla cemen t in t o it s r ect angu la r componen t s.A simplified design of a resolver is shown in Fig. 1015. In the plan

view, Fig. 115a, we see the materia lizat ion of a vector by a screw: theaxis of the screw poin ts in the direct ion of the vector , a t an angle X2 to azero line; the length Xl of the vector is established as the distance fromthe pivot O on which the whole screw is rota t ed to a pin T on the nut oft he scr ew.

To obtain the component s of the vector , slides are somet imes used,as in the case of the mult iplier in Fig. 1.10. In Fig. 115 there is sug-gest ed a solu t ion tha t gives much bet t er precision and saves space.Perpendicula r shafts pass through the block B tha t car r ies the pin P.These shafts a re car r ied by rollers on rails; their parallelism to givenlines is well assured by gears that mesh with racks fastened to the frame.For convenience of const ruct ion the axes of the shafts do not in tersectwith each other and with the axis of the pin T. This in t roduces a con-stan t term e in to the displacement of the shaftsthat is, it causes a dis-pla cemen t e in t he effect ive zer o posit ions of X3 a nd X4.It is of in terest to note how the parameter Xl is cont rolled from theinput shaft S, (Fig. 1.15b.). While the screw is rota t ed through theangle X2 on the shaft S2, it is necessary to cont rol the value of Xl by agear G tha t rota tes fr eely on this shaft . If such a gear is turned throughan angle propor t iona l to Xlis held fixed when Xl is constan tthescr ew will spin on it s axis wh enever X2 is cha nged; t he lengt h of t he vect orwill be a ffect ed by cha nge in XZ, and will not represent t he desired value ofXl. It is thus necessary to keep the screw without spin with respect to Szwhen only XZ is changedto keep the gear G moving a long with the shaftS2 whenever Xl is fixed. This is accomplished by the so-ca lled com-pensat ing different ia l, D. As is shown in the figure, the planeta ry gearof th is bevel-gear different ia l is geared to the shaft S2 in the ra t io 1 to 1;the differen t ia l thus receives an input X2. When the input shaft S6 isrot a ted through Xc, the output shaft S6 is rota t ed through an angle

x, = X6 2x,. (17)By gear ing the gear G to the shaft Sb in the ra t io 2 to 1, the angle turnedby G can be made to be

x. = 0.5X6 = 0.5X6 + x2.Then if S6 is sta t ionary, X. changes equally with X2, and the screw is notspun; X, remains constan t . If t he shaft Se is turned, the gea r G turns


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SEC.1.6] RESOLVERS 17with respect t o the shaft 82 through an equal angle. The change inXlis then propor t iona l to the rota t ion of the shaft S6:X6 = QXI, theconstan t Q depending on gea r ra t ios and the threading of the screw.

1 &+e I 1,

.(a )

(b)

T& ,,

IX,+e

l@. 1.15.Raeolver . (a ) P la n view. (b) E leva tion . Th e t eet h of t he r acks a re om it tedfrom the figu res .The design in Fig. 115 is so oversimplified tha t the resolver is sure to

be la cking in precision . In par t icu lar , t he flexibility of th e st r uctu r e sup-por ting the scr ew is excessive: shaft h s is easily bent and easily twisted.


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18 INTRODUCTION [SEC.1.6This can be r emedied by pla cing t he scr ew su bassembly on a circu la r plat ewith a la rge ball bear ing on its circumference, and using a dr iving shaft ofr easonable d iameter .

A bet t er constmct ion (but one that is not always usable) is presentedin Fig. 1.16. In the plan view, Fig. 10l&z, we observe the main clifferen cebetween th e subassembly of the screw in Fig. 1.15 and th e present desi.zn.

- A.t hese join t s va ry inversely with each other . It will be noted tha t Xl ist he sum of the lengths of the bases of two r ight t r iangles of a lt itude T andhypotenuses A and B respect ively, whereas X, is t he difference of thessbase lengths. We have then

X, = ~A1 T + ~~, (la )x, = d~ d~. (lb)

In th ese equat ions A, B, T, and th e square r oot s a re necessa rily posit ive.On mult iplying together Eqs. (la) and (lb) we obtain

XIXZ = AZ Bz, (2a)or A2 _ B,x,= xl . (2b)

There a re two var ian t s of this inversor , with A grea ter than B or withB grea ter than A. If B is grea ter than A (dashed lines in Fig. 2.3), X2 isa lways negat ive; t her e is no possibility of having X, equal X2. If A isgrea ter than B (solid lines in Fig. 2.3), it is possible t o h ave

X, = X2 = (A B2)~.1A concisesummaryof work in this fie ld,by R. L. Hippieley,will be found underLinkugm,n the Encyclopedia Britan nica, 14th ed.


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30 BARLIN KAGE COMPUTERS [SEC,2.2At this point the mechanism exhibit s an undesirable singularity; thejoints P and Q of Fig. 23 become coinciden t , and self-locking of thedevice may occur . These two forms of the Peaucellier inver sor a lsodiffer in their useful ranges. These a re

~


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SEC.2.3] DES IGN OF BAR-LINKAGE COMPUTERS 31Analyt ica l studiesl have been made of the th ree-ba r mot ion of a

poin t C r igidly a t tached to the centra l link AB of a three-bar linkage(Fig. 26). Three-bar mot ion is very useful in the design of complexcompu ter s, and will be discussed in Sec. 10.4.

To comdete this su rvey of the bar -linkage clit er at ur e in- En glish, it will ~u ffice t o men tion t ~epa per s of Emch an d H ippisley on closed lin kages.z A?-2.3. The Problem of Bar -linkage-com&ter

Design.-It is only recen t ly tha t much at tent ion ft i

,t(

I

hae been paid t o the problem of using bar linkages FIG. 243.Three-barin comput ing mechan isms. The litera ture in the $=~1$ ,t~~o~,~~ &field is especia lly rest r icted. The author knows of cent ra lbar .on ly one published work tha t employs the synthet ic approach to bar-linkage computer design~and this in a more rest r icted field than tha t oft he p res en t volume.

The basic ideas in the synthet ic approach to bar-linkage design aresimple, but quite differen t from the ideas behind the classical types ofcomputers. Bar linkages can be character ized by a large number ofdimensional constants, and the field of funct ions tha t they can genera teis correspondingly la rge-though not indefin itely so. Given a well-behaved funct ion of one independent var iable, one should be able toselect fr om t he field of fu nct ion s gen er at ed by ba r lin ka ges wit h on e degr eeof freedom at least one funct ion tha t differ s from the given funct ion by arela t ively small amount . The character ist ic problem of bar -linkagedesign is t hu s t ha t of select in g fr om a family of curves too numerous andvar ied for effect ive ca ta loging one that agrees with a given funct ionwith in specified tolerances.

The presence of a residual er ror set s bar linkages apar t from othercomput ing mechanisms. The er ror of a computer of classica l type ar isesfr om it s con st ru ct ion as an act ua l physical m echa nism, wit h u na voida bleimperfect ions. It is possible to r educe the er ror to with in a lmost anylimits by sufficient ly carefu l design-as, for instance, by enlarging the

I A. Cayley,On Three-bar Motion, Pm t. Math . S ot., L en d., 7, 136 (1875). R.L. Hippieley,A New Method of Descr ib ing a Threebar Curve , Pm t. Math . S oC.,Lad., 16,136 (1918). W. W. J ohnson,On Three-barMotion, Memengm of Mathe-matics,6,50 (1876). S. Rober ts , On Three-barMotion in PlaneSpace ,Proc. Math.Sac.,Luzd.,7,14 (1875).xA. Emch , I llu st r at ion of the E llip tic Int egra lof t he F ir st Kind by a Cer t a inLink-work,Annuls of Mathemat ics ,Ser ies2 , 1, 81 (1899-1900). R. L. Hippis ley,ClosedLinkagea,Pm t. Maih . S ot., L en d., 11,29 (1912-1913); Closed LinkagesandPoriet icPolygons,Proc. Math . S ot., L om i, 13, 199 (1914-1915).8Z. Sh.Blokh and E. B. Karpin, Practical Methods of DesigningFlat Four-sidedMechmisms/ Izdatels tvo Akademienauk SSSR,Moscow,Leningrad (1943). E. B.Kar@m,Atlas of Nomograms, Izdateletvo Akademienauk SSSR,Moscow, Lenin-grsd (1943).


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32 BARLIN KAGE COMPUTERS [SEC.2.4whole computer . In bar linkages there is usually a residual er ror tha tca nnot be eliminated by any ca re in const ruct ion , an er ror tha t is evidentin t he mat hemat ica l design of t he device, a s well a s in t he fin ish ed pr odu ct .This er r or will be called st ructura l er ror because it depen ds only on thest ructure of th e compu ter , and not on its size or oth er mechanica l proper -t ies. Reduct ion of st ructura l er ror requires a change in the st ructure ofthe computer-usually the addit ion of par t s. The grea t number ofa dju st able dimen sion al con st an ts gives gr ea t er flexibilit yy and ext ends t h efield of funct ions tha t the linkage can genera te; from this la rger field offu nct ion s on e ca n t hen select a bet ter a ppr oxim at ion t o t he given fu nct ion .

The fact tha t bar linkages can be used to genera te funct ions of a largeclass has been known for many years, and has been used (inst inct ively,r a ther than with a full development of th e th eory) by designer s of mecha-nisms. The field of funct ions tha t can be genera ted by some simple barlin ka ges has been a na lyt ically descr ibed. This, h owever , repr esen ts on lythe easier half of the problem; what one needs is to descr ibe the field offunct ions tha t can almost be genera ted by a given type of linkage. Thefir st a t tempts to solve this problem for one independent var iable havebeen tabular or graphica l. For very simple st ructures it is possible todevise graphs tha t a llow on e to determine wh ether a given funct ion can begenera ted approxima tely by such a st ructu re, and what st ructura l er roris inevit able. These methods a re pract icable if the linkage can bespecified by means of only two dimensional parameterstha t is, if thefield of funct ions depends upon ordy two adjustable parameters. Suchgr aph ica l m et hods a re difficu lt or a re n ecessa rily in complet e if t he field offunct ions depen ds upon three adjustable parameter s. Such a procedu recan hardly be at t empted when four or more dimensional parameters a reinvolved.

The design methods presen ted in th is book are in many cases basedon a gra phica l fa ct or iza tion of t he given fu nct ion in to fu nct ion s su ita blefor mechaniza t ion by simple linkages; the elemen ts of the mechanismdesigned in th is way can then be assembled in to the desir ed completelinkage. By such methods it is possible to design linkages having agrea t many adjustable parameters, but the solu t ion obta ined cannot becla im ed t o be t he best possible. Usua lly it is ea sy t o a pply t hese m eth odsto find bar linkages tha t have er ror s everywhere within reasonablet oler an ces. Th is is or din ar ily su fficien t for pr act ica l pu rposes.

2.4. Ch ar act er ist ics of Ba r-lin ka ge Compu ter s.Th e specia l pr oper -t ies of ba r-lin ka ge compu ter s ma y be summarized as follows.

Advantages.1. Ba r linka ges occu py less spa ce th an classica l t ypes of compu ter s.2. Th ey h ave n egligible fr ict ion .


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~EC.2-41 CHARACTERIS TICS OF BAR-LINKAGE COMPUTERS 333. They have small iner t ia .4. Th ey h ave grea t stability in per forma nce.5. Th eir complexit y does n ot n ecessa rily in cr ea se wit h t he complexit y

of t he a nalyt ica l formu la tion of t he problem .6. They a re easy to combine in to complex systems.7. Th ey a re rela t ively cheap.Disadvantages.1. Ba r finkages usua lly possess a st ructura l er ror .2. Th e field of mech an iza ble fu nct ion s is som ewh at r est rict ed.3. Th e complexit y of t he lin ka ge in cr ea ses wit h decr ea sin g t oler an ces.4. Lin ka ge compu ter s a re r ela tively difficu lt t o design . Th e difficu lt y

of th e design pr ocedu re increa ses with increasing complexity anddecreasing tolerances.5. The t r avel of the mechanism is usua lly limited to a few inches.Backlash er ror and elast icity er ror must be reduced by ca refu lcon st ru ct ion : t he u se of ball bea rin gs is essen tia l, a nd r igidit y of t hest ructu re perpendicu lar t o the plane of mot ion must be assured.The design should be such tha t mechanica l er ror s a re less than theawign ed t oler an ces for st ru ct ur al er ror .

Ba r linkages can at ta in ext en sive use as elements of com put ers only asefficien t m et hods of design a re est ablish ed. Th e complexit y a nd difficu lt yof the design procedure depends la rgely on the nature of the given func-t ion . It is usually easy to design a linkage with a st ructura l er ror tha tdoes n ot exceed 0.3 per cent of th e wh ole ran ge of mot ion of th e computer .It becomes r ela tively la bor iou s t o r edu ce t he st ru ct ur al er ror below 0.1 percen t . If the tolerances a re below 0.1 per centas a typica l va luea lterna t ives to the use of a bar linkage should be explored.

Bar Linkages can advantageously be combined with cams when thet olera ted er ror is small and a bar linkage a lone wou ld be excessively com -plex. For instance, if a given funct ion of one independen t var iable wereto be mechan ized with an er ror of not more than 0.01 per cen t , it might bedesirable to mechan ize this fu nct ion by a simple bar linkage with an er rorof, for example, 1 per cen t , and to use a cam to in t roduce the requ iredcor rect ion term. Since th is cor r ect ive term represent s on ly 1 per cen t ofthe whole mot ion of the linkage, it need not be genera ted with very highprecision ; for instance, if the working displacemen t of the cam is to be1 in., it can be fabr ica ted with a tolerance as rough as 0.01 in .

It is a fea tu re of bar -linkage computers tha t they can be used togenera te funct ions of two independent var iables in a very dir ect andm ech an ica lly simple wa y. Met hods for t he design of lin ka ges gen er at in gfunct ions of t hree independent var iables a re n ot n ow available when it is


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34 BARLIN KAGE COMPUTERS [SEC.2.5not possible to reduce the problem to the mechaniza t ion of funct ions ofone or two independent var iables; there is, however , some hope tha tpr act ica lly u sefu l m et hods ca n be fou nd.

Bar -linkage computers have grea t advantages when feedback is tobeu sedin t hedesign ofcomplex compu ter s. In compu ter s of t hecla ssica lt ype, feedback mot ion must be a small fra ct ion of t he t ot al out pu t mot ion.Lin ka ge compu ter s can, h owever , oper at e ver y close t o t he cr it ica l feed-backthat is, the degree of feedback at which the posit ion of the mecha-n ism becomes indete rmina te.

2.6. Bar Linkages with One Degr ee of Freedom.Bar linkages withone degree of freedom serve the same purpose as cams; they may beca lled lin ka ge cams. The parallelogram linkage of Fig. 2.2 and thelinkage inversors have mot ions expressed accura tely by very simpleformulas, but t hey a re not gener ally useful in t he mecha nizat ion of givenfunct ions. For this purposs, the following bar linkages are much moreinteresting.

Th e ha rmonic t ra nsjormer , shown in Fig. 21, establishes a r ela t ionbetween an angular parameter Xl and a t ransla t ional parameter X$. It isconvenient to disregard varia t ions in the form of this rela t ion due tochanges in scale of the mechanism-to consider as equiva lent two geo-metr ica lly s imila r mechan isms. The field of funct ion s

x, = F(X,) (4)genera ted by t he harmonic t ransformer then depends upon t wo rat ios ofdimensions: L/R and E/R, the rat ios to the crank length of the linklength and the displacement of the crank pivot from the cent er line of theslide. As L is increased from its minimum value, the plot of X, againstXI changes (in a typical case) from an isola ted point to a closed curve,then to a sinusoid, and finally, in the limit as L approaches infinity, t o apure sinusoid. From a pract ical poin t of view, the pure sinusoidal form isreached for links shor t enough for pract ical use. In the limit ing case,L = w, the equat ion of the harmonic t ransformer is

X,= Rsin Xl+C. (5)Su ch a h armon ic t ra nsformer will be ca lled idea l.

Only r ar ely is t he complet e r ange of m ot ion of a harm onic t ra ~formerused. When the range of the parameter X, is limited to Xl~ < X, < Xl..and the funct ions defined within t hese rest r icted limits a re taken as ele-ments of a new funct ional field, t here is obta ined a four-dimensiona lfunct iona l field depending on Xl~ and Xl,, as well as on L/R and E/R.Methods for the design of harmonic t ransformers will be discussed inChap. 4.


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SEC.2.5] BAR LINKAGES WITH ONE DEGREE OF FREEDOM 35Th e t hr ee-ba r lin ka ge sh own in F ig. 27 con sist s of t wo cr an ks pivot ed

to a frame and joined at their fr ee ends by a connect ing link. As acompu ter , t his ser ves t o compu te the parameter X2 as a funct ion ofthe parameter Xl. The linkage itself is descr ibed by four lengths: A 1, B1,AZ, Bt . The field of funct ions genera ted by this type of linkage is onlyt hr ee-dimen sion al, beca use two geomet rica lly sim ila r mech an ism s est ab-

FIG.2.7.Three-bar linkage. FIG. 2.8.Three-barlinkage mod ified byeccentric linkage.fish the same rela t ion between Xl and X2. The field of funct ions thusdepends on t hr ee ra t iosfor example, B1/A 1, Az/A,, and Bz/ A I. Usu-ally only a par t of the possible mot ion of the mechanism is used. Limitsof mot ion can be assigned for Xl or X2, though, of course, not independ-ent ly for the two parameters; for instance, one may fix Xl~ < Xl < X 1~.This increases t he number of indepen dent paramet er s by two; t he field offunct ions gen er at ed by a t hr ee-bar linkage oper at ing with in fixed limits

/ 1FIG. 2 .9 .Harmonic t ransformer modified by eccent r ic linkage .is five-dimensional. In Chap. 5 we shall see how to design a th ree-barlin kage for t he a ppr oxim at e gen er at ion of a given fu nct ion .

The euxn tr ic linkage is n ot a bar linkage, but is so convenien tly used inconnect ion with bar linkages that it should be ment ioned here. [email protected] shows a three-bar linkage modified by the inser t ion of an eccen t r iclinkage. One crank of the three-bar linkage car r ies a planetary gear tha tmeshes with a gea r fixed t o the frame. The centra l link is then pivotedeccentr ica lly to the planetary gear , ra ther than to the crank itself. Link-ages of th is t ype will be discussed in Sec. 7.9, wh er e their import ance will


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8EC.2.6] BAR LINKAGES WITH TWO DEGREES OF FREEDOM 372.6. Bar Lfnkages with Two Degrees of Freedom.Bar linkages with

two degrees of freedom can be used in the genera t ion of a lmost anywell-behaved function

X3 = F(xl, x2) (6)of t wo independent va riables. They pr ovide a mecha nica lly sa tisfact orysubst itu te for threedmensiona l cams, which have many disadvantagesand are to be avoided if possible.F igure 211 shows a linkage withtwo degrees of fr eedom, whichcon sist s of t h ree cr ank s conn ect edby two links and a lever . Thelever will degen er at e in to a simplelink if the pivots A and B aresuperposed; the resu lt ing st ruc-tu re of th ree links join ted at asingle pivot will be called a sta rlinkage. It s proper t ies a re dis-cussed in Chap. 9.

The bar-linkage adder shownin F ig. 2.12 con sist s of essen tia lly Fm. 2.12.Bsr-linka gedder.the same par t s as the linkage ofFig. 211, except that slides are used instead of cranks to const ra in thelinks. The dimensions obey the simple rela t ion

A, 1?, C,.A, B, = r , (7)It is easy to show that when this propor t iona lity holds, t he three pivotsP,, P%, and Pa lie on a st ra ight line. This device can, therefore, be usedto mechanize any alignment nomogram that consists of three parallelst ra ight lines; in par t icu la r , it can be used t o mechanize the well-knownnpmogram for addit ion. If Xl, X2, and X3 are t hree parameters measureda long these lines in t he same direct ion from a common zero line, then

(A, + A2)X3 = A,XI + A,X,. (8)This bar linkage is fr ee from st ructura l er ror .

In cont rast t o the adders, baT-linkage mult ipliers do not per form theoper at ion of mult iplica tion exa ct ly, but wit h a small er ror ; t he equ at ion ofsuch a mult iplier is RX3 = X,X, + 6, (9)where 8, the er ror of the mult iplier , is a funct ion of the two independentparameters Xl and X2. The design of mult ipliers will be discussed inChap. 9; a much simplified explanat ion of t he pr inciple will be given here.


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38 BAR -LIN KAGE COMPUTER S [SEC.2.6Figure 2.13 shows the essent ia l elements of one type of mult iplier .

Three bars of equal lengths, El = RZ = Rs = 1, are pivoted together .The first is pivoted also to the frame at the poin t O, the th ird to a slidewith center line passing through O. If the join ts A 1 a nd A z are placedat distances Xl and XZ from the cen ter line of the slide, the distanceOS = D will be exact ly

D=~l X; @-(X,-X,) +~1X;. (lo)Expanding in ser ies the terms on the r ight , one obta ins

wh ere X3 is the displacement of th e pivot S from the posit ion SOwhich itoccupies when Xl = X2 = O and

i the three links are coincident . Itis evident tha t X3 is equal t o theproduct XIXZ to the approxima-

X2 --- t ion in which the terms of four thand higher degrees can be neg-lected in compar ison with theterm of the second degree. For

FIG, su fficien tly sm all va lu es of Xl a nd2.13.E lemen ts of a ba r-lin kage X2 this mechanism is thus amult iplier . . mult iplier for these parameter s.Such a mult iplier is not pract ical, however , because of its small range ofmotion. If the er ror in the mult iplica t ion is to be kept below 1 per cent ,it is necessary to keep Xl, Xa s 0.2. [If X, = X, = 0.2, then

x3 = (0.2) + +(0.2)4 + . . . ,and the fract ional er ror is a lmost exact ly one per cent . ] Under thesecondit ions, however , one has Xa = 0.04, an impract icably small range ofmotion. q

There are in pr inciple two ways to improve this mult iplier . Witheit her m et hod it is n ecessa ry t o mak e t he st ru ct ur e mor e complica tedt oa dd n ew a dju sta ble pa ramet ers. On e possible a rr an gemen t is in dica tedin Fig. 214. Here the pa rameter Xa is a displacement of a slide (ofadjustable posit ion) tha t cont rols the posit ion of the join t A a through alink of adjustable length L,; X3 becomes an angular parameter , th e angleturned by a crank with adjustable length and pivot posit ion .

With the first method, the output parameter X3 is expressed interms of X, and X,, in the form of a ser ies with coefficien t s which dependon th e adjustable dimensions of th e mechanism. Th ese dim en sion s ca nthen be so chosen as to cause the terms of the four th degree in Xl and Xz


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SEC.2.6] BAR LINKAGES WITH TWO DEGREES OF FREEDOM 39to vanish . In th is way, the mult iplier can be made more accura te forsmall va lues of Xl and X2, and the domain of usefu l accuracy sub-stant ia lly increased. Toward the limits of th is domain , however , theinaccurac~ of the mult iplier willin cr ea se very rapid ly. Ap. . .

The second method for im-proving t h e mult ipliertha tfollowed in this bookcan be indl- ~,ca ted on ly very rough ly a t th is .. . - -poin t . It involves compar ison oft he idea l pr odu ct a nd t he fu nct ion ] [~ Iactua lly genera ted by the mult i-plier over the en t ire range of 1[!mot ion, and adjustment of thedim en sion al con st an ts. of t he sys- FIG.2.14.Modifiedbar-linkagemultiplier.t ern in such a way that the er ror of the mechanism is brough t with inspecified t oler an ces ever ywh er e wit hin t his domain . To see in pr inciplehow this can be done, let us consider the mechanism of Fig. 2.13. LetX3 and Xl be given a ser ies of va lues tha t have the fixed ra t io

(12)If t his lin ka ge wer e a n exa ct mult iplier , t he pivot A, wou ld in dica te a lwa ysthe same value of X2; it would move along a st ra igh t line a t constan tdista nce X; fr om t he line of t he slide. Act ua lly, t he pivot A zwill descr ibea curve tha t is t angent to th is st ra ight line for small va lues of Xl and X3,but will diverge from it as these parameters increase. To each va lue ofX2 there will cor r espond another curve; the curves of constan t X2 form afamily, each of which can be labeled with the associa ted value of th isparameter . Now we can make this mult iplier exact if we can int roducea const ra in t which, for any specified value of X2, will hold the pivot A ~on the cor responding curve of th is family. For example, if these curveswere all circles with the same radius Lz and cen ter s lying on a st ra ightline, it would be possible t o use the type of const ra in t illust ra ted in Fig.2.14. The X,-slide could then be used to br ing the pivot A a t o the cen t erof the circle cor responding to an assigned value of X2, and the pivotAz would stay on tha t circle, as required. Actua lly, the curves of con-st an t X2 will n ot form su ch a family of iden tica l cir cles. It will, however ,be possible to approxima te them by such circles in a way which will splitthe er ror and br ing it with in tolerances held fa ir ly uniformly over thewhole domain of act ion . Unlike the mult ipliers designed by the firstmet hod, a mult iplier t hu s design ed will n ot h ave u nn ecessa rily small er r or sin one par t of the domain and excessively la rge er rors in another par t .


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40 BAR -LINKAGE COMPUTER S [SEC.2.7Th is con cept of mult iplier design mu st be ver y gr ea tly ext en ded befor e

it can lead to the design of sa t isfactory computers. A power ful guide inbeginning the work is provided by the idea of nomograph ic mult ipliers,a lready discussed in Sec. 1.5. It is possible to design approximate inter -sect ion nomograms for mult iplica t ion that have as their mechanicala na logues ba r lin ka ges with t wo degr ees of fr eedom. F or in st an ce, IHg.814 sh ows a n omogr am for mu lt iplica tion obt ain ed by t opologica l t ra ns-format ion of the nomogram of Fig. 1.12; it consists of two families ofident ica l cir cles and a third family of curves that can be very closelyapproximated by a family of ident ica l circles. This nomogram cor -responds t o the bar-linkage mult iplier illust ra ted in Fig. 8.15, which, onimprovement of it s mechanical fea tures, takes on the form shown in Fig.8.16. The design techniques to be descr ibed in Chaps. 8 and 9 make itpossible to design mult ipliers with large domain of act ion and gooduniformit y of per formance t hr ou gh t his doma in .

Mult iplier s ca n be u sed t o per form t he in ver se oper at ion _ of division ;that is, they can be used to evaluate X2 = Xs/X1. It is, of course, notpossible to divide by zero; when a mult iplier is used in this way Xl willnever pass through zero. It is therefore useless to at tempt to reduce tozero the er ror of such a mult iplier for values of X1 very near to zero; itis a lso undesirable to a t tempt to reduce the error s of the device for nega-t ive values of Xl when only posit ive values can be int roduced. For th isr ea son t hr ee t ypes of mult iplier may be dist in gu ish ed.

1. Fu ll-r an ge mult iplier s, for wh ich bot h input pa ramet er s ca n ch an gesigns.

2. Half-range mult ipliers, for which only one parameter can changesigns.

3. Quar ter -range mult ipliers, for which neither input pa rameter canchange signs.

Dividers may be divided into two types.1. The plus-minus type, for which the numerator may change sign.2. Th e sin gle-sign t ype, for wh ich a ll-pa ramet er s h ave fixed signs.

An example of a pract ica l fu ll-range linkage mult iplier is shown in Fig.8.16; a h alf-r an ge mult iplier is sh own in Fig. 915.

2.7. Complex Ba r-lin ka ge Compu ter s.The elem en ta ry lin ka ge cellsa lready descr ibed may be combined to form complex comput ers. Sincesimple linkages can add, mult iply, and gen er at e funct ions of one and t woin depen den t va ria bles, ba r-lin ka ge compu ter s ca n solve a ny pr oblem t ha tca n be expr essed in a syst em of equa tion s in volvin g on ly t hese oper at ion s.The field of applica t ion of bar-linkage computers is quite large; they


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SEC.2.7] COMPLEX BAR-LIN KAGE COMPUTERS 41are especia lly usefu l if the computer must be light , as when it is to beca rr ied in a ir cr aft or gu ided m issiles.

An impor tant fea tu re of bar -linkage computers is the ease with whichthe cells can be assembled into a compact unit . It is na tura l t o spreadthe par t s of the computer ou t in a plane, to produce a ra ther fla t mecha-nism with its par ts easily accessible. The connect ions between cellsa repr ovided by shafts or con nect in g bars.

There is a simple t r ick tha t makes the connect ion of linkage cells evenea sier , and t he st ru ct ur e of some cells less complex. The simplificat ionof linkage adders is a cha racter ist icexample of th is t r ick. The bar-link- _ =age adder shown in Fig. 2.12 has nost r uct u ra l er r or . Any devia tion fr omthe pr inciple of th is design is likely tolead to a st ructura l er ror ; it is, how-

=ver , possible t o change the pr inciple -in such a way that the st ructu ra l er ror xis negligbly small. For instance, if FIG.2.15.Bw-linkageadder (appr oxi-the links B, and Bz are very long, mate),their lengths can be chosen at will without appreciably affect ing theaccuracy of the addit ion . F igure 2.15 shows such an approximate adder ;it s equ at ion is

(A, + A,)x, = AIX, + A,XZ. (13)The links L, and Lz must be so long tha t they lie near ly para llel to the linesof the slide, but t hey need not be exact ly para llel t o each other . Theact ion of th is device depends upon the essent ia l constancy of the pr ojec-t ion of the lengths of these bars a long the line of the slides. Let X4, x;,and X: be defined as the distances of the pivots PI, Pz, and P~ from somezero line perpendicu lar to the line of the slides. One then has, exact ly,

(A, + A,)X{ = A,X{ + A,X;. (14)NOW let 131be the angle between the bar L1 and the line of the slides.Then

X,= X;+ LICOS(?, +C, (15a)= X; L,(I COS6,) + (c + L,). (15b)

Except for an addit ive constant (which can be reduced to zero by properchoice of the zero poin t ), X{ and Xl cliffer only by the var iable termLl(l cos 8,). As LI is increased, 01 decreases with I/L,, (1 cos 01)decreases with l/L~, and L,(l cos 0,) decreases with l/Ll. Thus, bymaking L1 la rge and proper ly choosing the zero point , one can make Xland X; differ by a negligibly small t erm. In the same way X2 can bemade negligibly differen t from X4; Xa and X{ are ident ica l. Equat ion


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II

I1

III

42 BAR -LIN KAGE COMPUT ER S [SEC.2.7(13) follows as an approximat ion to Eq. (14). If 0, is kept less tha t0.035 radians (about 2) the difference between Xi and X{ will be about0.0006 LI. Thus if the bars devia te from parallelism with the slides by nomore than & 2 dur ing opera t ion of the adder , the result ing er ror in theou tput will not exceed 0.06 per cen t of the tota l length of the bars.

If the lengths of the bars in approximate adders are grea t enough, it iseven imma ter ia l wh et her t he slides move a lon g st ra igh t lin es; t he essen tia l-7L-

*~

FIG.2.16.Combina t ionof appr oxima teadder s.thing is tha t the parameters be measured as distances from a zero line.It is, t her efor e, possible t o con nect a dding cells t hr ou gh lon g con nect in gbars, and to omit some of the slides that would appear in the standardconstruct ion . Fig. 2.16 shows a combina t ion of th ree addhg cells tha twill solve (a ppr oximat ely) t h e equ at ion s

(A, + A,)X, = A,X, + A,X,,(D, + D,)X, = D,X, + D,X,,

1(16)

(E, + E,)X, = E=, + E,X,.


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CHAPTER 3BASIC CONCEPTS AND TERMINOLOGY

The present chapter will define the terminology to be employed indiscussing bar-linkage design and in t roduce some co~cepts with wideapplicat ion in the field., Of part icular impor tance are the concepts ofh omogen eou s pa ramet er s a nd h omogen eou s va ria bles, a nd a gr aphi-ca l calculus used in discussing the act ion of comput ing mechanisms inseries.

3.1. Defin it ions. Ideal Funct ional Mechanism.-Any mechanismcan be used as a computer if it establishes defin ite geomet r ica l rela t ionsbet ween it s pa rts-t hat is, if it is sufficien t ly r igid and free from backlash,

FIG.3.1.Crank terminal.

Zemposition

FIG.3.2.Slide terminal,slippage, or mechanical play. In the following discussion we shall becon cer ned on ly wit h su ch idea l fu nct ion al mech an ism s.

Terminuls.-The terminals of a comput ing mechanism are those ele-ments that , by their mot ions, r epresen t the variables involved in thecomputa t ion . The mot ion of all termina ls is usually specified withr espect t o some common fr ame of r efer en ce. If t he posit ion of a t erminalis cont rolled in order t o fix the configura t ion of t he mechanism, it may beca lled an input terminal; if its posit ion is used in cont rolling a secondmechanism, or is simply obser ved, it ma ybe ca lled an out put terminal.A terminal maybe suitable for use only as an input terminal, or only as anou tpu t t ermin al, or in eit her way, a ccor din g t o t he n at ur e of t he mech an ism .43


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44 BAS IC CONCEPTS AND TERMIN OLOGY [SEC..1Terminals tha t a re mechanically pract ica l a re of t wo kinds:1. Crank or rota t ing-shaft terminals (F ig. 31), which represen t ava ria ble by t heir a ngu lar m ot ion.

2. Slide terminals (F ig. 3.2), which represen t a var iable by a linearmotion.

Paranwters.-A parameter is a geom et r ica l quant ity tha t specifies theposit ion of a termina l. With a crank terminal, it is usually the angula rposit ion of t he t ermin al wit h r espect t o some specified zer o posit ion ; wit h aslide termina l, it is usua lly t he distance of t he slide fr om a zero posit ion.Parameters may be deiined in other ways for instance, as the distanceof a slide termin al fr om som e mova ble elem en t of t he mech anism -but suchpa ramet er s a re less gen er ally u sefu l t ha n t hose ju st m en tion ed.

An input pa rameter descr ibes the posit ion of an input termina l, anoutpu t parameter tha t of an outpu t termina l.Linkage Computer s.-A linkage compu ter establishes between its

parameters, Xl, X2, . . . X,, defin it e r ela tion s of t he formI,(X1, x2, x.) = o, r=l,2, ..., (1)

wh ich in volve on ly t hese pa ramet er s an d t he dim en sion al con st an ts of t hemechanism. With more genera l types of mechanisms these equa t ions ofmot ion may also involve der iva t ives of the parameters. Su ch mech a-nisms ar e usefu l in th e solu t ion of different ia l equa t ions, but t hey will beexclu ded fr om ou r fu tu re con sider at ion s; we shall be con cer ned on ly wit hlin ka ge compu ter s, wh ich gen er at e fixed fu nct ion al r ela tion s between t heparameters.

To descr ibe the configura t ion of linkage computers with n degrees offreedom, one must in genera l specify the va lues of n input parameters,x,, x,, . . . Xn. The values of any number of ou tpu t parameters cant hen be expr essed explicit ly in t erms of t hese n pa ramet er s:

X.% = G, (Xl, X2. - x.), r=l,2, ?n . (2)Dom.uin .-The parameters of a comput ing mechanism cannot , in

genera l, assume all va lues. Th e limitat ions may ar ise fr om th e geomet r i-ca l n at ur e of t he mech an ism (a lin ea r dim en sion will n ever ch an ge wit hou tlimit) or from the way in which it is employed. To each possible set ofva lues of the input parameters Xl, . . . Xm, there cor responds a poin t(x,, x2, . . . X,,) in n-dimensional space; to all set s of va lues tha t mayar ise dur ing a specific applicat ion of th e mechanism, t her e cor r espon ds adom ain in n-dimensiona l space, wh ich will be r efer red t o as t he doma in of the parameters. It must be emphasized tha t the domain of the param-et er s is not necessar ily determined by the st ructur e of the mechanism,but by the task set for it .


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SEC.3.1] DEFINITIONS 45In the most general case, the domain of the input parameters maybe of

arbit rary form-except , of course, tha t it must be simply connected,since all parameters must change cont inuously. In such cases the valuespossible for any one pa ramet er may depend on t he values assigned t o ot herparameters. A mechanism will be said to be a regula r mechanismwhen each input paramet er can var y independent ly of all ot her s, bet weendefin it e u pper a nd lower lim it s,

(3)which define the dainain of the parameter . With angular parameters,neit her of t hese limits is necessa rily fin it e: it is possible t o ha ve Xi~ = cu ,or Xi~ =+ Co.

The output parameters of a regular mechanism will vary betweendefinite (though not necessarily finite) limits as the input parameterstake on all possible values. These limits serve to defie a domain foreach output parameter . Although the input parameters vary inde-pendent ly t hrough t heir r espect ive domains, t his is hot always t rue of t heou tpu t par ameter s.

Travel.The range of mot ion of a terminal is ca lled its tr avel.This is

AXi = Xi~ _ Ximj (4)both for input and output terminals:

Vw-iu bfes.-Th e t erm var iable will denote the var iables of theproblem which the computing mechanism is designed to solve. A var ia-ble will be a ssocia ted wit h ea ch t ermina l of a mecha nism , a n inpu t va ria blewith an input terminal, an output var iable with an output terminal. Toeach value of a variable there will cor respond a defin ite configurat ion oft he t erminal; each va ria ble, t hen , will be funct ionally rela t ed t o a param-et er of t he mech an ism :

Xi= @i(Xi). ~=l,g,...o (5)It is impor tan t to keep in mind the dist inct ion between parameters,

which are geometr ica l quant it ies measured in standard units, and theva ria bles of t he pr oblem, which a re on ly funct iona lly r ela ted t o t he pa ram-eters. In this book, variables will be denoted by lower-case let ters,pa r ameter s by capit a ls .

Scu les.-The va lu e of t he va ria ble cor respon din g t o a given configu ra -t ion of a terminal can be read from a scale associa ted with tha t termina l.The calibrat ion of this scale is determined by the form of the funct ionalrela t ion between z and Xi. If xi is a linear funct ion of Xi the scale willbe emnthat is, even ly spaced calibrat ions will cor respond to evenlyBpSCedalues of xi. Such a scale may also be refer red to aa linear ,


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46 BAS IC CONCEPTS AND TERMINOLOGY [SEC.3.1in r eferen ce t o t he form of t he fu nct iona l rela tion r epr esen ted. (This termdoes n ot descr ibe t he geomet rica l form of t he sca le, wh ich maybe cir cu la r.)A linear terminal is a terminal with which there is associa ted a linearscale.

Range of a Var iable.As a parameter changes between its limits,Xin and Xi~, t he a ssocia ted var i~ble will a lso ch ange with in fixed, bu t n otnecessa r ily fin it e, limit s: Xims Xis X ild. (6)In the case of a regular mechanism, th is may be refer red to as the domain of the var iable; its range is

Ax~ = XW Z& (7)Mech an iza tion of a Fu nct ion .An idea l fu nct ion al mech an ism esta b-lish es defin it e r ela tion s between it s pa ramet er s:

F,(Xl, X2, . ) = O, r=l,2, ... (8)It maybe said to pr ovide a mechan izat ion of t h ese fu nct ion al r ela tion swith in th e given domain of t he in depen den t parameters.

Such a mechanism, togeth er with it s a ssocia ted scales, similar ly pr o-vides a mech an iza tion of fu nct ion al r ela tion s,

j,(zl, x*, ) = o, 7=1,2,, (9)between th e var iables xi, with in a given domain of th e indepen dent va ria-bles. The forms of these rela t ions may be der ived by eliminat ing thevalues of the parameters Xi between Eq. (8), which character izes themechanism, and Eq. (5), wh ich cha ra cter izes th e scales.

FIG.3.3.Input scale.If the ou tpu t var iables a re to be single-valued funct ions of the input

var iables, the inpu t parameter s must be single-valued funct ions of thein pu t va ria bles, an d t he ou tpu t va ria bles mu st be sin gle-va lu ed fu nct ion sof the outpu t parameters; it is not , however , necessa ry that the inverserela t ions be single-valued. Thus an input scale may have the form shownin Fig. 3.3, and an outpu t sca le that shown in Fig. 3.4, bu t not the rever se.

Lin ea r Mech an iza tion .A mech an iza tion of a r ela tion between va ria -bles will be termed a linear mechanizat ion if all scales a re linear .


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SEC.32] HOMOGENEOUS PARAMETERS AND VARIABLES 47A nonlinear mechanizat ion of a given funct ion may be usefu l when

input variables are set by hand, and only a reading of the output variablesis required. When a comput ing mechanism is to be par t of a more com-plex device, it is usually necessary tha t the terminals have mechanicalmot ion propor t iona l to the change inthe associated variablethat is, a linearmech an iza

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MIT Radiaton Lab Series V27 Computing Mechanisms and Linkages

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