Computing Mechanisms and Linkages
COMPUTING
AND
MECHANISMS
LINKAGES
ii
MASSACHUSETTS
INSTITUTE
RADIATION
OF TECHNOLOGY
LABORATORY
SERIES
Board of Editors
LOUISN.RIDENOUR,
Eddor-in-Ch@
GEOR~EB.
COLLINS, Deput!/ Edztor-in-Chief
BRITTON CHANCE, S. A. GOUDSMIT, R. G. HER~, HUBERT M. JAMWS,JULIAN K. KNIPP,
JAMES L. LAWSON, LEON B. LI~FORD, CAROL G. MONTGOMERY, C. NEWTON, ALEERT
M. STOISE, LouIs
A. TURNER, GEORC~ E. VALLEY, JR., HERBERT H. WHEATON
1. RADAIi SYSTEM l;KCXNE~R1~~ --l~zdenou~
2. R.4DAR .\IDS TO N.4v1GAmo-~a/L
3. R.ADAR BEAc’oNs—lioherls
4.
LoRA~—Pzmce,.T1cKenzfe,
and JVoodward
5. PULSE GEN ER.i’roRs<lasoe and Lebacqz
6. NIIcROwAVE
111.4
GNETR0N s—C011i71s
7. KLYSTRONS
AND kIICROWAVE
TRIol)Es—~amil/on,
8. PRIXCI?LES OF IVIICROWAVE CIRcrITs—,lIontgonzery,
9. h~IcROWAt-E TRANSMISSION
Knipp,
and Kriper
Dir!ie, and Purcell
CIRcrITs—h’agan
10. WAVEGUIDE HAY~Boo~-lfarcllL,itz
11. TECHXIQIE OF ~~lCROW.4VE h[EASGRE31EXTS—.~ fOntg0?lle,V
12. NIICROWAVE .\XTESNA THEORY .kND ~EslGN—Si&r
13. PRop.*G.k!rloxOF SHORT R.ADIO W’AVES-KW
14. LIICROW4VE nUPLEXERS—SnII{ Zlinand Montgomery
15. CRYST.AL Rectifiers—Torrey and ~~hif?ner
hIIx13Rs-P0u71d
16. NIICROW-AVE
17. COMPONENTS HANDBOOK—~/Uc!ibum
18. ~’ACIWM TUBE .kMPLIFIERs-~a//eq and ~~a~{fna?z
Hughes, McrcSichol. ,%yre, and ~f’illiams
19. ~~.kvEFoRMs-chunce,
20. ELECTRONIC
TIME ~fEASLrREMENTS-ChQnce,
HI(lsizer,
flfclc.~icho~,
and Williams
Holdam, and ~fac~ae
21. ~LECTRONIC IxsTRcMExTs~reenwood,
22. CATHODE R.4Y TUBE ~ISPL.*YS—L’3071W,Starr, and Valley
23. ~rICROWATE REcEIVERS—~’Ufl l’oorhis
24. THRESHOLD SIGx.&Ls—Lawson and L:hlenbeck
25. THEORY OF SERvoMEcHAN1s.Ms-~anles, .~irhok, and Phillips
26. RADAR SC.4iSNERSAND R.iDOMES-CU(~Jy, Karellfz, and Turner
27. COMPUTING lfECHAXISMS AND LIWK.4GES—&ot)Oda
28. INDEx—Henney
*
CO MPIJTING
MECHANISMS
AND
By
LINKAGES
AN
Eciitedby
OFFICE
OF
TONIN
SVOBODA
HIJB13RT
SCIEA-TIFIC
NATION.4L
M.
RESEARCH
DEFENSE
JAMES
AND
RESEARCH
D13VELOPM.ENr
COMMITTEE
FIRST EDITION
NEW
McGRAW-HILL
YORK
AND
BOOK
1948
*
LONDON
COMPANY,
INC.
.4
-=’
c“’
L.
COMPUTING
MECHANISMS
AND LINKAGES
COPYRIGHT, 1S48, EIY THE
MCGRAW-HILL
PRINTED
1>- THE
Boo < COMPANY,
l?N-ITEE ST LTES
INC.
OF .iMERI~A
All rights reserved.
This book, or
parts thereof, ntay z,oi be reproduced
in icn~j form dho,d
permission
of
the pubttshers.
THE
M.4PLE PRESS
(X)?I~PAh-Y,
YORK,
PA.
Forezuord
HE tremendous research and development effort that went into the
development of radar and related techniques during World War II
resulted not only in hundreds of radar sets for military (and some for
possible peacetime) use but also in a great body of information and new
techniques in the electronics and high-frequency fields. Because this
basic material may be of great value to science and engineering, it seemed
most important to publish it as soon as security permitted.
The Radiation Laboratory of MIT, which operated under the supervision of the National Defense Research Committee, undertook the great
The work described herein, however, is
task of preparing these volumes.
the collective result of work done at many laboratories, Army, Navy,
university, and industrial, both in this country and in England, Canada,
and other Dominions.
The Radiation Laboratory, once its proposals were approved and
iinances provided by the Office of Scientific Research and Development,
chose Louis N. Ridenour as Editor-in-Chief to lead and direct the entire
project. An editorial staff was then selected of those best qualified for
this type of task. Finally the authors for the various volumes or chapters
or sections were chosen from among those experts who were intimately
familiar with the various fields, and who were able and willing to write
the summaries of them. This entire staff agreed to remain at work at
MIT for six months or more after the work of the Radiation Laboratory
was complete.
These volumes stand as a monument to this group.
These volumes serve as a memorial to the unnamed hundreds and
thousands of other scientists, engineers, and others who actually carried
on the research, development, and engineering work the results of which
are herein described.
There were so many involved in this work and they
worked so closely together even though often in widely separated laboratories that it is impossible to name or even to know those who contributed
Only certain ones who wrote reports
to a particular idea or development.
or articles have even been mentioned.
But to all those who contributed
in any way to this great cooperative development enterprise, both in this
country and in England, these volumes are dedicated.
T
m
.$
#,. 4
L. A. DUBRIDGE.
v
-----
. ..—. ... . —- .
.—-----
Preface
HE work on linkage computers described in this volume was carried
out under the pressure of war. War gives little opportunity for the
advancement of abstract knowledge; all efforts must be concentrated on
meeting immediate needs. In developing techniques for the design of
linkage computers, the author has therefore been forced to concentrate on
tinding practical methods for the design of computers rather than on
developing a unified and systematic analysis of the subject.
The war has
thus given to this work a special character that it might not otherwise have
had.
The impulse to the development of the methods presented in this
volume for the mathematical design of linkage computers grew out of a
collaboration of the author with his friend, Dr. Vladimir Vand.
That collaboration was begun in France in 1940, and was brought to a premature
end by the progress of the war. Though these ideas and methods have
largely been developed by the author since that time, he wishes to
emphasize that credit for the initiation of the work is shared by Dr.
Vand. It must be mentioned also that the techniques described in this
book were for the most part developed before the author became associated with the Radiation Laboratory.
The author wishes to express sincere gratitude to Dr. H. M. James, the
editor of this volume, who gave the book its present form, contributing
many examples and many improvements to the methods.
(Sees.: 6.7,6.8,
6.15, 8.6.)
The book would never have been completed in such a short time without the assistance of Miss Constance D. Boyd, who read the manuscripts,
and Miss Elizabeth J. Campbell, Mrs. Kathryn G. Fowler, Miss Virginia
Driscoll, and Miss Patrica J. Boland, who calculated the tables and drew
nomograms. The author also wishes to thank Dr. I. Maddaus, Jr., for
bibliographical research.
The publishers have agreed that ten years after the date on which
each volume in this series is issued, the copyright thereon shall be
relinquished, and the work shall become part of the public domain.
T
A. SVOBODA.
.,
PSARA,CZECHOSLOVAmA,
June, 1946.
:.
vii
,.
1
Contents
FOREWORD BYL. A. DUBBIDGE,..
v
,,.
PREFACE.
vii
&AP. 1. COMPUTING MECHANISMS AND LINKAGES.
1
1
INTRODUCTION
1.1. Types of Computing Mechanisms .
1.2. Surveyof the Problem of ComputerDesign
1.3. Orgrmizationof the PresentVolume
I
2
5
ELEMENTARY
COMPUTING
MECHANISMS
1.4.
1.5.
1.6.
1.7.
I.S.
6
. . . .
. . . .
. . .
. . . .
. .
. . . . . . . . . . . . ...6
. . . . . . . . . . . . . . . 12
. .
.
.
15
. . . . . . . . . . . . ...19
,.23
CHAP.2. BAR-LINKAGE COMPUTERS
27
2.1.
2.2.
2.3.
2.4.
2.5.
2.6.
2.7.
CHAP. 3.
Additive Cells. . . .
Multipliers . . . . .
Resolvers, . . . . .
Cams. . . . . . . .
Integrators . . . . .
Introduction. . . . . . . . . . . . . . . . . . . . . ...27
HietoriealNotes. . . . . . . . . . . . . . . . .
. ...28
The Problemof Bar-linkage-computerDesign
Characteristicsof Bar-liikage Computers.
Bar Linkageswith One Degreeof Freedom
Bar Linkageswith Two Degreesof Freedom.
ComplexBar-linkageComputers.
BASIC CONCEPTS AND TERMINOLOGY.
3.1.
3.2.
3.3.
3.4.
3.5.
De6nitions . . . . . . . . . .
.
HomogeneousParametersand Variables
An OperatorFormalism.
GraphicalRepresentationof Operators
The Squareand Square-rootOperators
43
.
.
.
CHAP.4. HARMONIC TRANSFORMER LINKAGES.
THE HAB~ONXC
TRANSFORMED.
. . .
. .
.
3I
32
34
37
40
43
47
49
51
54
58
.
.
4-1. Definitionand Geometryof the Harmonic Transformer.
4.2. Mechanizationof a Function by a Harmonic Transformer.
ix
58
58
61
x
CONTENTS
4.3.
4.4.
45.
4.6.
4.7.
48.
HARMONIC
The Ideal Harmonic Transformerin HomogeneousParameters,
Tables of Harmonic TransformerFunctions
Total StructuralError of a Nonideal HarmonicTransformer
Calculation of the StructuralError Function ~H~of a Nonideal
Harmonic Transformer.
A Study of the StructuralError Function 6H~.
A Methodfor the Designof NonidealHarmonicTransformers.
62
63
67
6S
71
75
TRANSFORMERS
77
IN SERIES.
49.
Two Ideal Harmonic Transformersin Series. .
410. Mechanizationof a GivenFunctionby an Ideal Double Harmonic
Transformer. . . . . . . . . . . . .
. . .
4.11. PreliminaryFit to a Monotonic Function.
412. PreliminaryFit to a NonmonotonicFunction
4.13. Improvementof the Fit by a Method of Successive Approxirna tions.
4.14.
4.15.
. . .
79
82
89
!)1
. .
m
Nonideal Double Harmonic Transformers.
Alternative
Method for Double-harmonic-transformer
CHAP.5. THE THREE-BAR LINKAGE
THE
7?
.
.
Design
.
][)1
. .
107
5.1. FundamentalEquationsfor the Three-barLinkage.
5.2. Classificationof Three-barLinkages
5.3. SingularCasesof Three-bar Linkages,
1 ~2
5.4.
117
The Problem
NOMOGRAPHIC
55.
5.6.
5.7.
5.8.
of Designing
Three-bar
.
kfETHOD . . . . . . . . . . . . . . . . . . . . . 118
Analytic Basis of the homographic
The Nomographic
Chart . . . .
Calculation
of the Function
Linkage .,.,.......,,,
Complete
Linkages.
Representation
Nomogram,
,.,
.
Method.
. . . . . . .
Generated
of Three-bar-linkage
. . .
. . . . . . .
of the Design
Problem
for
. . . . . . . . . . . . . . .
Functions
. .
Restatement
Method,
5.10.
5,11.
Survey of the Nomographic
Method
.
Adjustment
of ba and a, for Fixed AX,, AX,, b,
512.
Alternative
Methods for Overlay Construction.
Choice of Best Value of b, for Given AX,, AX,.
An Example
of the Nomographic
118
. . . ...120
by a Given Three-bar
. . . . . . . . ...122
5.9
513.
5.14.
107
108
. . .
by the
. .
..1’25
the Nomographic
. . . . . . ...127
128
132
.
.
.
.
. .
Method.
THE GEOMETRIC
METHonFORTHR~E-BLR
LINKAGEDESIGN. .
.
136
137
.
139
. . 145
5.15. Statementof the Problemfor the Geometric Method. .
146
5.16. Solutionof a SimplifiedProblem. .
.
. . . 147
5.17. Solution of the Basic Problem.
.
151
5.18. Improvementof the Solutionby SuccessiveApproximations.
154
5.19. An Application of the Geometric Method: Mechanization of the
Imgarithmic
Function.
. . . . . . .
. .
.
.
. 156
k
xi
CONTENTS
CHAP. 6. LINKAGE
COMBINATIONS
DOM . . . . . . . . . . . . . .
OF Two
COMBINATION
AGE
.
.
.
.
.
.
WITH
ONE
.
DEGREE
OF FRf3E-
.
166
HARMONIC TRANSFORMERS WITH A THREE-BAR I.lNK-
.
.
.
.
.
.
of the Problem.
.,
6.1.
Statement
6.2.
Factorization
6.3.
6.4.
Example:
Example:
Factoring the Given Function
Design of the Three-bar-linkage
166
.,,.,
.
of the Given
166
Function
.
Harmonic
168
171
174
Component
6.5.
Redesign
of the Terminal
6.6.
Example:
Redesign
of the Terminal
Harmonic
Transformers
6.7.
Example:
Assembly
of the Linkage
Combination,
186
187
Transformers
193
THREE-BAR LINKAGES IN SERIES.
6.8.
CHAP. 7.
The Double
FINAL
7.1.
8.
Roles of Graphical
9.
195
OF LINKAGE
and Numerical
arameters
. . . . .
73.
Use of the Gauging
Parameter
7.4.
7.5.
Small Variations
Large Variations
.
CONSTANTS
Methods
in Linkage
. .
. . .
. . .
Linkage
Constants.
. .
in Adjusting
of Dimensional
of Dimensional
199
Design.
. .
199
..2o2
201
Constants
Constants
205
205
7.6.
Method
7.7.
Application
of the Gauging-parameter
Linkage . . . . . . . . . . . . .
Method to the Three-bar
. . . . . . . .
...207
7.8.
Application
Method
7.9.
Linkage.
An Example,...,..
The Eccentric Linkage as a Corrective
of Least Squares
LINKAGES
8.1.
82.
8.3.
Cu.
Linkage
ADJUSTMENT
72.GaugingP
CHAP.
Three-bar
195
206
of the Gauging-parameter
WITH
TWO
217
OF FREEDOM
223
Function.
223
226
228
of Grid Structures.
232
DEGREES
Analysis of the Design Problem
Possible Grid Generators for a Given
The Concept of Grid Structure
8,4.
Topological
8.5.
The Significance
Transformation
8.6.
Choice
8.7.
Uw of Grid Structures
of Ideal Grid Structure
of a Nonideal
BARLINKAGE
to the Three-bar
. .,........209
Device.
Grid Generator.
in Linkage
MULTIPLIERS
233
.
Design.
. .
.
.
.
238
. .
243
.
.250
9.1.
The Star Grid Generator,,
9.2.
A Method
9.3.
9.4.
Grid Generators for Multiplication.
A Topological
Transformation
of the Grid Structure
9.5.
Improvement
9.6.
9.7.
Design of Transformer Linkages.
Analytic Adjustment
of Linkage
Multiplier
Constants
271
277
9.8.
Alternative
the Error of a Grid Generator
281
for the Design
.250
of Star Grid Generators
Ideal Grid Structure.....,..
. .
of the Star Grid Generator
Method
for Gauging
with Almost
. . . . .
. ...251
of a Divider.
for Multiplication.
256
258
264
I
]
I
CONTENTS
xii
10. BAR-LINKAGE FUNCTION GENERATORS WITH TWO DEGREES OF FREEDOM . . . . . . . . . . . . . . . . . . ...284
~H,4P.
1
,
I
Summaryof the Design Procedure.
10.1.
10.2.
Example:
10.3.
Example:
10.4.
in Vacuum
Curve Tracing
Function
i
I
4
First
Approximate
in Vacuum
Improving
. .
. . . . . . . . .
the Mechanization
.,,.....
and Transformer
. . .
.
of
the
Ballistic
. . . .
. .
Mechanization
of the Ballistic
.
Linkages
.
284
...286
Function
for Noncircular
Scales
.292
295
1
>
APPENDIX A. TABLESOF HARMONIC
TRANSFORMER
FUNCTIONS
APPENDIX B.
. . . 301
PROPERTIES
OFTHETHREE-BAR-LINKAGE
NOMOGRAM
INDEX . . . . . . . . . . . . . .
. .
.
. . . .
.
. .
333
. 353
1
CHAPTER
1
INTRODUCTION
1.1. Types of Computing Mechanisms.—Computing
mechanisms
may be divided into two distinct types: arithmetical computing machines,
familiar to the layman through their common use in business offices, and
continuously acting computing mechanisms and linkages that range in
complexity from simple cams and levers to enormously complex devices
for the direction of naval and antiaircraft gunfire.
The arithmetical computing machines accept inputs in numerical
form, usually on a keyboard, and with these numbers perform the simple
and
arithmetical operations of addition, subtraction, multiplication,
division-usually
by the iteration of addition and subtraction in counting
devices. The results are finally presented to the operator, again in numerical form. In their simplest forms these machines have the virtue of
applicability in a wide variety of computations, including those requiring
very high accuracy.
By elaboration of these devices, as by the introduction of punched-tape control, their possibilities for automatic operation
can be greatly increased.
Characteristic of their operation, however, is
their production of numerical results by calculations in discrete steps,
involving delays which are always appreciable and may be very large
if the required calculation is of complex form.
Continuously acting computing mechanisms are less flexible and have
less potential accuracy, but their applicability to the instantaneous or to
the continuous solution of specific problems-even
quite complex ones—
makes them of great practical importance.
They may serve as mere
indicators of the solutions of a problem, and require further action by
human agency for the completion of their function (speedometer, slide
rule); or they may themselves produce a mechanical action functionally
related to other mechanical actions (mechanical governors, automatic
gunsight).
Continuously acting computers fall into two main classes: function
generators and differential-equation solvers.
Function generators produce mechanical actions—usually displacements or shaft rotations—that
are detite functions of many independent variables, themselves introduced into the mechanism as mechanical actions.
Simple examples of
such mechanisms are gear differentials, two- and three-dimensional cams,
slide multipliers and dividers, linkage computers, and mechanized nomograms. Computers of the second class generate solutions of some definite
1
2
INTRODUCTION
[SEC.1.2
differential or integrodifferential
equation—often
an equation that
involves functions continuously determined by variable external cirElementary devices of this type are the integrators, comcumstances.
ponent solvers, speedometers, andplanimeters.
From these elementary devices one can build up complicated mechanisms that perform elaborate calculations.
We may mention their
application in gunsights, bombsights, automatic pilots (for airplanes,
submarines, ships, and torpedoes), compensators for gyroscopic compasses, tide predictors, and other robots of varied types.
The present volume will deal only with the problem of designing continuously acting computing mechanisms.
1.2. Survey of the Problem of Computer Design.-There
is no set
rule or law for the guidance of a designer of complex mechanical computers. He must weigh against each other many diverse factors in the
problem: the accuracy required; the cost, weight, volume, and shape of the
computer; its inertia and delay in action; the forces required to operate it;
its resistance to shock, wear, and changes in weather conditions.
He must
consider how long it will take to design the computer, how easily it can
be built, how easily it can be operated by a crew, whether suitable sources
of power will be available, and so on. The complexity of the theoretical
and practical problems is so great that two designers working on a given
problem will never arrive at precisely the same solution.
For practical reasons, a designer should be asked to find a computer
that meets certain specified tolerances, rather than the best possible
computer for a given use. He should know what will be the maximum
tolerated error of the computer, the maximum cost, weight, and volume
occupied, the maximum number of operators in the crew, the maximum
number of servomechanisms allowed, and so on. Tolerances provide a
convenient means for controlling the development of the computer, and—
if established in a practical way—they permit some freedom of choice by
the designer.
Choice of Approach to the Design Problem.—The type of computer to be
built is sometimes indicated in the specifications.
If not, the first task
of the designer is to decide whether the computer is to be mechanical,
electrical, optical, or a combination of these. At the same time that this
important decision is made, the designer must weigh in his mind the path
that his thinking will follow.
There are two principal methods for designing a computer: the constructive method and the analytic.
The constructive method makes use of a small-scale model of the real
system with which the computer is to deal. For example, a constructive
antiaircraft fire-control computer might determine the elements of the
lead triangle by maintaining within itself and measuring the elements of a
small model of this triangle.
SEC.1.2]
SURVEY
OF THE PROBLEM
OF COMPUTER
DESIGN
3
In using the analytic method, the designer concentrates on the analytic
A relation between variables,
relations between the variables involved.
such as
(1)
can be given mechanical expression in terms of displacements or shaft
rotations, without regard to the nature of the quantity represented by the
variables x, y, and z. For example, one may possess two devices that
generate output displacements xy and x/y, respectively, given input displacements z and y. Combining these with a third device for adding their
output displacements, one can then produce a computer that, given input
displacements x and y, generates a final output displacement z having
continuously the value specified by Eq. (1).
The computer is then a
“mechanization”
of Eq. (1), rather than a model of any special system
involving variables x, y, and z thus related.
Computers designed by analytic methods consist of units (“cells”)
that mechanize fairly simple relations, so connected as to provide a
mechanization of a more complex equation or system of equations.
For
any given problem a great variety of designs is possible.
This variety
arises in part from the possible choice among mechanical cells mechanizing a given elementary relation, and in part from the variety of ways in
which the relation between a given set of variables can be given analytic
expression. Thus, each of the equations
Z=;
(yj+
l),
(2a)
2=X
()
(2b)
y+~,
Y
Z’7J= Z(yz + 1),
(2C)
[all equivalent to Eq. (l)] suggests a different method of connecting
This flexibility in analytic
mechanical cells into a complete computer.
design methods makes it possible to arrive at designs that are in general
more satisfactory mechanically than those obtained by constructive
methods.
In the present volume we shall be concerned entirely with mechanical
computers designed by the analytic method.
Block Diagram of the CompuLer.—To each formulation of the problem
in analytic terms there corresponds a block diagram of the computer.
in thk diagram each analytic relation between variables is represented
by a square or similar symbol, from which emerge lines representing the
variables involved; a line representing a variable common to two relations
will connect the corresponding squares in the diagram.
In mechanical
terms, each square then represents an elementary computer that estab-
4
INTRODUCTION
[SEC. 1.2
lishes a specified relation between the variables, and the connecting lines
represent the necessary connections between these elementary computers. By examination of block diagrams the designer will be able to
see the principal virtues of each computing scheme: the complexity of the
system, the workhg range of variables, the accuracy required of individual
components, and so on. On this basis he can make at least a tentative
selection of the block diagram to be used.
Selection oj Components jor the Computer.-Knowing
the accuracy and
mechanical properties required of each computing element, the designer
can select the elementary computers from which the complete device is to
be built.
As an example of the diverse factors to be borne in mind, let us suppose
that it is required to provide a mechanical motion proportional to the
product of two variables, X1 and X2. A slide multiplier of average size
will allow an error of from 0.1 per cent to 0.5 per cent of the whole range
of the variable; this error will depend on the quality of the construction
A linkage multiplier
-on the backlash and the elasticity of the system.
will have an error of some 0.3 per cent due to its structure, practically no
error from backlash, and a slight error due to elasticity of the system if the
unit is well designed; the space required by a linkage multiplier is small,
but its error cannot be reduced by increasing its size. If these devices do
not promise sufficient accuracy, the designer must use multipliers based
on other principles.
It is possible to perform multiplication by use of
two of the precision squaring devices illustrated in Fig. 1.23, by connecting these in the way suggested by the equation
X1X2 = +(X1 + x2)* – *(X1 – X2)2.
(3)
The error of such a multiplier may be as low as 0.01 per cent, but the
About the same accuracy is attainsystem has an appreciable inertia.
able by a multiplier based on the differential formula for multiplication,
d(xlxz)
= xl dxz + x2 dxl;
(4)
this employs two integrators, and is commonly used when two quantities
This scheme is useful only
are to be multiplied in a differential analyzer.
when it is possible to allow a slow change in a constant added to the
product XIX~a
change which will result from slippage in the integrators, negligible for a single multiplication but accumulating with repetition
of the operation.
From this discussion it should be evident that there is no “best”
multiplier.
Similarly, other components of a computer must be selected
with due regard for their special characteristics and the demands to be
made upon them.
Mathematical Design oj the fi@em.—From
the block diagram one
should proceed to the mechanical design of a system through an inter-
SEC.1.3]
ORGANIZATION
OF THE PRESE>T
VOLUME
5
mediate step-that
of establishing the “mathematical
design” of the
system. The mathematical design ignores the dimensions not essential
to the nature of the computation to be carried out-diameters
of shafts,
dimensions of ball bearings, dimensions of the fram~but
specifies the
dimensions of levers measured between pivots and joints, the size of friction wheels, tentative gear diameters and gear ratios. The properties of
this design should be studied carefully, because this usually leads to a
change in some detail of the design, and sometimes even to choice of a new
block diagram.
Final Steps in the Design.-From
the mathematical design of the
The elements
system one can proceed to the design of a working model.
of this model should be accessible rather than massed together, inexpensive, and quick to manufacture.
If the performance of the working model
is found to be satisfactog,
the first model can be designed.
Here the
The parts of
ingenuity of the designer must be used to the maximum.
the mechanism must be arranged compactly to decrease space requirements, weight, and the effects of elasticity and thermal expansion, but
they should not be massed in such a way that assembly is difficult, or
Sometimes division of the whole computer
repair or servicing impossible.
into several independent parts is advisable.
Finally, the computer can be
built and tested against specifications.
1.3. Organization of the Present Volume.—It is not possible to discuss in one volume all elements of the problem of computer design. This
book will deal principally with bar-linkage computers—specifically,
with
the mathematical design of elements for such computers.
Bar linkages
are mechanically very satisfactory, and computers built from them have
many important virtues, but the mathematical design of these systems
is relatively difficult and is not widely understood.
There are few standard bar-linkage elements for computers; it is usually necessary to design
the components of the computer, and not merely to organize standard
It is hoped that the design methods
elements into a complex assembly.
to be described here will lead to their more general use.
Bar linkages can be used in combination with the standard computing
mechanisms.
For this reason, and for the contrast with the bar linkages which are to be discussed later, this volume begins with a brief survey
of some more or less standard elements of mechanical computers.
Chapter 2 is devoted to a general discussion of bar linkages.
Chapter 3
establishes terminology and describes graphical procedures of which
extensive use will be made.
Chapters 4, 5, and 6 discuss, in order of their
increasing complexity, bar linkages with one degree of freedom—generators of functions of one independent variable,
Chapter 7 indicates some
mathematical methods of importance in bar-linkage design. Finally,
Chaps. 8, 9, and 10 develop methods for the design of bar-linkage gener-
[SEC.14
INTRODUCTION
6
ators of functions of two independent
linkages have very striking advantages.
ELEMENTARY
variables—a
COMPUTING
field in which bar
MECHANISMS
The remainder of this chapter will gi~’e a brief survey of elementary
Discomputing mechanisms, or “cells,” of more or less standard type.
cussion of bar-linkage cells will be deferred to Chap. 2.
1.4. Additive Cells.-”
Additive”
or “linear”
cells establish linear
relations between mechanical motions of the cell, usually shaft rotations
or slide displacements.
If these are described by parameters X1, X2, X3,
the cell will compute
X3=
Q. X,+
Q’. X,+C.
(5)
Here Q, Q, and C are constants depending on the design of the cell and the
choice of the zero positions from which Xl, XZ, and X3 are measured.
By
FIG. 1 1.—Bevel-gear clifferential.
proper choice of the zero positions, C can always be made to vanish; in
what follows it will be assumed that this has been done.
The bevel-gear deferential (Fig. 1.1) is a well-known linear CCI1f’[Jr
The parameter Xl is the rot:~which all three parameters are rotations.
tion of the shaft S1 from a predetermined zero position, X1 = O; the positive direction of rotation is indicated by symbols representing the hewl
and tail of an arrow with this direction.
The parameter X2 is the rotation
of the shaft SJ from a similar zero position; Xj is the rotation from its zero
position of the cage C carrying the planetary bevel gears G. The zero
positions are not indicated in the figure.
The equation of the bevel-gear differential is
X, = 0.5X,
+ 0.5XZ.
(6)
To derive this it is convenient to consider the value of X2 corresponding to
given values of X, and Xs. Let us consider the differential to be originally
in the position X1 = X2 = X3 = O. The parameters Xl and X3 can then
be given their assigned values in two steps, the first a rotation of both the
SEC,14]
ADDITIVE
CELLS
7
shaft S1 and the cage C through the angle X3, and the second a rotation of
the shaft S’1through an additional angle Xl – Xi.
In the first step the
differential moves as a unit; the shaft Sz is rotated through the angle X3.
In the second step, the cage is stationary and the movement of the shaft
S1 is transmitted to the shaft SZ with its sense of rotation reversed; the
rotation through angle Xl — X3 of the shaft SI causes rotation through
Xt – Xl of the shaft SZ. The total rotation of the shaft SZ is then
It is,
X2 = Xt + (Xt – X,), from which Eq. (2) follows immediately.
of course, essential that all rotations be taken as positive in the same
sense.
It is remarkable that Eq. (6) is independent of the ratio of the bevel
gearing of the clifferential; the essential characteristic of this type of
X3
FIG.1.2.—Cylindrica1-gear
clifferential.
differential is that the gearing of the cage transmits the relative motion of
the shaft S1 to the shaft SZ in the ratio 1 to 1, but with reversed sense. It
is not necessary to use bevel gears in the cage to obtain this result;
cylindrical gears can accomplish the same purpose.
A cylindrical-gear
differential is shown in Fig. 1.2. This differential is equivalent to the
It is
common bevel-gear differential, except in its mechanical features.
flatter, and easier to construct in large numbers, but there is one more
gear mesh than in the common type; there may be more backlash and
more friction.
It should be noted, however, that bevel gears are subject
to axial as well as radial forces in their bearings, and that these may also
increase friction.
The spur-gear differential shown in Fig. 1.3 has only two gear meshes,
and is quite flat. The planetary gears G in their cage C do not invert the
motion of the shaft SI when transmitting it to the shaft S3, but can be
made to transmit it at a ratio different from 1. The eauation of this
8
INTRODUCTION
[SEC.1.4
differential is
X, = QX, + (1 – Q) X,.
To prove this relation we can use the same method as before.
begin by considering the differential in the zero position,
(7)
Let us
X1=X,=X3=0.
We wish to find the value of X3 corresponding to given X, and X2. We
introduce the angles Xl and X2 in two steps, first turning both the shaft
FIG. 1.3.—Spur-gear clifferential.
FIG.1.4.—Differential
with axially displaced spiral gear.
S, and the cage C through the angle X2, and then the shaft S1 through an
In the first step the differential is turned as a rigid
additional Xl - X,.
ln the second
body; the shaft S, is also turned through the angle X2.
step the shaft S’s is turned through Q(xl — X2); its total motion is
X, = X, + Q(X, – X2), in agreement with Eq. (7).
If we make Q = Q’ = 0.5 by proper choice of the gear ratios, we can
The fact
obtain a differential equivalent to the” bevel-gear differential.
that the free choice of Q gives to this differential a larger field of applicability does not necessarily mean that this differential should be preferred
SEC.1.4]
ADDITIVE
CELLS
9
tothosewith
Q = 0.5; it inconvenient tousedifferentials
with Q =0.5
as prefabricated standard elements.
A deferential m“th axially displaced spiral gear is shown in Fig. 1.4.
The parameter,,
which measures the axial displacement of the spiral
gear andthepin PZ, invariable only ~\-ithinfinite limits.
The mechanical
structure of this differential is, however, much simpler than that of the
differentials already mentioned, for which all parameters can change without limitation.
The equation of this differential is
X3=X1*9X2,
(8)
where nis the number of threads perinch along the axis of the spiral gear
on the shaft Sz and m is the number of teeth on the gear with whichit
FIG.1.5.—Differential worm gearing.
meshes. Thehelical mgleofthe
gears should beat least 450 for smooth
action and small backlash.
Thediferential
worm gearing shown in Fig. 1.5 is used for the same
purpose asthe preceding differential, especially if the range of values of
X, corresponds to a large fraction of a revolution of the shaft S, or even
to several revolutions of this shaft. The equation of this d”%-entialis
x,=
++
X1+
+X2
(radians)
(9)
where t is the number of teeth of the worm gear, m is the multiplicity of
the 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 the
spiral or worm gear.
The screw differential shown in Fig. 1.6 combines an axial translation
Xl of a screw with a translation X2 of the nut N with respect to the screw;
x3 = xl + x,.
(lo)
10
[SEC.1.4
INTRODUCTION
To obtain the first translation, the pin P on which the screw turns is
displaced by X1. Therotation of thescrew comes fromthegear G,which
meshes with a cylindrical rack C and slides along it. The real input
IWILI
x,
i
I
*
‘“
‘c-k’
-+E=Eaii
FIG.1.6.—Screwdifferential.
parameter of the differential is not X,, but the angle X, through which the
rack is turned. The equation of the differential is then
X3 = x,
* kx4.
(11)
The sign depends on the sense of the screw; k is a constant determined by
the gear ratio, the number of threads per inch on the screw, and their
multipli~ity.
All three parameters of this
differential have constructive limits.
The belt deferential (Fig. 1.7) makes
use of the inextensibility of a belting on
several pulleys.
In practice,
chains,
strings, and special cables are used as belts.
The equation of the belt differential is
X3 = c – 0.5XI – 0.5X2,
(12)
where C is a constant depending on the
choice of zero points of the parameters.
.++
The tension in the belt must not fall
below zero at any time; if it does, the belt
FIG.1.7.—Belt differential.
will sag and the equation of the differential
will not hold.
To obtain positive action in the direction of increasing X3,
it is necessary to preload the belt by putting a load on the output pulley—
ior instance, by a spring that can exert a force large enough to produce the
desired action.
The maximum driving force required for thk dMerential will then be about twice the force necessary to operate it without
preloadlng.
The loop-belt di~erential (Fig. 1.8) has the belting in the form of a loop
with length independent of the position of the pulleys,
The belt can then
SEC-1.4]
MULTIPLIERS
11
be preloaded (turnbuckle B) without adding to the driving force of the
differential, except by the increased friction in the bearings.
Belt differentials are sometimes used to add a large number
of parameters; they are easily
combined in batteries, as indicated
schematically in Fig. 1.9. In such
an arrangement the parameter XT
may have so large a range that it
is impractical to use a slide as the
It is better
output terminal.
FIG.1S.-Loop-belt clifferential.
practice to use a drum (dashed
line in Fig. 1.9) on which the belt is wound on. and at the same time
wound off. To prevent slippage, the belt should make many turns on the
drum and be fastened to it; a chain
on chain sprockets may also be used
as the belt.
The above enumeration does not
exhaust the possibilities for linear
mechanical cells; there are many
variants the use of which may be
dictated by special circumstances.
As a rule, when a differential is
used in a computing mechanism, two
of its members (the input terminals)
are moved by external forces; this
results in movement of a third member (the output terminal) which is in
turn required to furnish an appreciable force.
If differentials were frictionless, any two of their
three
terminals could be used as input
terminals.
In reality, only a few of
the differentials described here have
FIG. 1.9.—Loop-belt differential for complete interchangeability
of the
the evaluation of
terminals.
For instance, with the
X1= C–X1–2XZ+2X,
–2X,
+ 2X, – 2X6. screw clifferential [Fig. 1.6) it is impossible to have Xd as the output
parameter if the helical angle of the screw is so low that self-locking of the
nut on the screw occurs; it is possible to use Xl as an output parameter,
and, of course, also X,.
With the chfferential worm gearing of Fig. 1“5,
X, is an impracticable output parameter.
12
INTRODUCTION
106. Multipliers.-Multipliers
three parameters a relation
are computers that establish between
RX3 = X, .X,,
(
[SEC.1.5
(13)
where R is a constant that depends on the type of multiplier and on its
dimensions.
The action of the slide multiplier shown in Fig. 1“10 is based on the
proportionality of the sides of two similar triangles.
These are triangles
with horizontal bases, and vertices at the central pin shown in the figure:
Iv;,:—’
.-
FIG.1.10.—Slidemultiplier.
the first has a base of length R and altitude Xl, the second a base of length
X2 and altitude X8. Thus
R
x,
T, = X’
(14a)
RX3 = X,X2.
(14b)
or
The figure gives a schematic rather than a practical design; the lengths of
the sliding surfaces as shown are not great enough to prevent self-locking
These lengths determine the
in all possible positions of the mechanism.
space requirements for multipliers of this type; they must be relatively
large in two directions.
It is difficult to make this type of multiplier
precise. The pins in slots, as shown in the figure, are mechanically
inadequate, and roller slides on rails must be used. One can not achieve
the same end by increasing the dimensions of the multiplier because the
SEC.1.5]
MULTIPLIERS
13
elasticity of parts comes into play, not only when the parts are operating
in a computer, but also when they are being machined.
The slide multiplier shown in Fig. 1.11 saves space in one direction.
There are fewer sliding contacts, and the slides are easier to construct.
Fm. 1.11.—SlidemultiplierwithinputsXI, XI - Xz.
.
I?m. 1.12.—Intmwction nomogram for multiplication z; = z j . z*.
This device cannot multiply Xl and X2 directly to compute RX3 = X1X2;
the input terminals must be given translations of Xl and X, – X2. The
difference is easy to obtain if the parameters are generated as shaft
revolutions before entering the multiplier; screws can then be used instead
14
INTRODUCTION
[SEC.1‘5
of the slides shown in the figure, and the required difference can be formed
by a gear differential.
Nomographic Multipliers.—A multiplier that is structurally related to
a nomogram for multiplication will be called a “nomographic multiplier. ”
Such multipliers can be derived from intersection or alignment nomograms;
the examples to be given here are related to intersection nomograms.
FIG.1.13.—An intersection nomogram for multiplication, obtained from the nomogram in
Fig. 1.12 by a projective transformation.
Figure 1.12 shows an intersection nomogram for multiplication in an
unusual form, the full significance of which will be made clear in the latter
part of this book.
Thk represents the formula
Xi = XiX