Mathematical Machines: Analog Devices, Volume 2

MATHEMATICAL MACHINES Volume II: Analog Devices By

Francis

J.

Murray

analogy demonstrates principle that diverse physical systems may often be described by the same mathematical and logical relationships. This principle is essen tial to engineering and scientific design arc! to such devices as guided missile controls, The

of

automatic airplane automatic transmissions.

pilot trainers,

pi!o?5,

ard

Three categories of mathematical ma chines which use the principle of analogy are discussed in this volume. The action prin ciples, characteristics, and use of continuous computers are presented in the first part.

These computers consist of components which represent individual mathematical opera tions. In their commercial form, these de vices are basic to the engineering design of complex machines such as airplanes and tanks, and they are incorporated into many control devices.

True analogs, discussed in the second are continuous devices wh?ch have

part,

been

utilized

by

individual

investigators to

obtain deeper insights into complex situa tions. The author examines the theory of

analogs and includes descriptions of Dimension Theory, Models, and principles of sp^fc elationships. Tha third part of the volume D-JJCM^ Me procedures and deigns which permit various 3 TiUiicji! ~rft JTrn+s to I liTjVh i 3cranced mathematical computations even though the devices themselves are simple. The instruments discussed include planim-

true

"

j

mat"

i

integrometers, and various geomet trigonometrical devices, and the author demonstrates that many of th-sie instruments appeared early in the de/esopment of mathematical machines and are the predecessors of devices now employed in continuous computers.

eters, rical

and

Aboul

the Author

Francis matics at

Murray is professor of mathe Duke University and Director of

J.

Special Projects

in

Numerical Analysis.

\m

8

MAR 24

1983

WY19

83

MAi

J984

510.81 Murray

M98m

v .2

6?

fethemtical machines

kansas city

Books

||||

will

public library

be issued only

on presentation of library card. D

!ease report lost cards and of

residence promptly. change Card holders are responsible for all

or other

books, records, films, pictu

library materials

checked out on their cards,

MATHEMATICAL MACHINES VOLUME

II

ANALOG DEVICES

MATHEMATICAL MACHINES FRANCIS

J.

MURRAY

VOLUME

II

ANALOG DEVICES

COLUMBIA UNIVERSITY PRESS

NEW YORK,

1961

Parts of this material were previously published under the

title

The Theory of Mathematical Machines Reproduction for the

Copyright

in

whole or

in

part of this

work

is

permitted

the United States Government purposes of

1947, 1948, 1961,

Columbia University

Library of Congress Catalog Card

Press,

New York

Number: 61-7812 Manufactured in the United States of America

CONTENTS OF VOLUME

II

Part III

CONTINUOUS COMPUTERS 1.

H.

Introduction A. B.

C.

A CONTINUOUS COMPUTER AMPLIFIERS AND STABILITY PROBLEM RANGE IDEA OF

D. CONTINUOUS

COMPUTATION

55

THEOREM 56

3

J.

COMPLEX CIRCUIT THEORY AND

4

S

ILLUSTRATIONS; FILTERS Electrical

and Multipliers INTRODUCTION TO MECHANICAL

A.

Computing INTRODUCTION

69

B.

POTENTIOMETERS

69

C.

ELECTRICAL ADDITION

75

D.

CONDENSER INTEGRATION

77

COMPONENTS

7

ADDERS

8

C.

MULTIPLYING BY A CONSTANT

12 7.

D. SIMILAR TRIANGLE MULTIPLIER

RESISTANCE

15

Amplifiers A. THE BASIC NOTION OF

AN AMPLIFIER VACUUM TUBES AS AMPLIFIERS

81

B.

81

C.

FEEDBACK AMPLIFIERS

AND MECHANICAL

COMPONENTS DIVISION

17

22

D. STABILITY 3.

Cams and Gears A. CAM THEORY

26

B.

FUNCTION CAMS

27

C.

INVOLUTE GEARS AND WRAPAROUNDS

31

D.

LOG AND SQUARE CAM MULTIPLIERS BACKLASH

33

8.

E.

E.

DRIFT COMPENSATION

F.

SUMMING AMPLIFIERS

G.

INTEGRATING AMPLIFIERS

Electromechanical Components A. MOVING WIRE IN FIELD

34 B.

4.

A.

INTEGRATORS

37

B.

DIFFERENTIATORS

39

C.

MECHANICAL AMPLIFIERS

43

95

97 101

103

MECHANICAL ANALOGS OF ELECTRICAL CIRCUITS

106

WATT HOUR METER

107

D.

SYNCHRO SYSTEMS

109

9. Electrical

Circuit

88 91

C.

Mechanical Integrators, Differentiators,

and Amplifiers

5.

63

6

B.

F.

THEOREM

APPLICATIONS OF THEVENTN

2. Differentials

E.

S

i.

6.

A.

THEVENTN

3

Multiplication

A.

INTRODUCTION

112

B.

TIME DIVISION MULTIPLIER

113

MODULATION MULTIPLIER

118

A.

Theory INTRODUCTION

48

C.

B.

NOTION OF A CIRCUIT

48

D. STRAIN

C.

THE CIRCUIT EQUATIONS

50

E.

STEP MULTIPLIER

119

D.

MESH EQUATIONS

52

F.

CATHODE-RAY MULTIPLIERS

120

SOLUTION OF THE CIRCUIT PROBLEM

53

E. F.

G.

THE MESH CURRENTS AS SOLUTIONS OF

10.

GAUGE MULTIPLIER

118

Representation of Functions

DIFFERENTIAL EQUATIONS

54

A.

FUNCTION TABLE

123

THE NATURE OF THE SOLUTION

54

B.

SCOTCH YOKE AND OTHER RESOLVERS

125

CONTENTS

VI

C.

THE ELECTRICAL REPRESENTATION 127

D. POTENTIOMETER

METHODS OF

REPRESENTING A FUNCTION

128

E.

MULTI-DIODE FUNCTION GENERATOR

129

F.

CATHODE-RAY TUBE FUNCTION

G.

INTRODUCTORY DISCUSSION AND SETUP

ANALYZERS

178

THE SHANNON THEORY FOR THE SCOPE

134

OF MECHANICAL DIFFERENTIAL

MAGNETIC MEMORY METHODS

135

ANALYZERS

1

178

OF MECHANICAL DIFFERENTIAL

180

D. REFERENCES

36

REPRESENTATION OF SPECIAL

E.

187

INTRODUCTORY DISCUSSION OF ELECTROMECHANICAL DIFFERENTIAL

140

Linear Equation Solvers

ANALYZERS

187

F.

PRELIMINARY SETUP

188

A.

INTRODUCTION

144

G.

SCALING AND LOAD CONSIDERATIONS

189

B.

TWO-WAY CONTINUOUS DEVICES

144

H.

WIRING AND OUTPUT CONNECTIONS

192

C.

MANUAL ADJUSTMENT

146

I.

GOLDBERG-BROWN DEVICE

149

D. E. F.

J.

MACHINES USING THE GAUSS-SEIDEL

MACHINE

FEEDBACK

Equation Solvers A. INTRODUCTION

E.

Error Analysis for Continuous Computers A.

INTRODUCTION

B.

THE TYPES OF ERROR

199

C.

LINEARIZATION

200

D.

THE NOTION OF FREQUENCY RESPONSE 202

198

156 159

Harmonic Analyzers and Polynomial

D.

196

153

H, STABLE MULTIVARIABLE

E.

A ERROR EFFECT

F.

THE a ERROR; SENSITIVITY

165

HARMONIC ANALYSIS AND SYNTHESIS

C. FINITE

195

EQUIPMENT

AUTOMATIC MULTIVARIABLE

FEEDBACK IN THE LINEAR CASE

B.

193

COMMERCIALLY AVAILABLE

K. REFERENCES

14.

G. STABLE

IMPLICIT SYSTEMS OF DIFFERENTIAL

EQUATIONS

POSITIVE DEFINITE CASE OF ADJUSTERS 151

METHOD AND THE MURRAY-WALKER

12.

B.

GENERATOR

FUNCTIONS

11.

Equation Solvers INTRODUCTION

C.

H. FOURIER SERIES REPRESENTATION I.

A.

13. Differential

OF FUNCTIONS

HARMONIC ANALYZERS

204

EQUATIONS

165

ERROR AND NOISE

G.

THE

H.

SOLUTION OF LINEAR DIFFERENTIAL

166

FOURIER ANALYSIS

203

ft

167

EQUATIONS WITH CONSTANT

CONTINUOUS ANALYZERS AND

208

COEFFICIENTS

169

SYNTHESIZERS F.

POLYNOMIAL REPRESENTATION BY

15.

SPECIAL DEVICES

Digital

Check Solutions

A.

USE OF DIGITAL CHECK SOLUTIONS

170

B.

STABILITY OF DIGITAL

172

C.

HARMONIC ANALYZERS; ZEROS;

G.

206

THE REPRESENTATION OF THE

SOLUTIONS

COMPLEX PLANE

Part

214

THE ACCURACY OF DIGITAL CHECK SOLUTIONS

172

H. CHARACTERISTIC EQUATIONS

212

CHECK

216

IV

TRUE ANALOGS 1.

Introduction to

"True

THE CONCEPT OF

B.

ANALOG APPLICATIONS

C.

MATHEMATICAL PROBLEMS SOLVED BY

ANALOGS

223

Dimensional Analysis and Models A. INTRODUCTION

224

B.

MEASUREMENTS

C.

DIMENSIONALLY COMPLETE RELA

2.

Analogs"

A.

"ANALOG"

225

TIONS

228

228

229

CONTENTS D.

BUCKINGHAM S THEOREM

229

E.

MODELS

231

F.

APPLICATIONS

231

7.

Electromechanical Analogies A. DEFINITION

B. 3. Electrolytic

VH

Tanks and Conducting

C.

Sheets

OF MECHANICAL

SYSTEM

273

CONNECTION DIAGRAMS

273

MATHEMATICAL RELATIONS IN CONNECTION DIAGRAMS

275

276

A.

INTRODUCTION

233

D. ELECTRICAL ANALOGIES

B.

ELECTROLYTIC TANKS

233

E.

MASS-CAPACITANCE ANALOGY

277

C.

CONDUCTING SHEETS

236

F.

IDEAL TRANSFORMERS

278

G.

MASS-INDUCTANCE ANALOGY

283

236

H.

ELECTROACOUSTIC ANALOGIES

286

I.

ELECTROMECHANICAL SYSTEMS

288

J.

APPLICATIONS

292

D. REPRESENTATION OF

THE COMPLEX

PLANE E.

F.

ELIMINATION OF ERRORS DUE TO FINITE SHEETS

239

POTENTIAL FLUID FLOW

241

G. SPECIAL

FLOW PROBLEMS

243

8.

Two-Dimensional

Electromechanical

Analogies 4.

Membrane Analogies

A.

245

A.

INTRODUCTION

B.

DIFFERENTIAL EQUATION OF

5.

9.

Network Representation of

Partial

250

Differential Equations

SOURCES OF ERROR

251

A.

INTRODUCTION

B.

SCALAR-POTENTIAL EQUATION

305

C.

RECTANGULAR LATTICE

306

NETWORK REPRESENTATION METHOD OF FINITE DIFFERENCES BOUNDARY CONDITIONS AND APPLI

307

INTRODUCTION

252

D.

B.

THEORY OF ELASTICITY

252

E.

C.

PHOTOELASTIC MODEL

255

F.

D. PHOTOELASTIC EFFECT

255

Analogies Between Two-Dimensional Stress

Problems

AIRY S STRESS FUNCTION

259

B.

BOUNDARY CONDITIONS

260

309

CATIONS

310

CURL RELATIONS

311

H.

RECTANGULAR LATTICE

312

CONNECTION DIAGRAM AND NETWORK 313

ANALOG j.

317

MAXWELL S EQUATIONS

K. DERIVATION OF EQ. IV.9.C.3,

265

ANALOGIES

305

G.

I.

A.

C.

297

MODELS

Photoelasticity A.

6.

247

RUBBER SHEET MODELS

D. SOAP FILM E.

294

ELASTICITY

245

MEMBRANE C.

B.

TWO-DIMENSIONAL LUMPED-

CONSTANT SYSTEMS

4, 5

320

PartV

MATHEMATICAL INSTRUMENTS 1.

Introduction A.

MATHEMATICAL INSTRUMENTS

329

D.

TRANSFORMATIONS OF THE PLANE

E.

GENERAL THEORY AND THE CONSTRUC TION OF CURVES

2.

335

337

Transcendental Algebraic and Elementary

Operations A. FIXED PURPOSE COMPUTERS B.

C.

3.

Instruments for the Differential and

331

Integral Calculus

SLIDE RULES

332

A. DIFFERENTIATORS

PLOTTING DEVICES

333

B.

INTEGRATION AIDS

340 341

CONTENTS C.

VARIABLE SPEED DRIVE DEVICES

D. AREA-MEASURING DEVICES

342 342

Planimeters A. BASIC PRINCIPLES

5.

Integraphs A.

INTRODUCTION

353

B.

INTEGRAPH INSTRUMENTS

353

C.

THE INTEGRAPH OF ABDANK

345

B.

EXAMPLES OF LINEAR PLANIMETERS

348

C.

INTEGROMETERS

350

D.

GRAPHS IN POLAR COORDINATES

351

ABAKANOWICZ WHEEL INTEGRATORS

D. STEERING

INDEX 357

354

354

Part III

CONTINUOUS COMPUTERS

Chapter

1

INTRODUCTION

m.l.A. Idea of a Continuous Computer In a

computer individual quantities as a sequence of digits. The registers

digital

appear in

is subject to another operation. Such operation a mathematical relationship is represented in the

computer by connecting the output of the com

and combinations of these

ponent associated with the first operation to the the second. In this way, the input of that of

operations provide approximate representation

mathematical equations are translated into a

for all the operations of analysis.

setup for the computer.

digital computer performs arithmetical opera

tions

on

these digits,

possible,

up a computing device in which

however, to set

quantities are represented instance,

It is

by magnitudes. For

the value of a variable x,

a rotation, a linear dis

either

represented by

may be

A

The above

may not be quantities

may

as

represented,

which can be

unknown

interconnected.

appropriately is

necessary,

it

is

desirable that variables throughout the device be

of approxi represented by physical quantities

mately the same

sort.

of Generally speaking, there are two types

computers in which in which magnitudes: mechanical computers, and have representation, geometrical quantities quantities are represented by

electrical

Mechanical computers

computers.

would normally contain such components "differentials"

"multipliers,"

tors,"

and

as

(which are used for addition), "gear

boxes,"

"integrators."

would have analogous

"function

quantities

the computer

may

properly set up,

it

On the

well be

itself

other hand,

such that when

can immediately generate the

quantities or functions as functions

of time.

The output of such a

device appears in one of

two forms. The desired output may be certain numbers which are obtained by measurements. one Frequently the desired output is a function of variable in which case the variable, independent will

the correspond to time, and the output of

computer will be a graph of the function desired. The output appears in a device which records the value of the ordinate either at intervals or continuously.

genera

Electrical

electrical

and the

have to be adjusted by the device

until certain relations hold.

interconnection

somewhat

immediately realizable between

a computing placement, a voltage, or current. device of this type consists of components mathematical operations representing specific

Because

is

since frequently the relation desired simplified

computers components. In

ffl.LB. Amplifiers and Stability

A

continuous computer, then, consists of

confine ourselves to computers in

various components to perform mathematical

which each component has clearly specified and outputs. There are other types of inputs continuous mathematical machines which we

means of connecting them operations with some in order to represent mathematical relations.

Part III

we

will treat in Part IV. this type is Normally, a computing device of in a given each operation by having

utilized

system of mathematical equations represented by mathematical a component in the computer.

A

the equations can normally be relationship in to mean that the result of one interpreted

The operation of

these

components requires

power. On the other hand, these components tend to be inaccurate when power is transmitted

through them from input to output. Con connections between components sequently, the

must be supplemented with sources of power called amplifiers. In most cases it is desirable that the

power transmitted through a component

INTRODUCTION be negligible. The power which appears output

is

at the

and the power used

normally negligible,

at the inputs should be adequate for the function

This means that corre ing of the component. to most of the outputs, there are

sponding

which

"amplifiers"

component

and produce a quantity

this signal

with considerable power.

representing

In

the

of

case

amplifiers

output of a

will receive the

as a signal

electrical

computers,

vacuum tubes or

use

electronic devices,

equivalent

and because of called

these

this,

these

computers are frequently In a mechanical continuous computer one may "electronic."

gear boxes

for

addition,

computer, certain

ing a

power

one of the

rules

stated above.

Considerable

outputs.

Multipliers

their

and automatic function

a considerable generators, in general, require

amount of

input power even

required at their output. puter,

then,

A

if

no power

mechanical

is

com

and function generator. The gear multiplier, boxes and differentials would then be part of the interconnection

system

between

these

com

the interconnection system con

contain amplifiers. sequently would not it is not customary, a similar setup Although

can be used for a continuous

A

electrical

number of mathematical

represented

When

this

occurs the

most

For

of the

translation

facile

relations of a

problem

it

Often the

mathematical

results in

an unstable

setup.

this

reason our discussion in this part will

be concerned greatly with questions of stability. However, this stability discussion has a far

than those range of applications mentioned here. Indeed, practically every non-

computing problem has stability questions it no matter what method is used

associated with

or what devices are used for solving

example,

in

the ideas

digital

For

it.

computation one often uses

and procedures of

stability analysis

developed for continuous computers.

normally would have amplifiers

associated with the output of each integrator,

ponents, and

in turn

considered unstable. In order

a stable setup. necessary to obtain

trivial

at

required

power

problem on a continuous computer

to solve a

the other hand, integrators almost invariably are is

is

computer setup

broader

when power

this

and these deviations

the desired computation.

of these power can be transmitted to the outputs devices from the inputs without inaccuracy. On

inaccurate

The output of

may be amplified by other power sources. This of unwanted signals can swamp repeated buildup

which permit one to add, integrators, multipliers, and function generators. Gear boxes and a certain extent to

source.

the desired output

machine

differentials are exceptional to

Weak

arise.

may

source then contains stronger deviations from

following components: which permit one to multiply by a constant, differentials which are combinations of gears

the

instabilities

accidental variations occur in the signal govern

is

find

and

integration,

multiplication,

functions. representation of Since power sources are present within the

by

passive

computer. can be

relations

networks

with

fair

networks accuracy, the outputs of these passive as signals for amplifiers which feed used being

The more customary arrangement based on components in which some mathe

IH.1.C. Problem Range

The design of a complex in general, plicated structure,

device or a is

preliminary theoretical investigations. Normally the objective of physical experiments and tests is either

to

theoretical

obtain

basic

investigations

information

obtained from them. Under present-day circum stances it is impractical, with regard to both time

and expense,

to

investigate

plicated action of

In such components the input appears as a signal while the output is associated with a quantity of relatively high power.

In electronic

differential

of this type are available analyzers, components

by

One can readily give examples of devices whose

is

combined,

possibilities

constructing full-scale experimental examples.

successful functioning depends

is

the

for

or to verify results

the networks.

matical operation and amplification

com

dependent upon

include

many

many

parts.

upon

the

com

These would

electronic devices, a large variety

of airplanes and guided missiles, motors of both and the nuclear types, machines

the customary

used in manufacturing processes, automatic factories, and chemical plants. Their action can

PROBLEM RANGE

III.LC.

be described in terms of systems of equations,

possible to set

either algebraic or differential. In

yield

many instances

up continuous computers which

approximate answers to problems in partial

impractical to solve these systems of equa tions in closed explicit form. The theoretical

differential

investigation of a device in question

of such complicated devices is an essential part of present-day technology. Most of these uses, it seems fair to say, have been developed during

it is

must be

based upon some computational method. It is often possible to use continuous computers for such investigations. Many solutions to a of equations can be obtained for different

and

values

initial

different values

system

and since the Second World War. These devices

of

have also been used for control purposes for There are many years. many problems which a

sets

of the design

Thus, one can gain an excellent notion of the behavior of a proposed device parameters.

and can base an

effective

design

upon

findings

supplemented with certain experimental

investi

gations.

Even when results

purely by experimental methods, such is

experimentation time consuming.

both extremely expensive and Furthermore, in extremely

complicated devices to

same

possible to obtain the

it is

it is

practically impossible

obtain the desired information by purely

experimental

information,

When

methods.

results are required to it

is

experimental

supplement theoretical

still

theoretical investigations.

necessary

to

make

To successfully design

a device, a designer must have a thorough under standing of the basis for its action. Both theoretical

and experimental

equations.

The use of continuous computers in the design

investigations are

continuous computer can solve accurately,

more

faster,

and more economically than a human

being can.

Fire control for artillery

is

the

example, and much of present-day

classical

computing equipment was developed in its modern commercial form for this one specific purpose. Automatic computation

is

essential for

antiaircraft fire control.

There are many other control purposes for

which continuous computers are used. Auto matic pilots both for sea and for air purposes utilize

continuous computation. There are many

industrial processes

whose success depends upon mathe

the fast and accurate solution of certain

To make

matical problems.

these processes

automatic, continuous computers are frequently used.

Frequently there

is

the situation in

which the

a certain device can only be

directed for this purpose.

desired action of

adequate.

described in a rather complicated mathematical way. Here either a continuous computer must be used or, when appropriate, the principle of a

Experimental results without a theoretical framework could never be

The

designer,

therefore,

uses a continuous

computer as a means to simulate a proposed device in the laboratory.

He may

think of the

continuous

computer

incorporated

into

device, for example, temperature controls.

the

Many

computer setup as the mathematical equivalent of a working model which, however, possesses

ideas developed for continuous computing or

and which,

such devices as automatic transmissions for

far greater flexibility of adjustment also,

is

far

more convenient

for observation.

Using the continuous computer, the designer can

control purposes have been incorporated into

automobiles. lay

down

One

cannot, therefore, practically

limits for

what one would

call

con

determine the value of design parameters and of the proposed device investigate the behavior

tinuous computing. In a complex technology the

under many

applied in very

different circumstances.

By omitting

principles

of continuous

many

computing can be

forms.

or varying terms in the equations, the designer

The above discussion has tended to emphasize

can determine factors important for the success of the design as well as factors to which the

the use of continuous computers for the solution

design

is insensitive.

of Frequently the system

in this equations considered

ordinary

way

differential equations.

is

a system of

However,

it is

of ordinary differential equations. For this purpose there are a number of commercial devices

on the market. There are mechanical

differential analyzers

and two types of electronic

INTRODUCTION One of

differential analyzers.

types

these electronic

suitable for systems of ordinary dif

is

equations with constant coefficients. is not subject to this restriction.

ferential

The other type

In addition to the commercial devices there are various university installations,

including

In general, a designer can

required.

utilize

a

continuous computer much more directly than he can a digital computer. The setup is more

and the equivalent of design changes can be made by the designer himself without intermediate coding. These advantages

readily understood,

however, part and parcel of certain dis

mechanical devices which can be used for solving

are,

ordinary differential equations and electronic devices capable of giving approximate solutions

advantages associated with the limited logical of continuous computers. con flexibility tinuous computer has some specified purpose, for

to

problems in partial differential equations. Devices have also been developed for solving

A

instance, the solution of differential equations.

modern commercial

simultaneous linear equations, polynomial equa

For

one unknown, and harmonic analyzers. Harmonic analyzers are devices which will either

and are very easy to code. They cannot, however, be used for most other computational purposes and, therefore, have to

tions in

permit the evaluation of the Fourier Series of a given function or,

when

the coefficients

are

give values for the function.

known,

are, nevertheless, certain limitations in

the use of continuous computers. technological

difficulties

limited accuracy.

For

There are

which prevent the con

struction of devices having

more than a

instance,

certain

one limitation

is

the accuracy with which measurements can be made. Another is associated with the length of

time an adjustment can be maintained. How ever, this accuracy limitation can be compensated for in

many

instances by using auxiliary digital

computation.

It is also true that there are

supplemented with digital computation. However, when proper supplementary digital

many

made.

The major

other. The continuous computer can be used for general exploration purposes for which it is faster and more convenient. The

ment each

precise investi

gations of regions for which higher accuracy

is

basic

for

is

the

these

theory of electrical components and the action of computers as a whole, one must understand basic circuit theory, which is developed in Chapter 5.

are

adequate. Frequently the two types of devices, that is, the continuous and digital, can be used to supple

of the

computers. In Chapters 2, 3, and 4, the com ponents used in mechanical computers are In order to understand both the described.

Among

is

objective of the present part

theory continuous computers, both with regard to the individual components and to the use of these

presentation

devices

computer can be used for

devices

flexibility

be

problems for which the limited accuracy of these

digital

purpose,

computation is available, full use of the speed and economy of continuous computers can be

UI.l.D. Continuous Computation

There

this

have great

commercial continuous computers there

many instances

of either electrical or electro

mechanical devices.

We

describe

the

corre

sponding components and their uses in linear equation solvers and harmonic and differential analyzers.

theory for

In Chapters 14 and 15 we present a the validity of the solutions of

continuous computers as well as an analysis of their joint use with digital equipment.

Chapter 2

AND MULTIPLIERS

DIFFERENTIALS

m.2.A.

Introduction to Mechanical Components

The purpose of Chapters

and 4

2, 3,

is

to

and, of course, the remaining angle also be measured.

amount can

Practically, this permits the

describe devices which are used as components in

use of scales divided into 100,000 or

continuous computers of a mechanical nature, In general, each of these components has a

for measuring rotations far in excess of

number of inputs and one output such that some

variable.

mathematical relationship is represented. For instance, a differential is a device for representing

quantities

Two

addition.

inputs are represented by the

rotation of two shafts, and the output

sented by the rotation of a shaft which

is

is

repre

equal to

the average of the rotations of the input shafts.

possible with

parts

what

is

any other type of continuous

methods

both

However,

more

have a number of

of

representing

difficulties.

From

the computer point of view probably the most

important objection flexibility

in the

is

the difficulty of obtaining

setup of the machine. It

is

with

addition, multiplication, integration, and repre

of setup that mechanical suffer in computers comparison with electrical ones. Nevertheless, a number of systems have

sentation of a function.

been

The

operations

represented

Probably the most frequent

will

consist

of

way of represent

ing input and output is by means of rotations of a shaft. On the other hand, in mechanical devices there

is

also the possibility of representing a

regard to

set

flexibility

in

up

which computing

mechanical devices. The output into

an

electrical

signal

which

is

done by

is

transformed

is

transmitted to

another component where it is again translated back into mechanical form. Unfortunately, this

two problems of time

quantity by the translation of a rod, by a linear

in turn introduces

or angular velocity, or by a force or a torque. Each of these has been extensively used. In the

One of these is

the average time delay of a system

as a whole.

The other and more important

present chapter we

time-delay problem

stress the

use of rotations and

translations to represent mathematical quantities. It is

and

more

efficient to treat

forces

Jn

the

the use of velocities

same way that electrical on the basis of the well-

parts

of a

is

delay.

the tendency of different

computing

device

to

represent

with different time delays. Direct mechanical connections for rotating

quantities

quantities are treated,

shafts are normally

made by means of gears.

If

known

the shafts are parallel, spur gears are used.

If

analogy. Indeed, velocities and forces are

used mostly in what we refer to as true analogs a detailed treatment will be given (see Part IV)

they are at an angle, bevel gears are used. By properly shaping the teeth on these gears, the

under that heading.

rotation of one shaft can be

;

Thus, for the

moment our major

interest will

be in devices in which numerical quantities are represented by

either translations or rotations.

For extremely inexpensive devices offer easy construction

methods.

translations

On

the other

hand, they are limited in scale. Rotation has the

advantage that large amounts of rotation can be

measured by introducing a counter on the shaft which counts the total number of revolutions,

made to be an

exact

rational multiple of the rotation of the other

when

the teeth are engaged. However,

motion

is

initiated there

be a

may when

before the teeth engage, and the rotation

is

reversed there

may

when

the

slight play

the sense of

be an interval

during which the teeth are not engaged. This is in "backlash," and must be taken into account the design of mechanical computers.

of gear teeth

is

The theory

a special case of the theory of

DIFFERENTIALS

AND MULTIPLIERS

Cams are used to make a given motion a function of the rotation of a shaft. (The basis of

cams.

cams

the theory of

will

be discussed in Section

III.3.A, backlash in Section III.3.E.)

Another

difficulty

move

the pair

AA along their line of centers an

amount x and point

2(x

R

+ y).

on

the pair

BB

an amount y, then a

the chain will

It is

move an amount

clear that this

can be applied

to

of mechanical devices arises

when it is

necessary to transmit a force or torque to the output. This usually results in slippage or friction which produces

from the inputs

inaccuracies in the desired mathematical repre

To

sentation.

devices

avoid

this,

must be introduced

power amplifying form of either

in the

torque amplifiers or servos.

However, there are many applications where flexibility

is

not of

A

interest.

problem

may

involve the repeated solution of precisely the

same mathematical problem, would want what

in which case, one

referred to as a

is

"special

purpose computer." In many such cases a simple mechanical computer may be most advantageous

on

the score of reliability, inexpensiveness,

and Fig. IIL2.B.1

general sturdiness. Examples of such computers are gunsights, automatic airplane pilots, and a

any number of addends. In Fig.

variety of ship installations.

In the present chapter, in Sections III.2.B and

D,

respectively,

we

Another way

consider devices for addition

and

multiplication. In Chapter 3 representation of functions by mechanical means is discussed;

III.2.B.1,

y

is

negative. to

do

this

would be

to have the

three parallel rods with a connection joining the three in such a way that three points, one fixed on

Chapter 4 considers mechanical integration and In these chapters, quantities are by rotation of shafts or by the

differentiation.

represented either

translation of a piece.

In Section III.8.D

we

will describe the selsyn

system which permits the transmission of a shaft to position from one mechanical

component

another by an

electrical

signal.

IH.2.B. Adders

Suppose given quantities are represented by the displacement of certain rods (from fixed positions) in an apparatus. We wish to obtain a displacement corresponding to the sum of two such displacements. There is a simple initial

Fig, IH.2.B.2

arrangement by which one uses an endless chain or tape to add displacements (see Fig. III.2.B.1). The chain passes around sprocket wheels A, A, B,

B

}

and

position;

C the

distance apart

C sprocket wheels are fixed in wheels A and A are a fixed as are the If we pair B and B

The

.

each rod, remain collinear. If one outside rod displaced an is

amount x and

displaced an

amount y, then the middle rod

displaced an amount \(x

is

the other outside rod is

+ y) (see Fig. III.2.B.2).

III.2.B.

ADDERS

There are a number of ways in which the three rods can be connected so that the specified three points will remain collinear. One may have a crossbar pivoted upon the central bar with a slot

on each

side in

which a pin which

is

fixed

on the

Owing to the equality of opposite sides, BCED is a parallelogram. Hence, BD and CFare parallel.

B is the midpoint of AC and BD is one CF in length, D is the midpoint of the line segment AF and, hence, A, D, and Fare always Since

half

There

collinear.

also a

is

"lazy tong"

gram arrangement, which we There

is

a third way of accomplishing the same

Two

objective.

parallelo

will discuss later.

racks are used instead of the

outer rods, and a pinion

middle bar (see Fig.

is

III.2.B.5).

mounted on

the

Again the output

+

#x y). The gear teeth are constructed in such a fashion that the movement of this system

is

Fig. IH.2.B.3

corresponding rod

slides (see Fig, III.2.B.3).

the pivot are constrained then pins and

The

to be

collinear,

This crossbar arrangement can be replaced by

a pantagraph (see Fig.

III.2.B.4).

The

bars

AC

Fig. ffl.2.B,5

is

similar to the

strictly

rods and a wheel which

and does not IIL2.B.6).

slip

movement of a is

relative to the

The motion of

pair of

in contact with

them

rods (see Fig.

the pair of rods and

?

and the wheel can be easily specified. Let C , , 2 denote a reference position for the center of the wheel Fig. HI.2.B.4

ing

CF are rigid and equal in length with mid B and E respectively, DE = BC and points = BD CE. AC and CF are hinged at C, AC and BD at B, BD and DE at D, DE and CF at E. and

and

its

contact to the points of

two

C1} PI, and Q l refer to the correspond the circle, P2 an(* 62 ^e points fixed on

bars. Let

on the bars (see Fig. corresponding points fixed moved to a new III.2.B.7). Now if the system is in which P and Q are the new points of position contact,

we

see that since the wheel did not slip

DIFFERENTIALS

10

relative to the bars that

Q 2 Q. 22

Since

and

QP = Q2,

P2 P

AND MULTIPLIERS

= P P = Q& = X

this yields that

P2 Cl5 ,

center of the wheel

is

the average of the other

displacements,

combination

This

are collinear.

will

conveniently

add

considerable size but normally displacements of a there will be backlash between the pinion and the

two racks.

The customary method of adding rotations is by means of a differential. This device is analogous to the rack and pinion adder with the by rotations. There are two types of differentials, the bevel gear differential and the spur gear differential. translations, however, replaced

Qo

Fig. III.2,B,8 illustrates the

Fig.

Note

arrangement of gears

m.2.B.6

that the displacement of the

P

rod has

the value

where 6

/.PjCjP in radians and

r

=

QP.

Fig. HI.2.B.8

in a bevel gear differential.

the input spur gear

A

around the axle

A A 9

which

A".

gear

The

B

is

,

bevel gear A, collar

C".

B

is

and the

free to rotate

rigidly

which

entire

is

connected to the bevel

rotation of this combination

input y.

The bevel

gears

C

are

around the axle C. However, the

combination of C and C",

The rotation of the com

constitutes the input rotation

A"

applied through a gear meshing with

shaft

constitutes the

C

is

connected to the axle

perpendicular to

C

,

so that this

combination may rotate around the axis of

the axle

C".

The

output \(x + y)

Fig. HI.2.B.7

The

and the connecting

are rigidly connected but are free to rotate

bination jc,

A"

rotation of the shaft

C"

is

the

of the combination.

Geometrically, the motion of the bevel gears Similarly the displacement of the

Q

rod

is

given

= rfl. Hence, by the expression x x + y = 2QC0, i.e., the displacement of the

QQ

is

equivalent to the motion of nonslipping right circular cones or frustums of cones.

We can even

replace each gear with a disk contingent to other

IIL2.B.

ADDERS

disks, the disk representing a cross section of the

cone

to the axis (see Fig. III.2.B.9). perpendicular

These disks must rotate without

11

(see Fig. III.2.B.11).

moves so

that

P

and

Let us suppose the system Q are the new points of

tangency.

slipping.

The rotation of the the arc

P2 P

,

i.e.,

x

disk

A

can be measured by Let z

=PP = 2

Pa

of the

Thus,

relative to circles

used for

many

and

straight lines, this

purposes.

can be

Chapter 3

CAMS AND GEARS

IH.3.A.

Cam Theory

The purpose of

point of contact.

chapter is to discuss of one variable by of functions representation shaft rotations. Ostensibly we want a device this

with two shafts such that when the turned an amount

first

shaft

is

the second shaft turns an

jc,

Then they must be cotangent

We will make certain

at this point.

These statements are

all

intuitively,

easily

comprehended

method of

x lamina, the y lamina

Another

is

the

a variety of

are, also,

method of cams. There

electrical devices for this

purpose. For the present cams of a special type only will

We suppose that the x and y mentioned above are parallel, and the x

be considered.

shafts

shaft drives the

Each of these

y

shaft

pieces

is

by certain metal

pieces.

planar in the sense that

it

has two plane faces perpendicular to the axis of the shaft connected to it, and the edges of the plane faces and the remaining surface consist of

of the shaft.

lines parallel to the axis

Mathe

matically such a solid is termed a "right cylinder."

The boundary of a plane "directrix." The figures in called

are called

bases

The

"bases."

is

lines

"elements."

called

face

on

The

termed the

the outer surface

distance between

When

"altitude."

is

the plane faces are

the altitude

small relative to the other dimensions,

customary

and the

to refer to this solid as a

altitude as the

"thickness"

it

in the

we continue will

the rotation of the

have to move. The

points of contact along the edge of the laminae

change, but the two directrices will remain cotangent at their point of contact. This motion involving the point of contact is

will

said to be rolling if there

no

is

relative

motion

between the laminae at the points of contact. For rolling motion the distance along the first directrix

between two points of contact Pl and ?/ on the second

equals the corresponding distance directrix.

there

may

In the more general case, however, be a relative motion between the

laminae at the point of contact in a direction tangential to the directrices. This is termed a sliding motion.

THEOREM

III.3.A.1.

Let

Q

C2

and

be two

laminae free to rotate around axes perpendicular to the common plane of their bases. They can rotate in contact provided that for each position

is

of one there

"lamina,"

of the lamina.

so that corresponding plane faces are

same plane. This

If after contact

is

Presumably there are two such laminae of equal thickness, one for each shaft. They are

mounted

but their

precise proof requires considerable mathematical care.

III.B.h).

possible,

give references for proof.

amount y =f(x). There are a number of methods for accomplishing this. One is the linkages (see also A. Svoboda, Ref.

geometrical

when

statements without proof, and,

is

a point

position of the

other

P

on

directrix

its

and a

such that the second

is tangent to the first at P. Let 0! be the angle of rotation of the lamina from some fixed position, and let

directrix

the angle of rotation of the second lamina

first 2

be

from a

possible if the axes are

reference position to the position described in

not too near each other. If the axes are not too

Theorem III.3.A.1, where it is cotangent with the

far apart, the

x

lamina

is

may

be rotated until

it

touches the y lamina. Ordinarily contact between the laminae would be just along an element.

Suppose both

directrices

have tangents

at a

first

lamina at ?. This determines

2

as a function

offli.

THEOREM

III.3.A.2.

Let

Q

and

laminae as in Theorem III.3.A.1, and

C2 let

be two

O x and

III.3.B.

refer

FUNCTION CAMS

respectively to the points

where the axes

of rotation meet the base plane.

Let n be the

2

common normal to let

A

the two directrices at P, and

be the intersection of this

common normal

will

to each other, but this corre

slip relative

sponds to a difference in the tangential com ponents of the motion of the curves at P.) Thus,

Aft is determined approximately by the equation

with Ofifr

Then

27

O^sin ft)Aft

^2 = M

Let