MATHEMATICAL MACHINES Volume II: Analog Devices By
Francis
J.
Murray
analogy demonstrates principle that diverse ph...
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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