Mathematical Machines: Analog Devices, Volume 2

385 Pages·1961·17.44 MB·English

by  Francis J. Murray

MATHEMATICAL MACHINES
Volume II: Analog Devices
By Francis J. Murray
The principle of analogy demonstrates
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,
pilot trainers, automatic airplane pi!o?5, ard
automatictransmissions.
Three categories of mathematical ma
chines which use the principle of analogy
are discussed in thisvolume. The action prin
ciples, characteristics, and use of continuous
computers are presented in the first part.
Thesecomputers consistof 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
part, are continuous devices wh?ch have
been utilized by individual investigators to
obtain deeper insights into complex situa
tions. The author examines the theory of
true 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 permitvarious
mat"i3TiUiicji! ~rft jJTrn+s to I liTjVhi 3c-
ranced mathematical computations even
though the devices themselves are simple.
The instruments discussed include planim-
eters, integrometers, and various geomet
rical and trigonometrical devices, and the
author demonstrates that many of th-sie
instruments appeared early in the de/esop-
ment of mathematical machines and are the
predecessors of devices now employed in
continuous computers.
Aboul the Author
Francis J. Murray is professor of mathe
matics at Duke University and Director of
Special Projects in Numerical Analysis.
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MAR 24 1983
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Murray
fethemtical machines
kansas
city |||| public library
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MATHEMATICAL MACHINES
VOLUME
II
ANALOG DEVICES

MATHEMATICAL
MACHINES
FRANCIS MURRAY
J.
VOLUME
II
ANALOG DEVICES
COLUMBIA UNIVERSITY PRESS
NEW YORK, 1961
Partsofthismaterialwerepreviouslypublishedunderthetitle
TheTheoryofMathematicalMachines
Reproductioninwholeorinpartofthisworkispermitted
forthepurposesoftheUnitedStatesGovernment
Copyright 1947,1948,1961,ColumbiaUniversityPress,NewYork
LibraryofCongressCatalogCardNumber: 61-7812
ManufacturedintheUnitedStatesofAmerica
CONTENTS OF VOLUME
II
PartIII
CONTINUOUS COMPUTERS
1. Introduction H. THEVENTNSTHEOREM 55
A. IDEAOFACONTINUOUSCOMPUTER 3 i. APPLICATIONSOFTHEVENTNSTHEOREM 56
B. AMPLIFIERSANDSTABILITY 3 J. COMPLEXCIRCUITTHEORYAND
C. PROBLEMRANGE 4 ILLUSTRATIONS; FILTERS 63
D. CONTINUOUSCOMPUTATION 6
6. ElectricalComputing
2. DifferentialsandMultipliers A. INTRODUCTION 69
A. INTRODUCTIONTOMECHANICAL B. POTENTIOMETERS 69
COMPONENTS 7 C. ELECTRICALADDITION 75
B. ADDERS 8 D. CONDENSERINTEGRATION 77
C. MULTIPLYINGBYACONSTANT 12
7. Amplifiers
D. SIMILARTRIANGLEMULTIPLIER 15
A. THEBASICNOTIONOFANAMPLIFIER 81
E. RESISTANCEANDMECHANICAL B. VACUUMTUBESASAMPLIFIERS 81
COMPONENTS 17
C. FEEDBACKAMPLIFIERS 88
F. DIVISION 22
D. STABILITY 91
3. CamsandGears E. DRIFTCOMPENSATION 95
A. CAMTHEORY 26 F. SUMMINGAMPLIFIERS 97
B. FUNCTIONCAMS 27 G. INTEGRATINGAMPLIFIERS 101
C. INVOLUTEGEARSANDWRAPAROUNDS 31
8. ElectromechanicalComponents
D. LOGANDSQUARECAMMULTIPLIERS 33
A. MOVINGWIREINFIELD 103
E. BACKLASH 34 B. MECHANICALANALOGSOFELECTRICAL
CIRCUITS 106
4. Mechanical Integrators, Differentiators,
and C. WATTHOURMETER 107
Amplifiers
A. INTEGRATORS 37 D. SYNCHROSYSTEMS 109
B. DIFFERENTIATORS 39
C. MECHANICALAMPLIFIERS 43 9. ElectricalMultiplication
A. INTRODUCTION 112
5. CircuitTheory B. TIMEDIVISIONMULTIPLIER 113
A. INTRODUCTION 48 C. MODULATIONMULTIPLIER 118
B. NOTIONOFACIRCUIT 48 D. STRAINGAUGEMULTIPLIER 118
C. THECIRCUITEQUATIONS 50 E. STEPMULTIPLIER 119
D. MESHEQUATIONS 52 F. CATHODE-RAYMULTIPLIERS 120
E. SOLUTIONOFTHECIRCUITPROBLEM 53
F. THEMESHCURRENTSASSOLUTIONSOF 10. RepresentationofFunctions
DIFFERENTIALEQUATIONS 54 A. FUNCTIONTABLE 123
G. THENATUREOFTHESOLUTION 54 B. SCOTCHYOKEANDOTHERRESOLVERS 125
VI CONTENTS
C. THEELECTRICALREPRESENTATION 13. DifferentialEquationSolvers
OFFUNCTIONS 127 A. INTRODUCTION 178
D. POTENTIOMETERMETHODSOF B. INTRODUCTORYDISCUSSIONANDSETUP
REPRESENTINGAFUNCTION 128 OFMECHANICALDIFFERENTIAL
E. MULTI-DIODEFUNCTIONGENERATOR 129 ANALYZERS 178
F. CATHODE-RAYTUBEFUNCTION C. THESHANNONTHEORYFORTHESCOPE
GENERATOR 134 OFMECHANICALDIFFERENTIAL
G. MAGNETICMEMORYMETHODS 135 ANALYZERS 180
H. FOURIERSERIESREPRESENTATION 136 D. REFERENCES 187
I. REPRESENTATIONOFSPECIAL E. INTRODUCTORYDISCUSSIONOF
FUNCTIONS 140 ELECTROMECHANICALDIFFERENTIAL
ANALYZERS 187
11. LinearEquationSolvers F. PRELIMINARYSETUP 188
A. INTRODUCTION 144 G. SCALINGANDLOADCONSIDERATIONS 189
B. TWO-WAYCONTINUOUSDEVICES 144 H. WIRINGANDOUTPUTCONNECTIONS 192
C. MANUALADJUSTMENT 146 I. IMPLICITSYSTEMSOFDIFFERENTIAL
D. GOLDBERG-BROWNDEVICE 149 EQUATIONS 193
E. POSITIVEDEFINITECASEOFADJUSTERS 151 J. COMMERCIALLYAVAILABLE
F. MACHINESUSINGTHEGAUSS-SEIDEL EQUIPMENT 195
METHODANDTHEMURRAY-WALKER K. REFERENCES 196
MACHINE 153
14. ErrorAnalysisforContinuousComputers
G. STABLEAUTOMATICMULTIVARIABLE
A. INTRODUCTION 198
FEEDBACKINTHELINEARCASE 156
B. THETYPESOFERROR 199
H, STABLEMULTIVARIABLEFEEDBACK 159
C. LINEARIZATION 200
D. THENOTIONOFFREQUENCYRESPONSE 202
12. Harmonic Analyzers and Polynomial E. AERROREFFECT 203
EquationSolvers F. THEaERROR; SENSITIVITY
A. INTRODUCTION 165
EQUATIONS 204
B. HARMONICANALYSISANDSYNTHESIS 165
G. THE ERRORANDNOISE 206
C. FINITEFOURIERANALYSIS 166 ft
H. SOLUTIONOFLINEARDIFFERENTIAL
D. HARMONICANALYZERS 167
EQUATIONSWITHCONSTANT
E. CONTINUOUSANALYZERSAND
COEFFICIENTS 208
SYNTHESIZERS 169
F. POLYNOMIALREPRESENTATIONBY 15. DigitalCheckSolutions
HARMONICANALYZERS; ZEROS; A. USEOFDIGITALCHECKSOLUTIONS 212
SPECIALDEVICES 170 B. STABILITYOFDIGITALCHECK
G. THEREPRESENTATIONOFTHE SOLUTIONS 214
COMPLEXPLANE 172 C. THEACCURACYOFDIGITALCHECK
H. CHARACTERISTICEQUATIONS 172 SOLUTIONS 216
PartIV
TRUEANALOGS
1. Introductionto"TrueAnalogs" 2. DimensionalAnalysisandModels
A. THECONCEPTOF"ANALOG" 223 A. INTRODUCTION 228
B. ANALOGAPPLICATIONS 224 B. MEASUREMENTS 228
C. MATHEMATICALPROBLEMSSOLVEDBY C. DIMENSIONALLY COMPLETERELA
ANALOGS 225 TIONS 229