Analog Computing
Contents
Acknowledgments
VII
Preface to the 2nd edition
IX
1
1.1
1.2
1.3
Introduction
1
Outline
1
The notion of analog computing
Direct and indirect analogies
4
2
2.1
2.2
2.3
2.4
2.5
2.5.1
2.5.2
2.5.3
2.5.4
2.6
2.7
2.8
Mechanical analog computers
9
Astrolabes
9
The Antikythera mechanism
9
Slide rules
11
Planimeters
13
Mechanical computing elements
17
Function generation
18
Differential gears
20
Integrators
21
Multipliers
24
Harmonic synthesizers and analysers
25
Mechanical fire control systems
29
Differential analysers
32
3
3.1
3.1.1
3.1.2
3.2
3.3
3.4
3.5
41
The first electronic analog computers
Helmut Hoelzer
41
The “Mischgerät”
42
Hoelzer’s analog computer
48
George A. Philbrick’s Polyphemus
57
Electronic fire control systems
61
MIT
67
The Caltech Computer
68
4
4.1
4.1.1
4.1.2
4.2
4.3
4.4
Basic computing elements
73
Operational amplifiers
73
Early operational amplifiers
77
Drift stabilisation
80
Summers
85
Integrators
89
Coefficient potentiometers
92
2
XII
Contents
4.5
4.5.1
4.5.2
4.5.3
4.5.4
4.5.5
4.5.6
4.5.7
4.6
4.6.1
4.6.2
4.6.3
4.6.4
4.6.5
4.6.6
4.6.7
4.6.8
4.7
4.8
4.9
4.10
4.11
4.12
4.13
97
Function generators
Servo function generators
97
Curve followers
98
Photoformers
99
Varistor function generators
100
Diode function generators
100
Inverse functions
102
Functions of two variables
103
Multiplication
105
Servo multipliers
105
Crossed-fields electron-beam multiplier
Hyperbolic field multiplier
107
Other multiplication tubes
108
Time division multipliers
109
Logarithmic multipliers
110
Quarter square multipliers
111
Other multiplication schemes
114
Division and square root
114
Comparators
115
Limiters
116
Resolvers
117
Time delay
117
Random noise generators
120
Output devices
121
5
5.1
5.2
5.3
5.4
5.5
5.6
The anatomy of a classic analog computer
Analog patch panel
123
Function generators
124
125
Digital patch panel and controls
Readout
127
Control
129
Performing a computation
131
6
6.1
6.2
6.3
6.4
6.5
6.6
Some typical analog computers
133
Telefunken RA 1
133
GAP/R analog computers
136
EAI 231R
138
Early transistorised systems
141
Later analog computers
147
THE ANALOG THING
150
106
123
Contents
7
7.1
7.2
7.3
7.4
7.4.1
7.4.2
7.5
151
Programming
Basic approach
151
Kelvin’s feedback technique
153
Substitution method
155
Partial differential equations
157
Quotient of differences
158
Separation of variables
160
Scaling
162
8
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
8.10
8.11
Programming examples
165
Solving ÿ + ω 2 y = 0
165
Sweep generator
166
Mass-spring-damper system
168
Predator and prey
172
Simulation of an epidemic
174
Bouncing ball
176
Car suspension
181
Lorenz attractor
187
Mathieu’s equation
190
Projection of rotating bodies
194
Conformal mapping
196
9
9.1
9.2
9.3
Hybrid computers
201
Systems
201
Programming
205
Example
206
10
Digital differential analysers
209
210
10.1
Basic computing elements
10.1.1
Integrators
210
10.1.2
Servos
213
10.1.3
Summers
214
10.1.4
Additional elements
214
10.2
Programming examples
215
10.3
Problems
216
10.4
Systems
217
10.4.1
MADDIDA
217
10.4.2
Bendix D-12
220
10.4.3
CORSAIR
224
10.4.4
TRICE
225
XIII
XIV
Contents
229
11
Stochastic computing
12
12.1
12.2
12.3
12.4
Simulation of analog computers
233
Basics
234
DDA programming system for the IBM 7074
CSMP
237
Modern approaches
240
13
Applications
243
13.1
Mathematics
243
13.1.1
Differential equations
243
13.1.2
Integral equations
244
13.1.3
Roots of polynomials
247
13.1.4
Orthogonal functions
247
13.1.5
Linear algebra
248
13.1.6
Eigenvalues and -vectors
249
13.1.7
Fourier synthesis and analysis
250
13.1.8
Random processes and Monte-Carlo simulations
13.1.9
Optimisation and operational research
252
13.1.10
Display of complex shapes
253
13.2
Physics
253
13.2.1
Orbit calculations
255
13.2.2
Particle trajectories and plasma physics
255
13.2.3
Optics
258
13.2.4
Heat-transfer
258
13.2.5
Fallout prediction
262
13.2.6
Semiconductor research
262
13.2.7
Ferromagnetic films
264
264
13.3
Chemistry
13.3.1
Reaction kinetics
264
13.3.2
Quantum chemistry
265
13.4
Mechanics and engineering
266
13.4.1
Vibrations
267
13.4.2
Shock absorbers
268
13.4.3
Earthquake simulation
268
13.4.4
Rotating systems and gears
269
13.4.5
Compressors
270
13.4.6
Crank mechanisms and linkages
271
13.4.7
Non-destructive testing
272
13.4.8
Ductile deformation
272
13.4.9
Pneumatic and hydraulic systems
273
13.4.10
Control of machine tools
276
234
251
Contents
13.4.11
13.5
13.6
13.6.1
13.6.2
13.6.3
13.6.4
13.6.5
13.7
13.7.1
13.7.2
13.7.3
13.7.4
13.7.5
13.7.6
13.7.7
13.7.8
13.7.9
13.8
13.8.1
13.8.2
13.8.3
13.9
13.10
13.10.1
13.10.2
13.10.3
13.10.4
13.10.5
13.10.6
13.10.7
13.10.8
13.11
13.11.1
13.11.2
13.11.3
13.11.4
13.11.5
13.12
13.12.1
13.12.2
13.12.3
277
Servo systems
Colour matching
278
Nuclear technology
278
Research
279
Reactor/neutron kinetics
280
Training
281
Control
282
Enrichment
283
Biology and medicine
284
Ecosystems
284
Metabolism research
285
Cardiovascular systems
285
Closed loop control studies
286
Neurophysiology
286
Epidemiology
288
Aerospace medicine
289
Locomotor systems
289
Dosimetry
290
Geology and marine science
290
Oil and gas reservoirs
290
Seismology
291
Ray tracing
292
Economics
292
Power engineering
295
Generators
295
Transformers
296
Power inverters and rectifiers
296
Transmission lines
297
297
Frequency control
Power grid simulation
298
Power station simulation
300
Dispatch computers
301
Electronics and telecommunications
301
Circuit simulation
301
Frequency response
303
Filter design
304
Modulators and demodulators
304
Antenna and radar systems
305
Automation
305
Data processing
306
Correlation analysis
306
Closed loop control and servo systems
307
XV
XVI
Contents
13.12.4
13.12.5
13.13
13.13.1
307
Sampling systems
Embedded systems
308
Process engineering
308
Mixing tanks, heat exchangers, evaporators, and distillation
columns
309
Adaptive control
311
Parameter determination and optimisation
311
Plant startup simulation
312
Transport systems
313
Automotive engineering
313
Railway vehicles
317
Hovercrafts and Maglevs
317
Nautics
318
Aeronautical engineering
320
Landing gears
321
Aircraft arresting gear systems
323
Jet engines
323
Helicopters
323
Flutter simulations
324
Flight simulation
325
Airborne simulators
334
Guidance and control
336
Miscellaneous
338
Rocketry
338
Rocket motor simulation
338
Rocket simulation
339
Real-time data analysis
343
Spacecraft manoeuvres
343
345
Mercury, Gemini, and Apollo
Military applications
347
Education
348
Arts, entertainment, and music
349
Arts
349
Entertainment
352
Music
354
Analog computer centers
354
13.13.2
13.13.3
13.13.4
13.14
13.14.1
13.14.2
13.14.3
13.14.4
13.15
13.15.1
13.15.2
13.15.3
13.15.4
13.15.5
13.15.6
13.15.7
13.15.8
13.15.9
13.16
13.16.1
13.16.2
13.16.3
13.16.4
13.16.5
13.17
13.18
13.19
13.19.1
13.19.2
13.19.3
13.20
14
14.1
14.2
14.3
Future and opportunities
Challenges
359
Applications
361
Recent work
362
357
Contents
Bibliography
Index
427
365
XVII
“An analog computer is a thing of beauty and a joy forever.”1
1 John H. McLeod, Suzette McLeod, “The Simulation Council Newsletter”, in Instruments
and Automation, Vol. 31, March 1958, p. 488.