Analog Machines Research Articles

Analog computer synthesis and error matrices

AN ANALOG COMPUTER or differential analyzer programmed for a system of linear differential equations ideally exhibits solutions of the mathematical problem. Such performance necessarily can only be approximated, inasmuch as the networks in an electronic analog machine are far more complex than the mathematical problem equations to be solved imply.

Numerical integration of differential equations on the 704 EDPM

To provide a convenient means for solving complex and nonlinear ordinary differential equations, a system has been devised which causes a 704 EDPM to behave like an analogue differential analyzer. Use of this system involves the interconnection of appropriate subroutines in a manner analogous to the interconnection of operational amplifiers in an analogue machine. The subroutines are designed so that the user may compile the required connections automatically, without much knowledge of digital computer coding. Solving differential equations by this method takes no more setup time than that required by analogue machines and in many cases may be a good deal faster. Integration is done by using Milne's formulas. The system automatically adjusts the integration interval to take advantage of portions of the solution which can be computed with large intervals at no sacrifice of accuracy. Nonlinear functions may be expressed by a table of values in storage.

Matrices in analogue mathematical machines

This paper initiates a rational account of the methods of solution of linear differential equations on electronic analogue machines from the matrix point of view. The 1-to-1 reciprocal correspondence between the mathematical problem matrices and the electronic synthesis networks is established. The systematic matrix viewpoint bearing on both the mathematical and physical systems yields theoretical and practical advantages, most of which remain to be exploited.

Electrical Analog Computing Machine for Solving Linear Equations and Related Problems

A new iteration method of solving linear equations is described. The convergence of this method depends on a single scalar parameter which can be predetermined. A brief description of an electrical analog machine for solving linear equations with ten variables based on this iteration method is given. In addition to the solution the machine gives the largest eigenvalue of the matrix of the linear system.

The physical and computing significance of an electrical analogue of creep and recovery

The instrument described is basically a simple non-inductive transmission line, made up of identical sections, with the capacitance divided into recoverable and irrecoverable elements, in series, by the use of rectifiers. A voltage programme can be set up and applied automatically to simulate the stressing, and the charge taken by the network, with time, is integrated and recorded as the creep strain analogue. The simple power-type creep curve exhibited by solids (in particular the cubic type) is accurately reproduced. The time-functions which relate the character of the creep in metals with the times of stress and off-stress are very similar to those obtained with the analogue. Creep recovery, permanent flow and creep under changed stress (including reversed stress) can all be satisfactorily handled. The character of transient creep and recovery does not, therefore, demand special models involving arbitrary distributions of relaxation times. Problems must be set up on the machine using data obtained from short-term tests: calculations of prolonged intermittent creep can then be quickly completed. No account can at present be taken of temperature changes. On this basis, a comprehensive analogue creep computer, of a much higher operating speed, appears possible. This would be complex and expensive, but no more so than many analogue machines now in use for other purposes, and the cost and delay of extensive long-term testing would be much reduced.

Some comparisons between analogue and digital computers

The paper discusses the reasons for using computers in industry and the considerations which should apply to the choice of machine for different types of work. Examples are given of both analogue and digital machines and the uses to which they can be put. The advantages and disadvantages of each type are mentioned, with the aim of encouraging a rational approach to the specification and design of computing devices. A representative large-scale machine of each class is described briefly.

Analog Computer for the Roots of Algebraic Equations

An electronic analog machine for computing the roots of algebraic equations of degrees through the eighth is described. The coefficients of the equation may be complex. The roots are indicated as points in the complex number plane light points on the screen of a cathode-ray tube. Once per second the roots are displayed, and when some of the coefficients are varied the corresponding curves described by the roots are made visible by the persistence of light emission from the screen. An accurate determination of the roots is obtained by means of a comparison-root which can be moved all over the number plane and thus be brought to coincide with a root. The comparison-root is displayed by a scanning system, which also controls the positions of the roots, and so it is possible to obtain an accurate reading not influenced by distortion in the cathode-ray tube. By means of a normalizing device, the voltage level in the machine is kept high throughout the computation, and so all the roots are determined with about the same percental accuracy. The modulus of an error will normally be of the magnitude of 1 per cent of the corresponding root-modulus.

An Analogue Multiplier

IN practically all analogue computer work there seems to arise the problem of multiplying two time-varying voltages. There are quite a number of methods of performing this multiplication and some of them fulfil the requirements of general analogue machines: (1) preservation of sign; (2) frequency response at least as good as other components in the machine; (3) voltage-levels high and output impedance low; (4) error having a maximum of 0.1–1.0 per cent.

A Machine for Synthesizing Two-Dimensional Fourier Series in the Determination of Crystal Structures

An electromechanical analog machine is described which performs a two-dimensional Fourier synthesis for use primarily in x-ray crystal structure calculations, ρ(xy)=Σ lim hΣ lim kFhkocos2π(hx+ky).The analog to a term in this summation is obtained through the use of a pair of selsyns as a synchro-variable transformer somewhat similar to its use in servomechanisms. ρ(xy)hk is represented by the output voltage, Fhko by the input voltage, and 2πhx and 2πky are the angles of rotation of the primary and secondary rotors respectively. A system of crossed shaft and cascading gear trains turns the rotors to the appropriate angles for each selection of point (x, y) in the unit cell. The present machine handles h and k values up through 8, but provision is made for future extension through 16. Computations are shown for two known structures to indicate the results obtainable with the machine.

The Differential Analyser Computing Mechanisms and Linkages

THE growing importance of numerical methods in scientific research is reflected in the beginnings of a separate literature. These two books are concerned not with methods, but with tools, and are welcome additions to the small number of books dealing with this field. They are both concerned with what have been called analogue machines, and not with the digital type of calculator which is now exciting a good deal of attention. Although they are both concerned with analogue devices, the scope of the two books is very different. Mr. Crank describes the differential analyser, which is used for the solution of differential equations, and is purposely designed to cover a wide variety of problems, and Prof. Svoboda is concerned with the design of simple mechanisms to represent specific functions. The Differential Analyser J. Crank. Pp. viii + 137 + 4 plates. (London, New York and Toronto: Longmans, Green and Co., Ltd., 1947.) 10s. 6d. net. Computing Mechanisms and Linkages Antonin Svoboda Hubert M. James. (Massachusetts Institute of Technology: Radiation Laboratory Series, No. 27.) Pp. xii + 359. (New York and London: McGraw-Hill Book Co., Inc., 1948.) 27s.