A Small Differential Analyzer with Ball Carriage Integrators and Selsyn CouplingR. E. Meyerott and G. Breit Citation: Review of Scientific Instruments 20, 874 (1949); doi: 10.1063/1.1741419 View online: http://dx.doi.org/10.1063/1.1741419 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/20/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Phase integral approximation for coupled ordinary differential equations of the Schrödinger type J. Math. Phys. 49, 053523 (2008); 10.1063/1.2919888 A small electrostatic retarding field energy analyzer with compensating differentiation circuit J. Appl. Phys. 51, 1431 (1980); 10.1063/1.327841 An Electronic Differential Analyzer Am. J. Phys. 21, 53 (1953); 10.1119/1.1933343 A Differential Analyzer for the Schrödinger Equation Rev. Sci. Instrum. 21, 411 (1950); 10.1063/1.1745603 An Electronic Differential Analyzer J. Appl. Phys. 19, 148 (1948); 10.1063/1.1698382

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THE REVIEW OF SCIENTIFIC INSTRUMENTS VOLUME 20, NUMBER 12 DECEMBER, 1949

A Small Differential Analyzer with Ball Carriage Integrators and Selsyn Coupling

R. E. MEYEROTT AND G. BREIT

Yale University,* New Haven, Connecticut

(Received June 3, 1949)

A small scale differential analyzer is described which uses ball carriage integrators instead of the usual Kelvin disk integrators, and selsyn angle transmission instead of mechanical coupling of the rotating parts. The machine is inexpensive, easily constructed and is moderately accurate.

I. INTRODUCTION

T HE practical value of a small scale differential analyzer was discovered by D. R. Hartree and A.

Porterl when a model originally intended to be an aid in the design of a larger machine was assembled and tested. It worked so well that it was kept in operation. Since that time, several other model analyzers have been built following essentially the design of the earlier larger machines.t. 2- 4 The construction of all these small

scale machines has been light and subject to vibration but worked well as long as no more than four or five integrators were coupled together. Their over-all ac­curacy has been stated to be around 2 percent.

The machine built in this laboratory differs from those previously built mainly in two ways: (a) the integrators are of the ball carriage type instead of the Kelvin disk type; (b) electrical transmission by means of selsyns replaces mechanical coupling of rotating

FIG. 1. Photograph of the differential analyzer. The independent variable shaft shown in Fig. 2 is below the table. The meter scale indicates relative dimensions.

* Assisted by Joint Program of the ONR and the AEC. 1 D. R. Hartree and A. Porter, Mem. Proc. Man. Lit. Phil. Soc. 79, 5 (1935). t For a more general discussion of differential analyzers see reference 4. 2 Massey, Wylie, Buckingham, and Sullivan, Proc. Roy. Irish Acad. 45, 1 (1938). 3 Lennard-Jones, Wilkes, and ·Bratt, Phil. Soc. Proc. 35, 485 (1939). 4. V. Bush and S. H. Caldwell, J. Franklin Inst. 240, 255 (1945).

874

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SMALL DIFFERENTIAL ANALYZER 875

FIG. 2. Schematic diagram of the differential analyzer. The gear train used to position the ball carriage is here shown as a screw and a half-nut. Broken lines represent selsyn signal wires.

parts which is often accomplished by means of shafts. Advantages of the ball carriage type integrators are: (1) a good output torque which makes it possible to do without torque amplification, (2) a convenient way of adjusting the ratio of rotations of roller and disk; only the ball carriage has to be moved and the integrator table assembly remains fixed. Selsyn transmission has the following convenient features: (1) economy in pre­cise shop work, (2) ease in changing connections be­tween rotating parts. The latter feature is a time saver when the machine is being coupled for a new problem.

The main object of the present note is to call atten­tion to the relative ease of construction of a differential analyzer of small or moderate accuracy which is being made possible by the availability of war surplus equip­ment in the form of ball carriage type mechanical integrators, servo mechanisms and other parts.

II. CONSTRUCTION

The machine contains three ball integrators, three function units and an adder unit. The term "function unit" is used here to denote a device capable of carrying an input function which can be coupled to other parts of the machine. It is also used to describe a device capable of recording an output function.

The function units were provided by a slight re­modeling of two discarded kimographs and by the addi­tion of a drum to a ruling engine. The adder unit is a differential geared to two selsyns. This unit is so ar­ranged that when the adder is not used, one selsyn is disconnected and the differential then serves as an

ordinary gear. This design feature makes it possible to permanently attach an adder to the output of an inte­grator and thus simplify the change-over for different equations.

A photograph and a plan of the machine are shown in Fig. 1 and Fig. 2 respectively. The connections are shown for the differential equation:

(d2y/dx2)- (dy/dx)+g(x)y=O.

Inspection of Fig. 2 shows that this equation requires the first integrator to drive the integrator table of the second. This means that sufficient torque must be available in the first integrator to drive the second integrator, five selsyns, and an adder unit. No slipping of parts was observed in ordinary operation. Inspection of Fig. 2 will show that the selsyns most likely to introduce errors are those which set the linear displacement of an integrator. Selsyns when used in the above manner, may introduce several degrees error. This effect has been eliminated by introducing an accurate gear reduc­tion of 2500 to 1 between the selsyn and the integrator.

III. PERFORMANCE

The over-all backlash was tested by setting the ma­chine to generate sine functions and noting the in­crease in amplitude per cycle. Sine functions were generated by making g(x) = constant on the input func­tion and disconnecting selsyn labeled y on the adder unit shown in Fig. 2. The increase in amplitude per cycle is about 0.2 percent which is only about 1/10 that reported for other small machines2 and compares favor-

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876 M. W. MAKOWSKI

ably with that of some more elaborate machines. 5 This error has been found to be due to the backlash of the integrator itself. To reduce this error it would be neces­sary to improve the integrator or else to provide a cor­rection for backlash. No attempt was made to do either.

In order to investigate the performance when the in­put function was not constant it was found necessary to take considerable care in plotting the function g(x). If this is done and g(x) is large over most of the range of integration so that the first integrator is kept at a reasonably large displacement, the errors are about the same as those for sine functions. When no precautions are taken the error seldom exceeds 1 percent.

IV. CONCLUSIONS

The machine has proved satisfactory and convenient for the solution of problems with a nominal accuracy of one percent as described in the preceding ,section. Ball

Ii See for example H. P. Kuehni and H. A. Peterson, Elec. Eng. 63, 221 (1944).

THE REVIEW OF SCIENTIFIC INSTRUMENTS

carriage type mechanical integrators do not require torque amplification when used with precautions de­scribed. Electric ("selsyn" or "synchro") angle trans­mission has proved reliable and convenient as a means of coupling functional units in the computing device.

ACKNOWLEDGMENTS

We wish to acknowledge the assistance of Mr. P. S. Jewett in making preliminary tests, of Mr. A. G. Pet­char of the Sloane Physics Laboratory shop in the construction of the machine and of Mr. S. Geltman in performance tests. The Applied Physics Laboratory of The Johns Hopkins University has kindly supplied three mechanical integrators. Thanks are due also to Professors A. G. Conrad and J. L. Bower of the Elec­trical Engineering Department of Yale University for the loan of several selsyns, and to Dr. D. T. Sigley of the Applied Physics Laboratory for performance char­acteristics of mechanical integrators in similar comput­ing devices.

VOLUME 20, NUMBER 12 DECEMBER, 1949

A Slide Rule for Radiation Calculations

M. W. MAKOWSKI

Polish University College, London, England (Received February 17, 1949)

A slide rule has been developed to facilitate rapid calculations based on the Planck radiation formula. Quantities such as the radiant flux density in a given wave-length region, the spectral radiant flux density at a given wave-length, or the corresponding quantities expressed in photon units, can be obtained readily for a black body over a range XT= 2X t()2 to XT=4X 106 micron degrees with an accuracy of about t percent. Simple extension rules can be used for larger values of XT.

I. INTRODUCTION

I N the course of certain radiation calculations it was found that the available tables based on the Planck

radiation formula were neither sufficiently compre­hensive nor convenient for frequent reference. The need was felt for a device that would facilitate rapid calculations without involving too great a sacrifice in accuracy. Accordingly it was decided to develop a nomogram, based on the Wien displacement law, capable of giving rapidly approximate values for the radiation quantities most frequently encountered. Typical of such quantities is the radiation per unit area emitted by a black body at a given temperature in a given wave-length interval. The general features of the nomogram are shown in Figs. 1 and 2.

As the work proceeded the advantages of a slide rule arrangement became apparent, and it was decided to extend the computations already carried out with a view to the development of a radiation slide rule. This paper gives a description of the slide rule and the

methods used in the associated calculations. The final form of the rule, the general arrangement of which can be seen in Fig. 3, includes the scales required for calcu­lations involving spectral radiant flux density and radiant flux density, together with the corresponding quantities in photon units. The over-all dimensions are 19t in.X3t in. The stock carries the scales of maximum spectral radiant flux density, HAmax.,a total radiant flux density, H, maximum spectral photon flux density, QAmax., total photon flux density, Q, temperature (centigrade and absolute), and wave-length. One side of the slide carries the scales required for calculations involving the first two of the above quantities, while the scales necessary for photon calculations are on the

& The symbols used throughout this paper have the usual significance. The subscripts A, AX, II, and A ... , refer to wave-length, wave-length interval, wave number, and wave number interval respectively. The symbol em is used to denote unit of length referring to the surface of a black body. The values of the con­stants used follow Birge (Ct=2".hc2=3.7403XIQ-12 watt cm2 ;

ct'=2".c= 1.88355 X 1011 em sec.-I; C2= Itc/k = 1.43848 em deg. A).

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