US2398238A - Integrating apparatus - Google Patents

Description

April 9, 1946. M. M NATT 2,398,238

INTEGRATING APPARATUS Filed May 1942 1 l I 7 .22 2| n 23 ZLF 'SOALER SHARPENER IN VENTOR.

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ATTORNEY Patented Apr. 9, 1946 INTEGRATING APPARATUS Eugene M. McNatt, Tulsa, :11., assig'nor to Standard Oil Development Company, a corporation of Delaware Application May 9, 1942, Serial No. 442,351

4 Claims.

The present invention is directed to a method and apparatus for integrating mathematical functions.

In many physical and mathematical problems it is necessary to perform repeatedly integrations of the form white-area wherein p(a:) and q(x) may be known only as curves and 11:0 or a constant. Such integration are commonly carried out arithmetically by plotting the two curves, measuring the value of each function at equally-spaced values of x, mul. tiplylng the values of p(:ry) and q(a:) for each measured value of m, and adding the resulting products. Since this process is laborious and time consuming, various machines have been devised for performing this operation mechanically or electrically. A representative type of such machines is. described in The Cinema Integraph, Hazen, Brown and Hedeman, Journal of the Franklin Institute, July 1940, in which may also be found illustrations of the type of integrals, to the solution of which the present invention is directed.

The principal object of the present invention is the provision of a relatively simple method and apparatus for performing integrations of the character referred to with any desired degree of accuracy. Briefly, the method involves the creation of light flashes, the number of which is 'a function of the curves represented by p(a:y)

and q(:c) and, more specifically, a function of the product of these two curves.

The present invention may be more clearly understood from the following detailed description of the accompanying drawing in which Fig. 1 is a plan view of one embodiment of the present invention; I

Fig. 2 is a detail thereof;

Fig. 3 is a detailed view of a light mask used in the arrangement shown in Fig. 1; and

Fig. 4 is a similar view of a different light mask.

Briefly, the arrangement shown in Fig. 1 consists of a pair of revolving masks interposed between a light source and a photocell or other device capable of registering lighte flashes. One of the masks has drawn on it a curve represented by the equation z=p(a:) and the other has drawn on it a curved represented by the equation z=q(.'c). Both masks have the :z: axis parallel to the axis of rotation and their axes of rotation are parallel to each other and perpendicular to the beam of light. Each mask is so constructed that for each revolution it will pass a number of flashes of light which will be a function of the value of z for the value of a: at the point on the mask where the light beam strikes. The R. P. M. of one mask is made a large multiple of the R. P. M. of the other mask, and upon the size of this multiple will partially depend the accuracy of the integrations.

Referring to Fig. 1 in detail, C1 and C2 designate two hollow, transparent cylinders supported by bearings I, 2, 3 and 4. These bearings are the conventional ring type ball bearings. It will be observed that the driving shafts 6 and I, respectively, for these cylnder are journaled'in bearings I and 2, while the outer surface of the cylinders themselves are journaled in bearings 3 and 4'. This-is necessary because the outer ends of the cylinders are. open. The bearings are suitable mounted on a base plat 5. Shaft 6 is driven by a suitable motor, and is geared to shaft 1 in the manner shown. It will be apparent that any gear ratio may be employed in this transmission.

In the open ends of the cylinders, tubes 8 and 9 are mounted so as to have their axes coincident with the axes of rotation of cylinder C2 and C2, respectively. These tubes are rigidly mounted in a block l0 which is arranged to slide on spaced tracks ll attached to the base plate 5. The block Ill carries a screw threaded rider l2 which is in threaded engagement with a screw l3, one

end of which is journaled in a, bearing l4 and the other end of which terminates in a gear in the gear box in mesh with a driving gear fixed to the shaft 1. The pitch of the screw l3 can be arbitrarily selected, but th smaller it is made the greater will be the accuracy of the integration.

Fixed in the tube 8 is a suitable light source l5, such as an automobile lamp bulb, and a mirror l6 adapted to receive a beam of light from source l through a pin hole I! and reflect it through a pin hole l8 in the wall of tube 8 in a line perpendicular with the axes of rotation of cylinders C1 and C: and in the plane thereof. Tube 9 is provided with a hole l9 to pass said beam of light. Fixed in tube 9 in register with said hole is a light sensitive device 20, such as a photocell, which in the embodiment shown i connected electrically to an amplifying and sharpening cir cult 2|. A suitable sharpening circuit for this purpose is that describedby Huntoon and Strohmeyer in an article in the Review of Scientific 65 Instruments, 12, 35 (1941) entitled A hard vacuum tube pulse equalizing sharpening circuit. The purpose of this circuit is to convert the pulses of electricity generated by the photocell 20 into a suitable form to operate a scaling circuit 22 to which they are delivered. Connected to the scaling circuit is a counter 23 which displays a numercial summation of the pulses delivered to it.

As shown in Fig. 2. the surface of the transparent cylinder C1 carries a series of opaque bands 24 parallel to each other and to the axis of the cylinder. In general these bands will be very narrow and so spaced that the transparent and opaque strips are of equal width. A preferred method of producing the stripes is to form them on transparent material, such as Cellophane, with India ink and then fasten the transparent sheet on the cylinder C1 by means of glue of transparent type. It will be observed that only cylinder C1 is so striped.

In Figs. 3 and 4 are shown masks which may be mounted on cylinders C1 and C2. For the practice of the invention it is immaterial which mask is mounted on which cylinder. In Fig. 3, for example, the mask has a transparent portion 25, the upper boundary of which follows a curve designated by the equation z=p(:c) in which 2 represents the ordinate and a: represents the abscissa. The remaining portion 28 of the mask is opaque. This mask is wound on the cylinder with its axis parallel to the axis of rotation of the cylinder.

In Fig. 4 the mask illustrated has a transparent portion 21. The upper boundar is a curve represented by the equation z=q(a:). ing portion 28 of this mask is opaque. It will be obvious that the portions 25 and 21 of the respective masks can be cut out entirely. The masks are preferably made by tracing the curves on transparent sheets and then blackening the portions 26 and 28.

The masks are of such dimensions that the maximum value of z is equal to the circumference of the corresponding cylinder, and th maximum value of a: is not greater than the length of the corresponding cylinder. As has been previ ously indicated, either mask can be placed on either cylinder, but for the purpose of discussion it will be assumed that the mask of Fig. 3 is placed on cylinder C1 and the mask of Fig. 4 is placed on cylinder C2.

It should now be clear that, as the cylinders rotate, the beam of light passing through C1 and C: will be chopped into flashes by the stripes on C1 and that this flashing beam will be interrupted at intervals both by C1 and C: as the opaque portions of the masks intercept the beam. The resultant pulsating signal from the photocell is amplified and the pulses sharpened as previously described by circuit 2! and counted.

To operate the device the masks are placed on the cylinders and the tubes 8 and 9 are positioned so that the beam of light passes through x=-u and z=0 on the mask on C1, and a:=z=0 on the mask on C2. Then, with the amplifier and scaler ready to function, the reading of the electric counter is noted or is set at zero. Then the motor is started and allowed to run until the screw it has advanced the beam of light to the point corresponding to z=t--u on the mask on cylinder Cl and :c=t on the mask on C: where t is the upper limit of the integral, At this point the electrical counter is disconnected from the the scaler, the motor is turned on, and the reading of the counter is noted.

The manner in which the machine performs The remainasoaass the desired multiplication and summation will now be considered. Assume that the apparatus has been started with the ray of light passing through the point :z:=9 and z=0 on C1 and :c=0=z on Ca. Let the total number 01 clear strips on Ci be- N (disregarding the mask) and let the circumference of C1 be D1 and the circumi'erence of C: be D2. Let the gear ratios be such that C1 rotates n revolutions to one revolution of C2, and let the pitch of screw it be such that for one revolution of C: the beam of light advances a distance S=c.A:z: in the positive a: direction, where A: is the increment in a: corresponding to Sv and c is the scale; that is, c is the ratio of the length of a given increment in the :c direction to its value in the units in which a: is expressed. For purposes of clarity, it will be assumed in the following discussion that put-11) and q(:::) remain constant over any given increment, As. The maximum error .introduced by this approximation will clearly be proportional to A2.

During the first revolution of C1 the number of flashes of light, n, passing through C1 is proportional to where a is the constant ratio 01' the height of p(:r1l) as measured on C1 from the a: axis to the numerical value of p(a:1/). In other words, it is a constant which is a characteristic of the mask itself, and represents the scale to which the curve on the mask is plotted.

During the different revolutions of C1, C2 rotates Dz/n times in the positive 2 direction and the light beam advances a distance 8/11 in the positive 1: direction.

Assuming that b times q(0) is greater than Da/n, where b is a constant for C: corresponding to a for C1, 1 flashes of light will also pass through C: and actuate the photocell and be registered by the sealer and counter. Furthermore, 1 flashes of light will continue to be registered per revolution of C1 until C: has rotated through the distance b times 11(0). The number of revolutions, m1, of C1 corresponding to this fraction of a revolution of C: is the number of revolutions of C1 per revolution of C: times the ratio of b times qua) to the total circumference of C2. That is During the remaining part of the revolution of where numerator and denominator have been multiplied by S=c.A:c. Since during subsequent revolutions of C: the above process will be repeated, we may write where the constants have been lumped into K, and where the subscript refers to the i revolution of C2. The grand total number of flashes asaaasa of light registered as the light beam advances from a:= to r=t will then be r r' EFF right-mam i t I ginrr}; pe-wqw x where R. is the difference between the final and initial counter readings and d is the scaling factor. The error in the results will evidently be at least roughly proportional to Ax/nN, and it is consequently desirable to make this quantity as small as practicable. It is for this reason that a scaling circuit is used, for by this means a much larger value of N can be used than would otherwise be possible to register and still operate the machine at a reasonably fast rate. Examples of practical values of these constants for moderate accuracy are Az=1 mm., n=N= 100.

In the foregoing discussion, reference has been made to an equation containing the values put-y) and q(:z:) These values have been represented on the masks shown in Figs. 3 and 4 as being curves of a certain shape. It will be obvious that the curve may assume any shape, depending upon the specific functions of a: represented by these values.

Other methods for generating pulses of light of a number which will be a function of the product of the values of p(:r-y) and q(a:) and will represent the summation of these products for different values of a: between zero and it than that specifically disclosed will occur to those skilled in the art. One obvious modification is to omit the stripes from cylinder C1 and generate the initial light flashes by inserting a shutter in the form of a toothed wheel between the.-

light source i5 and the mirror It in the manner disclosed in my copending application Serial No. 405,910. In this case, both the shutter and the light source will be arranged outside the tube 8 so that means for driving the shutter at a speed which bears a fixed ratio to the R. P. M.s of the cylinders C1 and C2 can be utilized.

As has been pointed out in the foregoing description, the accuracy of the device can be increased by increasing the gear ratio between Cr and C2, decreasing the width of 'the stripes on C1, and decreasing the pitch of the thread on screw I3. It can be seen, therefore, that the machine is capable of considerable adjustment, depending upon the type of problem to be dealt with and the degree of accuracy desired.

In the above-description it has been assumed that p(a:-y)' and q(:r) are both always positive. In the event that this is not true, separate masks may be made for the positive and negative portions of the two functions. then be made up of four parts:

(a) the positive products corresponding to the positive portions of each function;

(b) the negative products or the negative por- The integral will tion of the first times the positive. portion of the second;

(c) the negative products of the positive portion of the first times the negative portion of the second; and

(d) the positive products corresponding to the negative portion or the first times the negative portion of the second.

It is clear that these separate parts of the integral may be obtained by four separate integrations, and the total integral is then determined by the algebraic sum.

The nature and objects of the present invention having been thus described and illustrated, what is claimed as new and useful and is desired to be secured by Letters Patent is:

1. An apparatus for integrating an equatiorr of the type where y is 0 or a constant and (a) and Ike) can be represented by curves, comprising a light source, a light receiver spaced from said source and in alignment therewith, a plurality of rotative screens interposed between said light source and said receiver, markings on said screens to cause the light of said source to be transmitted to said receiver in a number of flashes, the numintegrating an equation where 11 is 0 or a constant and p(a:) and qua) can be represented by curves comprising a cylinder having a transparent portion and an opaque portion, a light source arranged along y the axis of said cylinder adapted to cast a beam of light normal to the wall thereof, a second cylinder arranged with its axis parallel to the axis of the first cylinder and having an opaque portion and a transparent portion, the transparent portion of one of said cylinders being an area defined by the curve z=p(.'c) and the v transparent portion of the other being an area defined by the curve z=q(a:), the :c axis in each case being parallel with the axis of the cylinder, a photocell arranged on the axis of said second cylinder in a position to receive a light beam from said source of light, means for rotating both cylinders while maintaining a selected gear ratio between them, means for chopping said light into a series or flashes whereby said photocell generates periodically pulses of current, and means for counting said pulses or current.

4. An apparatus according to claim 3 in which the means for chopping the light beam into flashes constitutes spaced opaque longitudinal stripes on one of said cylinders. I

EUGENEM. MoNA'I'I.

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