US2716522A - Calculator for computing volumes of revolution - Google Patents

Description

Aug. 30, 1955 M. D. BRAlD CALCULATOR FOR COMPUTING VOLUMES OF REVOLUTION Filed Oct. 9, 1955 5 Sheets-Sheet l liizranlsir Murray D. Braid M. D. BRAID Aug. 30, 1955 CALCULATOR FOR COMPUTING VOLUMES OF REVOLUTION 5 Sheets-Sheet 2 IFZFEWZET Murray .2). Braid MMWW M HZLZHE Flled Oct 9 1953 Aug. 30, 1955 M. D. BRAID 2,716,522

CALCULATOR FOR COMPUTING VOLUMES OF REVOLUTION Filed Oct. 9, 1953 5 Sheets-Sheet 3 0, 1955 M. D. BRAID 2,716,522

CALCULATOR FOR COMPUTING VOLUMES OF REVOLUTION Filed 001;. 9, 1955 5 Sheets-Sheet 4 W Q 7 1% w 4 Y EYE 272277 Murray .D. .Bmz'a M MM-M M 9 W b H H ZZ 175 I Aug. 30, 1955 M. D. BRAID 2,716,522

CALCULATOR FOR COMPUTING VOLUMES OF REVOLUTION 157.273 17ZUT Murray .23. Bra/I0 wva MM W Z:/7 7 HZZL LE United States Patent 0 CALCULATOR FOR COMPUTING VOLUMES OF REVOLUTION Murray D. Braid, Mentor, Ohio, assiguor to Thompson Products, Inc., Cleveland, Ohio, a corporation of Ohio Application October 9, 1953, Serial No. 385,211

10 Claims. (Cl. 235-61) The present invention relates to calculating machines, and more particularly relates to calculating machines for use in drafting rooms as an aid in determining volumes of revolution.

In modern industry the necessity of determining intricate volumes of revolution, or volumes of revolution of intricate parts, arising constantly. For example, in the manufacture of forged metal parts such as automotive engine valves, valve tappets and other similar metal parts having a symmetrical configuration, it is desired that the amount of metal in the finished part be known. When the finished volume is known, the initial billet or blank from which the part is to be forged may be cut to the proper size so that flash and waste material may be substantially eliminated.

The desirability of the prevention of such waste has long been understood in industry and the relatively tedious methods of approximation heretofore utilized in determining volumes, of revolution have been followed in the design and drafting rooms in order to permit the elimination to the greatest possible extent of waste material in the manufacturing processes. While, of course, simple volumes of revolution such as for example those of cylinders, cones and spheres may readily be determined without diliiculty, in most instances the configuration of the part is made up of curves, angles and other irregularities which make it substantially impossible to calculate the volume accurately in a reasonable period of time.

An object of the present invention is, therefore, to provide an easily manipulated apparatus of convenient size which will automatically determine the volume of revolution of any curve that can be drawn on flat paper.

A further object of the invention is to provide an extremely compact calculating apparatus capable of use on ordinary drafting boards for the determination of volumes of revolution.

Still a further object of the present invention is to provide an extremely compact calculator capable of automatically solving the equation for predetermined range of values of x and r and in which r is non-uniformly variable.

And yet another object of the present invention is to provide a dual integrating apparatus capable of determining the volumes of revolution of a predetermined curve by the manual manipulation of a portion of the apparatus to trace the curve.

A further object of the present invention is to provide a calculating apparatus capable of determining the volume of revolution of any curve by an operator having no knowledge whatever of the mathematics involved.

A further object of the present invention is to provide a calculating machine constructed to move at a uniform rate along a carrier positionable in adjusted relation to 2,716,522 Patented Aug. 30, 1955 a coordinate of a drawn curve and having an indicator adjustable to follow the curve in the direction of the other coordinate whereby tracing the curve by said pointer will automatically determine the volume of revolution of the curve about the first coordinate.

Still other and further objects of the present invention will, of course, become apparent to those skilled in the art from a consideration of the attached sheets of drawings in which a preferred embodiment of the present invention is shown by way of illustration only.

On the drawings:

Figure 1, is a perspective view of the calculator constructed according to the present invention.

Figure 2, is a plan view of the body of the apparatus, with the cover removed to disclose the internal structure thereof.

Figure 3, is a front elevation view in detail shown in partial cross section.

Figure 4, is a sectional view taken along the line IV- IV of Figure 2 showing the gearing connections of the first integrator in detail.

Figure 5, is sectional view in detail taken along the line V-V of Figure 2.

Figure 6, is a schematic showing of the basic interrelationship between the various components of the present invention.

Figure 7 is a gearing schematic of the apparatus, further illustrating its construction; and

Figure 8 is a diagrammatic drawing of the mechanical integrator units utilized in the present invention.

As shown on the drawings:

The problem faced in the present instance, and solved by the computer of the present invention, is the value of the volume of revolution of a curve such as the curve 8 in Figure 1, about an axis of revolution denoted X. Mathematically, utilizing the symbols: r=radius of a particular point, or its distance away from the axis of revolution x, 2:: distance along the axis x-x, and dr and dx equal small increments in the respective values of r and x respectively, the volume of revolution V may mathematically be stated as follows:

V=21rf f rdrdx In solving the above double integral equation the present invention contemplates the utilization of two ball and disk type mechanical integrators. Such an integrator is shown in diagrammatic form in Figure 8 and although its operation is generally known in the art, the fundamental theory thereof Will hereinafter be reviewed. Assuming the shaft 0 is arranged to drive a second cylindrical shaft through a gear ratio N, then for a rotation d" of the driving shaft 0, the rotation of the driven shaft will be Ndli. When the gear ratio changes during the driving action of the shaft 0, the total rotation of shaft 1: equals the integral of Ndfi.

In the use of an integrator of the ball and disk type the gear ratio N equals the distance U of the transmitting ball B from the axis of rotation of the disc divided by.

R, the radius of the driven shaft cylinder Since the value R is a constant it will thus be seen that for a rotation of 110 with a changing gear ratio, the total rotation of the cylinder will be 1 fUde Since, in mechanical integrators, it is desirable to pro vide a change in the position of the ball B by means of a lead screw or a similar threaded apparatus which provides axial movement by rotation, the factor U may be expressed in terms of rotation of a shaft. Thus U can be said to equal where P is the pitch of the screw measured in terms of turns per inch of displacement along a shaftv which is turning in accordance with the variable of 1ntegration.

Substituting in the equation Ude Utilizing this with an integrating apparatus in actual practice, 1 preferably provide a mechanical integrator we therefore have constructed to provide two complete revolutions of the disk 0 per inch of travel or distance r as viewed in Figure 1 thus making r directly proportional to 0. Movement of the pointer P .is thus translated into two revolutions of the disk per inch of r. In the calculator of the present invention, which is solving for the integral of rdr, r is the 3 variable of integration and is also the input into the first disk 0. Therefore, for the first integrator v equals 01 which is in turn directly proportional to r. In this connection it is noted that hereinafter the subscript 1 indicates that the component is for the first integrator of the series and subscript 2 indicates that component under consideration applies to the second integrator of the series.

In view of the above relationships,

1 4 1 EFT! f l l or, integrating,

Since as was calculated above,

, hi 21mm then 2 wRlPlPzRzl The exact solution of this equation is dependent upon the limitations imposed by the calculator range requirements. The limits of the values r and x in the equation V=21rffr drdx are governed by the size of object to be measured. For the sake of the purposes for which the computer of the present invention was constructed, it was decided that the maximum value of x=3 inches and the maximum value of r=2 inches were satisfactory. However, in order to make the computer easier to handle, the gearing components of a computer constructed according to the present invention were designed to produce a reading at the output shaft 92 equal to cubic inches X100 when the pointer traces a layout of the curve which is five times its regular size. Thus, the calculator mentioned will operate over a range of x equals 15 inches and 1' equals 10 inches maximum. While this calculator has proven very satisfactory, it is noted, of course, that the actual maximum dimensions can be varied in constructing the apparatus.

Since, as was noted above, in the present instance the input to the first disk, or 61 is directly proportional to r, or, in other words 61:10, and since the values of RiliRzPz are all constant, it is therefore clear that where dti'z is the angular input to the second integrator disc and is directly correlated in a constant manner to movement of the computer in the xx direction as viewed in Figure 1. On these bases it is clear that the dual integrators may be arranged through appropriate constant gearing to provide the solution to the integral, as set out above.

While variations may of course be utilized in constructing the gear trains correlating the integrators with the 1' pointer P and with the movement of the entire mechanism in the xx direction, for purposes of illustration a sample gear train is hereinafter set forth.

A commercially available integrator of the type above discussed and having a ball carriage travel of .75 inch and an output factor of 1 may be used. Since, for the sake of convenience, both of the integrators utilize the same structural components, the values of R1 and R2 are equal. Further, the lead screws utilized for modifying the position of the ball carriage are also equal and are chosen to be 32 threads per inch. Under these circumstances, the equation will equal, with substitutions,

In order to obtain a direct relationship between (162 and x, in terms of x, it is necessary to apply the computer range limits to the above equations. Since, from Q 2R P 20 above, and since the output 1 of the first integrator equals rdr 2 then the equation V=21rjfrdrdx Due to the chosen ratio of 2:1 between inches of r and turns of the input shaft 61, r equals its maximum of "10 inches then 61 equals 20 turns. Therefore, at the limit position the equation 2 t e 200 f i z becomes 400 d d0 2d0 Since the rotation of the output shaft is to equal the volume of revolution V or a multiple thereof, and since the rotation of the output shaft of the second integrator equals 02, then Applying the limit 0:20, and inserting the value of d2 equals 21102, we have 112 equals 2d02=801rdx, or, in other words we find that the rotational input 1102 to the disc of the second integrator should bear a relationship of 401r turns per inch of travel in the x direction.

Substituting this value in the previous equation of When 0:20, or the maximum of the assumed sample conditions adopted, then 2=801rdx.

From the above equations it will be apparent that the actual gearing values of the various elements in the mechanical integrator may be as follows, referring to the schematic showing of Figure 6: The input shaft 10 l to the first integrator will be geared by a two to 1 ratio to the pointer 11 so that one inch of movement of pointer 11, measuring 1', will cause 2 turns of the shaft 10. Rotation of the shaft 10, which rotation is termed 0, is fed directly to the input shaft 12 of the disk 13 of the first integrator generally indicated at 1. As will be recalled from the calculations above the quantity 61 is placed in one to one ratio onto the disk 13 and hence the gearing connection between the shafts 10 and 12 is a one to one direct drive. The quantity v, which indicates the movement of the ball carriage relative to the disk 01, will be recalled from the above calculations to be controlled by a lead screw having a pitch equal to 32 turns per inch. This lead screw is shown at 14 and connects the shaft 10 to the integrator 1 to provide 1 inch of movement of the integrator 1 for each 32 turns of the shaft 10.

The output shaft 15, the rotation of which has been indicated in calculations above to be will form this mechanical arrangement provide an output to a connecting shaft 16. The connecting shaft 16 provides the input v2 into the second integrator generally indicated at 2 by providing the variable of integration to the ball carriage through the lead screw 17 again having a pitch of 32 turns per inch.

The quantity 02, or the input into the disk 18 of the integrator 2, as was found above, bears a relationship with movement along the x axis of 401r turns of the shaft 19 per inch of movement of the pointer 11 in the x direction. Thus the shaft 19 is geared to a rack lying in the x direction to provide the 401r turns per inch and represents the dx quantity which will, in the mechanical integrator, be a uniform rate of movement in the x direction. The output from the second integrator, 2, is taken from the cylinder shaft 21 which represents, in the calculations above, 2. This output equals, according to the development above,

1 2 2 fthdG; OI KfT dx and is applied through suitable gearing indicated at 22 of Figure 6 to a counter shaft 23 which provides a numerical indication capable of multiplication by a constant to provide the actual volume of revolution of the curve in its exact dimensions. It is of course to be understood that the counter can be adapted to provide a direct reading requiring no multiplication by a constant, through the appropriate use of gear trains between the counter and the output shaft 21.

While of course, various gear trains can be utilized in obtaining the necessary ratios between the components, the train shown in the Figure 7 gearing schematic diagram has proven very satisfactory in the example calculator. As may there be seen, the pointer 11 is provided with rack teeth 25 which engage a gear 26 fixedly secured to the shaft 27. The gear 26 is provided with 20 teeth and the gear 28 secured to the opposite end of the shaft 27 is provided with 40 teeth. Gear 28 in turn is enmeshed with the 14 tooth gear 29. The shaft 30 which carries the gear 29, carries also a gear 31 which drives the shaft 10A through the gear 32 having 32 teeth. The rack or pointer 11 has teeth out on a diametral pitch of 32, as do all of the remaining gears in the above gear train. Thus, the rack has teeth per inch of its length. On this basis it is clear that movement of the rack 1 inch will provide 2 turns.

I in the gearing schematic Figures 6 and 7 wherein the output 1 is fed through a screw connection 17 having 32 threads per inch into the ball carriage of integrator 2. The quantity dx which is a uniform rate of change along the axis x in the present example, is provided by means of a constant speed motor 35. According to the above calculations the motor is constructed to turn the shaft 20 at a rate which is equal to 401:- times the travel in inches along the x dimension.

In mechanically constructing this arrangement it is therefore desirable that the motor be utilized to provide simultaneously the power to the shaft 20 and also the power to move the entire apparatus in the x direction. This. may be accomplished through the gearing shown by way of example in the gearing schematic Figure 7. There, the motor 35 drives a pinion gear 36 having 30 teeth and which is enmeshed with a gear 37 mounted on the countershaft 38. The gear 37 is provided with 32 teeth and a second gear 39 also secured to the shaft 38 is provided with 44 teeth and meshes with the gear 40 on the shaft 20. The gear 40 is provided with 14 teeth and hence for each revolution of the motor shaft gear 36, the shaft 20 will rotate times.

Parallel gearing is provided for driving the entire carriage in the x direction along a rack designated 41 and comprises a worm 42 aflixed to the shaft 38 and in contact with a worm wheel 43. The worm wheel 43 is caused to rotate once for each 40 revolutions of the worm 42 and the gear 43 is caused to rotate l revolution for each 1 inch travel of the rack 41 relative thereto. This one revolution of gear 43 per inch of travel of the rack 41 is provided by the gear 44 fixedly secured to the gear 43, and idler gear of 45 and a rack gear of 46 having 44 teeth which in turn is directly engaged with the rack 41 by means of a pinion 47 having 42 teeth. The rack is provided with teeth per inch, having a diametral pitch of 32. It will thus be seen that the shaft 20 which turns at a rate of 1r times the rate of shaft 38 which in turn rotates at 40 times the rate of rotation of the gear 43 will necessarily rotate at a rate of 401.- revolutions per inch of travel on the rack 41. The input motor 35 may, of course, turn at any speed desired since the ratios necessary to the proper performance of the apparatus relate only to the relationship between the shaft 20 and the rack 41 rather than to the rotation of the motor shaft.

Arrangement of the gearing shown in the schematic Figure 7 into a compact mechanical apparatus may be clearly seen from a consideration of Figures 2, 4 and 5. In these views which show the calculator with the cover, 8, removed, the numbers of the various components utilized in the schematic diagrams in the Figures 6 and 7 have been utilized, although the physical arrangement of the component parts is modified to provide a very compact unit.

The carriage board or base upon which the calculating components are mounted is generally indicated at 3 and is movable in the directions of the arrow 4 on the carriage rails 5 and 6 by means of the wheels 7. The pointer P connected to the rack 11 is slidably carried in the slide blocks 9 secured to the housing base 3 and may be manually reciprocated in the guides as the pointer P is guided over the surface of the curve being traced. This reciprocating movement transversely of the rails 5 and 6, of the rack 11, constitutes the variable input to the calculator in the r direction and is fed into the integrator 1 at the shafts 12 and 14 which comprise the shaft connected to the integrator disk and the ball carriage respectively. The connections between the rack and the shafts 12 and 14 comprise the gearing shown in the schematic Figure 7 and are mounted in the actual mechanical construction as follows as may be seen from Figures 2, 4 and 5.

The gear support 50 carries the shaft 27 having the gear 26 mounted thereon for cooperation with the rack gear teeth 25. At its opposite end the shaft 27 carries a gear 28 meshed with the gear 29 carried by the stub shaft 30. The stub shaft 30 carries the gear 31 which turns the gear 32 and also turns the gear 32a through the intermediate idler 33. The output of the gear 32a is fed through the shaft b and the gearlng 34 to the shaft 12, while the output gear 32 reciprocates shaft 14 by means of the threaded lead screw 13 having threads which cooperate with internal threads on the gear. The output shaft of the integrator 1 1s directlycoupled to a sleeve nut 16 which is threaded on to the shaft 17 of the integrator 2. As explained above, the shaft 17 has 32 threads per inch thereon and is moved axially upon rotation of the shaft 15. Thus the integral of rdr is fed into integrator 2 as a variable effecting the position of the ball carriage relative to the disk 18 of the second integrator.

The dx component which comprises the constant rate movement of the carriage base 3 relative to the rack 41 on the rail 6 is fed into the second integrator at the shaft 20. This gearing may best be seen from a consideration of Figures 2 and 5 wherein the gear support 51 carries the gear 39 on the shaft and it further carries aifixed thereto the worm 42 meshed with the gear 43. The support 51 also carries the shaft 43 which supports the gears 43 and 44 at right angles to the worm 42. The gear 44 meshes with the rack 41 through the idler gear 45, mounted on the stub shaft 49, and the gears 46 and 47.

Power input for moving the carriage along the rail 6 is, as was above noted, provided electrically in form of the motor 35 which drives the shaft 38 through the gear 36 and 37 after passing through the reduction gear indicated at 35a. The output of the second integrator shaft 21 is fed into the revolution counter 23 by a pair of gears 52 having equal numbers of teeth. As explained above this relationship provides a numerical figure at the counter equal to the volume to be determined times 100. Of course a to 1 gear ratio could be utilized in place of the gear 52 to provide a direct reading at the counter 23 if so desired. Further, since varying the magnification of the original drawing will change the volume by a factor not influenced by integration, varying magnifications of the original drawings may be used merely by changing the gearing 52 or my modifying the reading at 23 by an appropriate constant factor.

In operation, the apparatus is set up as shown in Figure 1. As may there be seen, the rails 5 and 6 are placed parallel to the x axis, or axis of revolution, of the curve S. The pointer P is then positioned at the origination point 0 of the curve S. The motor 35 may then be energised and as the base 3 is moved along the rails 5 and 6, the pointer P is moved along the curve S manually. When the pointer reaches the end of the curve S, the reading of the counter 23 is taken. For convenience sake, a reset knot 23b is provided on the counter to permit setting the counter at zero at the initiation of tracing.

It has been found that the above apparatus will calculate volumes of revolution with an average error of approximately .3 to 1.2% maximum depending mainly upon the ability of the operator of the machine to follow the curve exactly with the pointer. Of course in setting the apparatus up to determine the volume of revolution it is essentially that the line xx of the drawing, namely the axis of revolution, be exactly parallel with the carriage rails 5 and 6, and that the pointer P be positioned on the axis xx at the end of the curve S.

The electric motor 35 and the reduction gearing 35a were, in the example apparatus, constructed to provide a rotary input to the gears 36 of approximately revolutions per minute. Using the gearing set forth in the example above, this provides a movement of approximately 2.7 inches per minute of the carriage 3 in the xx directional on the rails 5 and 6. This feed may of course be slowed by providing a motor 35 of a different speed or a transmission 35a of a different gearing ratio. It has been found however that a movement of 2 to 3 inches per minute is slow enough for the average operator to trace the curve S with reasonable accuracy. It is to be noted, of course, that the electric motor 35 may be completely eliminated and the carriage 3 moved manually if so desired. However, in view of the care with which the pointer P should be manipulated, it is desired that the operator be required to manipulate it only rather than provide movement in the x direction also.

As an aid in the manipulation of the pointer, I have provided an auxiliary hand wheel 60 which is geared to the rack 25 through the shaft 61 and idler gear 33. The hand wheel is supported conveniently on the carriage 3 by supports 62, 63 secured thereto. By rotational movement of the hand wheel, the pointer P may be caused to follow the curve S and, simultaneously, the input will be fed to the integrator 1.

It will thus be apparent that I have provided an extremely efficient compact, and easily operated calculating apparatus. This will be more fully appreciated when it is understood that the dimensions of the baseboard 3 of the housing are, in the example above set out, only 7% inches wide and 13 /2 inches in lengh. The rails 5 and 6 may of course be any length desired but in view of the fact that the apparatus set out in the example above was constructed to provide a maximum travel of 15 inches in the xx direction, it is clear that a length of approximately 30 inches is entirely satisfactory for the rails. Under these circumstances it is clear that the entire apparatus is substantially less than 3 feet in length and hence may readily be utilized with the ordinary drafting boards in use today.

Further, the apparatus in the present invention is extremely simple to operate since it is only necessary that the operator trace the curve properly in order to provide an extremely accurate reading of the volume of revolution. Approximately 5 minutes time is all that is required to completely trace a curve and hence to calculate the accurate volume of revolution. Since the calculation of volumes of revolution having irregularly curved surfaces by approximation requires several hours time when carefully done, and even then the answer arrived at is only approximate, it will be apparent that the use of the calculator of the present invention has greatly facilitated the accurate computation of volumes of revolution and has hence materially accelerated the design time in preparing structures for manufacture.

It is of course to be understood that the specific gear trains set out in the above example are exemplary only and may be modified according to space requirements. Further, it is also clear that modifications could be made in the ratio of movement of the pointer relative to the rotation of the input shaft 12 of the integrator 1 and that threads of different pitch could be utilized at 14 and 17. These changes, which provide different constant factors in the equations, would necessarily require a modification of the gearing which accompanies these changes. However, utilizing the formulas set out and the interrelationship between the integrators above described, it is clear that the volume of revolution may accurately be obtained through the mere tracing of the curve S by reciprocating rack 11 and the movement of the carriage 3 in the xx direction. It is of course, to be further understood that other modifications and variations may be made within the scope of the concepts of the present invention.

I claim as my invention:

1. A calculator for the automatic computation of the volume of revolution of a curve S about an axis x and having a varying radius 1', which comprises a tracer point for tracing said curve, said tracer being connected to the disk and the ball carriage of a ball and disk integrator to provide a varying output at a rotating output shaft, a rail means for carrying said integrator in a direction parallel to the x axis, means for moving said carriage and said integrator along said rail, a second ball and disk integrator having its ball carriage connected to the output shaft of the first integrator and having its disk geared to said rail to correlate the rotation of said disk with the travel of said carriage relative to said rail, and a revolution counter secured to the output shaft of said second integrator for indicating the volume of revolution of said curve upon completion of the tracing thereof.

2. A calculator for the automatic computation of the volume of revolution of a curve about an axis comprising a first mechanical integrator having a rotating disc drivingly connected to a rotating output shaft by an adjustably positioned ball, means for tracing said curve and positioned for movement perpendicular to said axis, second means connecting said tracing means to said ball for varying its position in response to the variation in distance of points in said curve from said axis, third means connecting said tracing means to said disk for rotating said disc in response to said distance fourth means connecting said output shaft to the ball of a second integrator, fifth means for driving the disc of said second integrator in direct relation to travel of said tracing means along said axis and counter means driven by the output shaft of said.

second integrator for recording a value directly proportional to the volume of revolution of said curve about said axis.

VII

3. Apparatus for solving the integral Kfr dx comprising first and second ball and disc type integrators arranged in series with the output of the first providing the variable of integration for the second, and tracing means movable along perpendicular axes r and x, first means translating movement of said tracing means along the r axis to the ball and to the disc of the first integrator and second means translating movement along the x axis to the disc of the second integrator and counting means secured to the output of the second integrator for indicating the solution.

4. Apparatus for solving the integral comprising first and second ball and disc type integrators arranged in series with the output of the first providing the variable of integration for the second, and tracing means movable along perpendicular axes r and x, first means translating movement of said tracing means along the r axis to the ball and to the disc of the first integrator and second means translating movement along the x axis to the disc of the second integrator and counting means secured to the output of the second integrator for indicating the solution, said first means comprising a rack reciprocatable along the r axis and geared directly to the disc of said first integrator to cause rotation thereof and geared to the ball of said first integrator to cause reciprocation thereof responsive to movement of the rack.

5. Apparatus for solving the integral comprising first and second ball and disc type integrators arranged in series with the output of the first providing the variable of integration for the second, and tracing means movable along perpendicular axes r and x, first means translating movement of said tracing means along the r axis to the ball and to the disc of the first integrator and second means translating movement along the x axis to the disc of the second integrator and counting means secured to the output of the second integrator for indicating the solution, said second means comprising a fixed rack extending parallel to the x axis and gears connecting said rack to the disc of said second integrator whereby movement of the tracing means along said x axis will cause rotation of the disc of said second integrator in proportion to the movement along said x axis.

6. Apparatus for solving the integral comprising first and second ball and disc type integrators arranged in series with the output of the first providing the variable of integration for the second, and tracing means movable along perpendicular axes r and x, first means translating movement of said tracing means along the r axis to the ball and to the disc of the first integrator and second means translating movement along the x axis to the disc of the second integrator and counting means secured to the output of the second integrator for indicating the solution, said first means comprising a rack reciprocatable along the r axis and geared directly to the disc of said first integrator to cause rotation thereof and geared to the ball of said first integrator to cause reciprocation thereof responsive to movement of the rack, said second means comprising a fixed rack extending parallel to the x axis and gears connecting said rack to the disc of said second integrator whereby movement of the tracing means along said x axis will cause rotation of the disc of said second integrator in proportion to the movement along said x axis.

7. A calculator comprising a platform mounted for reciprocation along a fixed guide rail, a first rack mounted on said platform for reciprocation along an axis perpendicular to said rail, a first integrator having a rotating disc, an output shaft rotating at right angles to the axis of the disc and a movable ball positioned for providing a variable speed drive between said disc and said output shaft, first gears connecting said first rack to said disc for rotation thereof, second gears connecting said first rack to said ball for moving its position, a second integrator having a second output shaft, a second movable ball and a second disc, at second rack secured to said rail, third gears mounted on said platform and connecting said rail to said second disc to rotate said second disc, fourth gears connecting said first output shaft to said second ball and a counter actuated by said second output shaft.

- 8. A calculator comprising a platform mounted for reciprocation along a fixed guide rail, a first rack mounted on said platform for reciprocation along an axis perpendicular to said rail, a first integrator having a rotating disc, an output shaft rotating at right angles to the axis of the disc and a movable ball positioned for providing a variable speed drive between said disc and said output shaft, first gears connecting said first rack to said disc for rotation'thercof, second gears connecting said first rack to said ball for moving its position, a second integrator having a second output shaft, a second movable ball and a second disc, a second rack secured to said rail, third gears mounted on said platform and connecting said rail to said second disc to rotate said second disc, fourth gears connecting said first output shaft to said second ball and a counter actuated by said second output shaft, said first and second discs being rotatable about parallel axes and said first and second output shafts being rotatable about parallel axes.

9. A calculator comprising a platform mounted for reciprocation along a fixed guide rail, a first rack mounted on said platform for reciprocation along an axis perpendicular to said rail, a first integrator having a rotating disc, an output shaft rotating at right angles to the axis of the disc and a movable ball positioned for providing a variable speed drive between said disc and said output shaft, first gears connecting said first rack to said ball for moving its position, a second integrator having a second output shaft, a second movable ball and a second disc, a second rack secured to said rail, third gears mounted on said platform and connecting said rail to said second disc to rotate said second disc, fourth gears connecting said first output shaft to said second ball, a counter actuated by said second output shaft, and power means for moving said. platform along said rail at a constant rate.

10. A calculator comprising a platform mounted for reciprocation along a fixed guide rail, a first rack mounted on said platform for reciprocation along an axis perpendicular to said rail, a first integrator having a rotating disc, an output shaft rotating at right angles to the axis of the disc and a movable ball positioned for providing a variable speed drive between said disc and said output shaft, first gears connecting said first rack to said disc for rotation thereof, second gears connecting said first rack to said ball for moving its position, a second integrator having a second output shaft, a second movable ball and a second disc, a second rack secured to said rail, third gears mounted on said platform and connecting said rail to said second disc to rotate said second disc, fourth gears connecting said first output shaft to said second ball, a counter actuated by said second output shaft, and auxiliary rotatable manual means for reciprocating said first rack.

References Cited in the file of this patent UNITED STATES PATENTS 1,503,824 Fry Aug. 5, 1924 1,875,019 Koeppen Aug. 30, 1932 2,678,772 Imm May 18, 1954

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