The substitution of analogues for digits makes it considerably easier for the amateur to build a computer of his own. An analogue computer capable of solving most of the problems that are likely to interest amateurs need occupy no more than two cubic feet of space and can cost less than $50. Such a computer, specially designed for amateur construction, has been developed by Gordon F. Pearce, professor of mechanical engineering at the University of Waterloo in Canada, who writes:

"My computer is entirely mechanical, consisting of screws, shafts, disks, wheels, sprockets and chains. All parts except the disks, shafts and fittings come ready-made. The machine can be assembled with hand tools, although a lathe is useful for making the disks and reducing the diameter of shafts. Various combinations of these parts perform by their motions the equivalent of addition, subtraction and multiplication. Two of the subassemblies also compute the growth of variable quantities-the operation known as integration.

"Addition and subtraction are done by a set of gears similar to those in the differential of an automobile. In the computer a bevel gear is fixed to the inner end of a pair of shafts that are equivalent to the rear axles of an automobile. The bevel gears mesh with a differential gear that is free to turn on a short shaft fixed at right angles to its supporting shaft [below right]. Power is applied to the shafts that carry the bevel gears, which drive the shaft attached to the differential gear.

"The rotation of either bevel gear turns the differential shaft a proportional amount. When both bevel gears turn in the same direction, the rotation of the differential shaft is equal to the sum of their rotations. When the bevel gears rotate in opposite directions, the rotation of the differential shaft is equal to the difference of their rotations.

"Mathematically the action can be expressed as W = c(x + y), in which W is equal to the sum or the difference of x and y multiplied by a constant c, which is determined by the ratio of external gears that drive the unit. Either x or y can be considered a negative quantity depending on the direction in which its shaft rotates. By convention, clockwise rotation symbolizes a positive quantity, counterclockwise rotation a negative quantity.

"The machine is designed so that all input and output shafts are parallel and extend from the front panel, where they can be fitted with sprockets for interconnecting the several mechanisms. The sprockets, which are coupled as needed by chains of the ladder type, serve not only to drive the mechanism but also to multiply one quantity by another. For example, an input shaft equipped with a sprocket of 12 teeth that makes one revolution will impart a three-quarter turn to an output shaft equipped with a sprocket of 16 teeth. The action is equivalent to multiplying a quantity x by .75. The input quantity x can have any value.

"Multiplication by a negative number is accomplished by crossing the chain so that the sprockets rotate in opposite directions. A pair of meshed gears would work, but gears cannot be interchanged as handily as sprockets in a machine equipped with shafts in fixed positions.

"The mechanism that performs the mechanical equivalent of integration consists of a wheel that rolls on the face of a smooth disk and a carriage that can shift the path of the wheel between the center and the edge of the disk [see Figure 1]. Mechanisms of this type find occasional use as speed changers. The disk is driven by a hand-operated crank that represents the variable input x. The carriage that transports the wheel is moved by a sprocket-driven screw, whose rotation represents the second variable quantity, y.

"The rate at which the wheel turns with respect to the rotation of the disk depends on the point at which the wheel makes contact with the disk. The wheel stands still when it is at the center of the disk but rotates at an increasing rate as it is shifted toward the edge. In a sense the device totals the product of two quantities that can vary in size continuously while they are being multiplied. The rotation of the shaft that carries the wheel represents the output of the integrator.

"In the symbolism of the calculus the output W is written ydx, the long s meaning 'the sum of.' The product of any two quantities, including that of the variables y and dx, can be depicted as an area. For example, the product of 3 times 4 can be represented by a rectangle that contains four rows of three squares. Similarly, the integration of a variable quantity with respect to a constant can be represented by an area bounded on one side by a curve. In this regard the integrating mechanism of the computer resembles a simple planimeter, the instrument used for measuring areas enclosed by a curving line [see 'The Amateur Scientist; SCIENTIFIC AMERICAN, August, 1958].

"All the mechanisms except the sprockets and chains are enclosed in a box of sheet metal. The top of the box serves as the base of an automatic recorder fitted with two movable pens that write on a rectangular sheet of graph paper [Figure 1]. The paper rests on a flat table that is moved sideways along the x coordinate of the graph by means of a screw. The screw is driven by a pair of bevel gears equipped with a sprocket that is accessible on the front of the machine. Similarly equipped screws drive two pen carriages at right angles to the movement of the paper, along the y coordinate of the graph. Hence two output quantities, y₁ and y₂, can be recorded simultaneously. The machine includes two integrators and one unit for adding and subtracting.

"The fabrication and assembly of the machine can be performed easily by anyone with the basic shop experience acquired in high school. The necessary gears, sprockets and chain are all standard items and can be bought locally from dealers in mill supplies. The accuracy of the computer will depend in part on the workmanship and the care used in assembly.

"The housing of the addition-subtraction unit can consist of a channel section formed by screwing sheet-metal sides to a block of wood. Holes through the sheet metal serve as bearings for three shafts. The shafts of my computer are 4 1/2-inch lengths of drill rod 3/16 inch in diameter. The x and y shafts each carry a spur gear (Boston Y4884) locked in place by setscrews. The x + y shaft carries four loosely fitting gears, consisting of two bevel gears (Boston G463Y) each of which is soldered to a companion spur gear (Boston Y4842).

"To solder the gears, I tinned the mating faces of a spur gear and a bevel gear, slipped the gears onto a scrap of rod to align the holes and heated them until the solder melted. When the gears cooled, I reamed the hole so that the assembly ran smoothly on the x + y shaft. Incidentally, all gears may as bought have slightly undersized holes. If so, ream them to fit their shafts.

"The differential gear (Boston G463Y) turns on a shaft that extends from a carrier block locked to the x + y shaft by a setscrew. I made the block of 1/2-inch hexagon bar stock 3/4 inch long, but square stock would do. A shallow hole is drilled in the center of one end. Into the hole is soldered a length of drill rod 3/16 inch in diameter. The rod is then cut off at a point 5/8 inch from the end of the block. This stub shaft supports the differential gear.

"At right angles to the stub shaft drill a hole 3/16 inch in diameter through the block. Space the center of this hole exactly 1/4 inch from the end of the block to which the stub shaft is attached. This transverse hole should make a snug but sliding fit with the x + y shaft.

"In the end of the block opposite the stub shaft drill a hole that penetrates the transverse hole and thread it for a setscrew. Finally, near the outer end of the stub shaft drill a transverse hole 1/16 inch in diameter through the stub shaft. After the differential gear has been assembled on the stub shaft slip a cotter pin through the 1/16-inch hole.

"When assembling the unit in the housing, pass the x + y shaft through the outer bearing and into one of the spur-bevel gear subassemblies. Slip the assembled carrier block and differential gear on the x + y shaft. Pass the shaft through the remaining spur-bevel gear and the rear bearing. Lock the carrier block to the shaft by its setscrew. The x + y shaft should turn freely, even when either of the spur-bevel gears is held stationary.

"The two integrators are made exactly alike. The rotating disk consists of two parts, a base of 1/4-inch plywood and a disk of 18-gauge sheet metal. The metal is facing for the plywood. The two parts are fastened together at the edge by four countersunk machine screws with flat heads and attached at the center to a flange that fits a shaft 1/4 inch in diameter. I made the flange with a lathe, but a spur gear about two inches in diameter could be substituted.

"The wheel that rolls on the disk consists of a steel pulley fitted with an O ring of rubber slightly more than two inches in diameter. The effective radius of the wheel becomes one inch when the O ring is pressed against the disk by a helical spring that exerts a force of eight ounces. A collapsing coupling links the shaft of the wheel to a second shaft that extends through the front panel of the machine [Figure 4].

"The screw that drives the carriage of the wheel is 1/4 inch in diameter and has 20 threads per inch. All threaded shafts of the computer are made from the same stock. Bevel gears (Boston G464Y) couple the input shaft x of the integrator to the drive shaft of the disk. Similarly, a Boston G481-Y-P bevel gear, which is locked to input shaft x of the recorder, meshes with a Boston G481Y-G bevel gear to drive the table sideways. The movement represents the x coordinate of the graph. I find that pens equipped with plastic nibs are more reliable than ball-point pens, although they draw wider lines.

"A set of 15 sprockets in various sizes enables the user to program the machine for solving problems that are of most interest to amateurs. All the sprockets in my machine, as well as the gears, were made by the Boston Gear Company. My set consists of two CBA6 sprockets, two CA9, one CA10, two CBA12, four CBA18B, two CBA24B and two CBA36 sprockets. (The numeral in these code numbers indicates the number of teeth on the sprocket.) About six feet of ladder chain that matches the sprockets is required.

"After assembly all traces of grease must be removed from the rubber O ring of the integrator wheels and the faces of the disks, preferably by a degreasing agent such as Varsol. The wheels must not slip on the disks. All shafts must turn easily in their bearings. A lubricant such as Molyslip is useful during the initial running period.

"The operation of the computer is perhaps best explained by an example. Consider the physical problem of determining how the position of a stone, dropped from a building 100 feet high, will vary with time. In elementary physics the equation that describes the motion of a stone in free fall states that the acceleration a is equal to the acceleration of gravity -g. In this example upward acceleration is assumed to be positive. Downward acceleration is indicated by the negative sign.

"In this computer quantities are represented by the rotation of shafts. To feed a problem into the machine, the programmer begins by specifying the number of revolutions a shaft must make to represent one unit of the quantity under consideration. This ratio is known as the scale factor.

"In the case of the falling stone the y shaft of the first integrator is selected to represent acceleration. It can be arbitrarily assumed that one revolution of the y shaft should equal an acceleration of one foot per second per second. Some other scale factor could have been selected. It could have been assumed, for example, that 10 revolutions were to equal one unit of acceleration, but a scale factor of one is practical.

"Having made this decision, the programmer can feed into the y shaft the acceleration of gravity: -32.2 feet per second per second. (Gravity acts in the downward direction, so that the quantity is negative.) This quantity is entered by rotating the y shaft 32.2 turns in the counterclockwise or negative direction, which moves the wheel away from the center of the disk, its zero position.

"The time during which the stone falls will be fed into the x shaft of the first integrator. A scale factor must also be selected for this shaft. In making the selection account is taken of the fact that the scale factor of the output shaft of the integrator is determined by the scale factors of the x and y shafts. The integrators of the machine are designed so that the output shaft's scale factor (S₃) is equal to the product of the scale factors of the x shaft (S₁) and the y shaft (S₂) divided by 20. (S₃ = S₁S₂/20).

"A scale factor of 40 can be assigned to the x shaft, that is, 40 revolutions of the x shaft will represent one second of time. The integrator combines time in seconds with acceleration in feet per second per second to determine velocity in feet per second, which is represented by the rotation of the output shaft. As explained, the scale factor of the output shaft is determined by multiplying the scale factors of the inputs and dividing by 20: 40 x 1/ 20 = 2. Therefore two turns of the output shaft will represent a velocity of one foot per second.

"The position of the stone at any instant after it has been dropped depends on how long it has been falling. To find any position the velocity of the stone's fall must be combined with the interval of time through which it has been falling. The calculation can be made by feeding the output of the first integrator into the y shaft of the second integrator, and time into the x shaft of the second integrator. The x shafts of the two integrators, which represent time, are therefore coupled by a ladder chain, which links sprockets of equal size.

"However, sprockets chosen for a speed reduction of 1:4 are used between the output of the first integrator and the shaft of the second integrator. This arbitrary reduction in speed lowers the torque imposed by the load on the output shaft of the first integrator and decreases the tendency of the wheel to slip on its driving disk. The arbitrary reduction is taken into account when computing the scale factor of the output shaft of the second integrator: 40 x 40 x 1 / 20 x 20 x 4 = 1, indicating that one revolution of the output shaft is equivalent to a displacement of one foot in the position of the falling stone. The initial velocity of the stone is zero. This information is fed into the machine by rotating the y shaft of the second integrator by hand until the wheel stands at its zero position in the center of the disk.

"The scale of the graph that will be drawn by the movement of the pens and the table must also be determined. The screws that drive the pens and the table have 20 threads per inch. Assume that the horizontal or x coordinate of the graph will display time-the quantity represented by the rotation of the x shafts of the integrators. The scale factor of these shafts is 40. By coupling either x shaft to the screw of the table by sprockets of equal size the graph can be moved two inches to represent one second.

"The vertical scale of the graph is similarly determined. A displacement of one foot in the position of the stone is represented by one revolution of the output shaft of the second integrator. By falling through 100 feet the stone causes the output shaft to make 100 revolutions. Coupling the output shaft directly to the shaft of the pen would cause the pen to move five inches along the y coordinate of the graph during 100 revolutions. The movement is equal to a displacement of 20 feet per inch. It is desirable, however, to minimize the torque imposed on the output shaft of the integrator. A combination of sprockets that reduces the speed by a ratio of 1:2 results in a scale of one inch per 40 feet of displacement and a graph of reasonable size. The chain that couples the output of the second integrator to the pen must be crossed to indicate a negative quantity, because the stone falls downward.

"A sheet of graph paper ruled with rectangular coordinates is placed on the table and attached at the corners by adhesive tape. The shafts of the pen and the table are turned by hand until the nib of the pen rests on the y coordinate at a point 2 1/2 inches from the x coordinate. If desired, the x and y coordinates can be lettered to indicate seconds and feet respectively.

"The recorder is then coupled by chains to the driving sprockets. A11 chains should fit snugly, being neither so tight as to cause excessive wear or friction nor so loose that backlash results The links of the chain can be opened and closed easily with a pair of long-nosed pliers. A hand crank bent from drill rod is attached to any x shaft by means of a coupling equipped with setscrews. The accompanying diagram [Figure 7] shows the pattern of chains and sprockets used for programming the problem of the falling stone. The numerals in the diagram indicate the number of teeth on the sprockets.

"To solve the problem turn the crank counterclockwise. The resulting graph will take the form of a smooth, downward-sloping curve that begins at the 100-foot point on the y coordinate and ends just short of the 2.5-second point on the x coordinate. The graph is identified by a zero in the accompanying illustration [above] to indicate that the velocity of the stone was zero at the moment it was released.

"The computer can be programmed to display the behavior of the stone thrown upward or downward with various velocities. For example, to solve the problem for an assumed upward velocity of 60 feet per second insert 60 into the velocity shaft (y) of the second integrator. The scale factor of this shaft is V/2. The initial setting of the shaft is therefore 60/2, or 30 turns in the clockwise direction. Turning the handle of the computer now generates the new solution. The consequences of assuming other initial velocities, such as -20, 20 and 40 feet per second, were similarly computed and resulted in graphs as shown.

"The behavior of a bouncing ball can also be computed. Assume that the ball is projected horizontally at known velocity from a predetermined height, that friction between the ball and the air is negligible and that the ball loses 40 percent of its kinetic energy at each impact with the ground. The problem is to determine the path of the ball. Two pens can be used, one to display the velocity of the ball and the other the height. The pattern of interconnected sprockets that solves this problem appears in the accompanying illustration [above].

"An interesting procedure is to solve the problem and plot both the vertical displacement and the vertical velocity simultaneously against the horizontal displacement. When the ball strikes the ground, stop the computer and read the velocity as recorded by the pen connected to the output of the first integrator. Multiply it by -.774 to find the lowered velocity caused by the impact and set the new velocity as the revised initial condition in the second integrator. Restart the computer by turning the handle counterclockwise and repeat the procedure each time the ball strikes the ground. The accompanying illustration [below] shows how the ball bounces.

"To reset an integrator without disturbing the chains lift the wheel from contact with its disk by pulling the carriage against the force of the spring. While holding the wheel away from its disk, rotate the sprocket that drives the velocity pen to the point at which the pen indicates the revised velocity. This operation simultaneously shifts the carriage of the second integrator to its proper revised position.

"Other interesting problems that can be programmed include the trajectories of projectiles; the decomposition of chemicals such as dinitrogen pentoxide; the steady-state conduction of heat; damped vibration; the response of instruments such as thermometers and thermocouples to abrupt changes in temperature, and the paths of rockets.

"The amateur who undertakes the construction of this computer will find many uses for it. Probably he will discover that it can become an interesting experimental project in its own right."

Programs for using the computer to investigate problems in mechanics, chemistry, thermodynamics and mathematics will be included in a manual to be published in the near future. The manual will include detailed working drawings of all the parts. For details concerning the availability of the manual write to Gordon F. Pearce, 43 George Street, Waterloo, Ontario, Canada.

Bibliography

ANALOG METHODS: COMPUTATION AND SIMULATION. Walter J. Karplus and Walter W. Soroka. McGraw-Hill Book Company, Inc., 1959.

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