General Organization of Mit Differential Analyzer No. 2
A differential analyzer is basically made up of shafts that turn. When we set up the machine to solve a differential equation, we assign one shaft in the machine to each quantity referred to in the equation. It is the job of that shaft to keep track of that quantity. The total amount of turning of that shaft at any time while the problem is running measures the size of that quantity at that time. If the quantity decreases, the shaft turns in the opposite direction. For example, if we have speed, time, and distance in a differential equation, we label one shaft “speed,” another shaft “time,” and another shaft “distance.” If we wish, we may assign 10 turns of the “time” shaft to mean “one second,” 2 turns of the “distance” shaft to mean “one foot,” and 4 turns of the “speed” shaft to mean “one foot per second.” These are called scale factors. We could, however, use any other convenient units that we wished.
By just looking at a shaft or a wheel, we can tell what part of a full turn it has made—a half, or a quarter, or some other part—but we cannot tell by looking how many full turns it has made. In the machine, therefore, there are mechanisms that record not only full turns but also tenths of turns. These are called counters. We can connect a counter to any shaft. When we want to know some quantity that a shaft and counter are keeping track of, we read the counting mechanism.
The second differential analyzer, which MIT finished in 1942, went a step further than any previous one. In this machine, a varying number can be expressed either (1) mechanically as the amount of turning of a shaft, or (2) electrically as the amount of two voltages in a pair of wires. The MIT men did this by means of a mechanism called an angle-indicator.
Angle indicators have essentially three parts: a transmitter, a receiver, and switches. The transmitter (Fig. 10) can sense the exact amount that a shaft has turned and give out a voltage in each of two wires which tells exactly how much the shaft has turned (Fig. 11). The receiving device (Fig. 12), which has a motor, can take in the voltages in the two wires and drive a second shaft, making it turn in step with the first shaft. By means of the switchboard (Fig. 13), the two wires from the transmitter of any angle-indicator can lead anywhere in the machine and be connected to the receiver of any other angle indicator.
In a differential analyzer, we can connect the shafts together in many different ways. For example, suppose that we want one shaft b to turn twice as much as another shaft a. For this to happen we must have a mechanism that will connect shaft a to shaft b and make shaft b turn twice as much as shaft a. We can draw the scheme of this mechanism in Fig. 14: a box, standing for any kind of simple or complicated mechanism; a line going into it, standing for input of the quantity a; a line going out of it, standing for output of the quantity b; and a statement saying that b equals 2a.
One mechanism that will make shaft b turn twice as much as shaft a is a pair of gears such that: (1) they mesh together and (2) the gear on shaft a has twice as many teeth as the gear on shaft b (Fig. 15). On the mechanical differential analyzer that MIT finished in 1930, a pair of gears was the mechanism actually used for doubling. To make one shaft turn twice as much as another by this device, we would: go over to the machine with a screwdriver; pick out from a box two gears, one with twice as many teeth as the other; slide them onto the shafts that are to be connected; make the gears mesh together; and screw them tight on their shafts.
On the MIT differential analyzer No. 2, however, we are better off. A much more convenient device for doubling is used. We make use of: a gearbox in whichthere are two shafts that may be geared so that one turns twice as much as the other, and two angle-indicator transmitters and receivers. Looking at the drawing (Fig. 16), we can see that: shaft a drives shaft c to turn in step, shaft c drives shaft d to turn twice as much, and shaft d drives shaft b to turn in step. Here we can accomplish doubling by closing the pairs of switches that connect to the gearbox shafts.
Angle indicators: T, transmitters, and R, receivers
Above, we have talked about a mechanism with gears that would multiply the amount of turning by the constant ratio 2. But, of course, in a calculation, any ratio, say 7.65, 3.142, ···, might be needed, not only 2. In order to handle various constant ratios, gearboxes of two kinds are in differential analyzer No. 2. The first kind is a one-digit gearbox. It can be set to give any of 10 ratios, 0.1, 0.2, 0.3, ···, 1.0. The second kind is a four-digit gearbox. It can be set to give any one of more than 11 thousand ratios, 0.0000, 0.0001, 0.0002, ···, 1.1109, 1.1110. We can thus multiply by constant ratios.
Adders
We come now to a new mechanism, whose purpose is to add or subtract the amount of turning of two shafts. It is called an adder. The scheme of it is shown in Fig. 17: an input shaft with amount of turning a, another input shaft with amount of turning b, and an output shaft with amount of turning a + b. The adder essentially is another kind of gearbox, called a differential gear assembly. This name is confusing: the word “differential” here has nothing to do with the word “differential” in “differential analyzer.” This mechanism is very closely related to the “differential” in the rear axle of a motor car, which distributes a driving thrust from the motor to the two rear wheels of the car.
A type of differential gear assembly that will add is shown in Fig. 18. This is a set of 5 gears A to E. The 2 gears A and B are input gears. The amount of their turning is a and b, respectively. They both mesh with a third gear, C, free to turn, but the axis of C is fastened to the inside rim of a fourth, larger gear, D. Thus D is driven, and the amount of its turning is (a + b)/2. This gear meshes with a gear E with half the number of teeth, and so the amount of turning of E is a + b.
We can subtract the turning of one shaft from the turning of another simply by turning one of the input shafts in the opposite direction.
Integrators
Another mechanism in a differential analyzer, and the one that makes it worth while to build the machine, is called an integrator. This mechanism carries out the process of integrating, of adding up a very large number of small changing quantities. Figure 19 shows what an integrator is. It has three chief parts: a disc, a little wheel, and a screw. The round disc turns horizontally on its vertical shaft. The wheel rests on the disc and turns vertically on its horizontal shaft. The screw goes through the support of the disc; when the screw turns, it changes the distance between the edge of the wheel and the center of the disc.
Now let us watch this mechanism move. If the disc turns a little bit, the wheel pressing on it must turn a little bit. If the screw turns a small amount, the distance between the edge of the wheel and the center of the disc changes. The amount that the wheel turns is doubled if its distance from the center of the disc is doubled, and halved if that distance is halved. So we see that:
(the total amount that the wheel turns) equals the sum of all the small (amounts of turning), each equal to: a bit of (disc turning) multiplied by the (distance from the center of the disc to the edge of the wheel) applying to that bit
If we look back at our discussion of integrating (p. 72), we see that the capital words here are just the same as those used there. Thus we have a mechanism that expresses integration:
(the total amount that the wheel turns) equals the integral of (the distance from the center of the disc to the wheel) with respect to (the amount that the disc turns)
The scheme of this mechanism is shown in Fig. 20.
For example, suppose that the screw measures the speed at which a car travels and that the disc measures time. The wheel, consequently, will measure distance traveled by the car. The mechanism integrates speed with respect to time and gives distance.
This mechanism is the device that Lord Kelvin talked about in 1879 and that Dr. Bush made practical in 1925. The mechanical difficulty is to make the friction between the disc and the wheel turn the wheel with enough force to do other work. In the second differential analyzer, the angle indicator set on the shaft of the wheel solves the problem very neatly.
Function Tables
The behavior of some physical quantities can be described only by a series of numbers or a graphic curve. For example, the resistance or drag of the air against a passing object is related to the speed of the object in a rather complicated way. Part of the relation is called the drag coefficient or resistance coefficient; a rough graph of this is shown in Fig. 21. This graph shows several interesting facts: (1) when the object is still, there is no air resistance; (2) as it travels faster and faster, air resistance rapidly increases; (3) when the object travels with the speed of sound, resistance is very great and increases enormously; (4) but, when the object starts traveling with a speed about 20 per cent faster than sound, the drag coefficient begins to decrease. This drawing or graph shows in part how air resistance depends on speed of object; in other words, it shows the drag coefficient as a function of speed (see Supplement 2).
Now we need a way of putting any function we wish into a differential analyzer. To do this, we use a mechanism called a function table. We draw a careful graph of the function according to the scale we wish to use, and we set the graph on the outside of a large drum (Fig. 22). For example, we can put the resistance coefficient graph on the drum; the speed (or independent variable) goes around the drum, and the resistance coefficient (or dependent variable) goes along the drum. The machine slowly turns the drum, as may be called for by the problem. A girl sits at the function table and watches, turning a handwheel that keeps the sighting circle of a pointer right over the graph. The turning of the handwheel puts the graphed function into the machine. Instead of employing a person, we can make one side of the graph black, leaving the other side white, and put in a phototube (an electronic tube sensitive to amount of light) that will steer from pure black or pure white to half and half (see Fig. 23).
We do not need many function tables to put in information, because we can often use integrators in neat combinations to avoid them. We shall say more about this possibility later.
We can also use a function table to put out information and to draw a graph. To do so, we disconnect the handwheel; we connect the shaft of the handwheel to the shaft that records the function we are interested in; we take out the pointer and put in a pen; and we put a blank sheet of graph paper around the drum.
We have now described the main parts of the second MIT differential analyzer. They are the parts that handle numbers. We can now tell the capacity of the differential analyzer by telling the number of main parts that it holds:
| Shafts | About 130 |
| One-digit gearboxes | 12 |
| Four-digit gearboxes | 16 |
| Adders | About 16 |
| Integrators | 18 |
| Function tables | 3 |
On a simpler level, we can say that the machine holds these physical parts:
| Miles of wire | About 200 |
| Relays | About 3000 |
| Motors | About 150 |
| Electronic tubes | About 2000 |