First, the distinction between numerical and analog computers is recalled. Numerical computers perform calculations on numbers. Analog computers use physical phenomena associated with continuous variables. Numerical computers work independently of the problem they are solving. Analog computers, on the other hand, exploit the analogy between a problem that cannot be addressed directly and a physical system that can be measured. Therefore, measuring instruments are used to measure input data and the final result. The problem of generating a function using cams is then considered. A simple but significant example is then given. It involves calculating the length of one leg of a right triangle when the lengths of the hypotenuse and the other leg are known by measurement. An important use of analog computers was in the fire control systems of naval guns. Here it is necessary to perform complex but repetitive calculations quickly, a task at which analog computers excelled during World War II. The chapter ends with a note about spiral cams.

1. 1.

   It is well known that the real potential of modern computers now goes far beyond mere number processing. This does not affect the definitions given here.

Authors and Affiliations

1. University of Bologna, Bologna, Italy

   Umberto Meneghetti

Appendix: Spiral Cams

To better understand the meaning and possibilities of using a cam to generate functions, it may be useful to take a quick look at spiral cams (Rothbart1956).

Suppose you have a measuring instrument of any physical entity with a non-linear output, and you want to report the result on an appropriate linear scale using a mechanism in which both the driving and driven elements are cams. We callx the value of the quantity to be measured, i.e. the value of the instrument's output. The linearized valuey that we want to read is related tox by a known function:

This problem can be solved by using two spiral cams, one drive and one follower, whose rays are defined as a function of their respective angles of rotation. The independent variablex is the angle of rotationθ_d of the driving cam, while the variabley is the angle of rotationθ_c of the other cam:

Leta be the distance between the cam axes, see Fig. 11.8. The two cams must always touch at a pointP, which can vary, but which is always aligned with the axes of rotation of the cams, or rather with the traces of these axes. In fact, when they touch, the two cams have speeds that are both perpendicular to this orientation and, as it must be, parallel to each other. This means that the sum of the cam radiir_d andr_c at the point of contactP must always bea. It is easy to see (Rothbart1956) that this condition is satisfied if

where\(f^{\prime } (\theta_{d} ) \, = {\text{d}}\theta_{d} /{\text{d}}\theta_{c}\) and, of course,r_d = a − r_c. By drawing the cams with these laws, the rotationθ_c of the follower cam is linked to the rotationθ_d of the drive cam in the desired way, i.e.θ_c = f(θ_d).

Spiral cams. The illustration is a generic one and the cams do not refer to the numerical example

Full size image

Let us now look at a numerical example—for which we will not specify the units of measurement—that also illustrates the possible problem of scales.

Numerical example. Let us suppose that we measure a quantityp with a measuring instrument, through which we want to go back to a quantityt, which is related top by the relation:

We also want to return the value oft on a linear scale;p varies from 10 to 60, and the encoder causes a corresponding 45.5° rotation of the drive cam;t can vary from 0 to 200, but we want the reading scale to be linear in the range between 0 and 30. For the base circle of the cam, we take the valuea = 100 mm. Forp andt we then have the following scale factors:

From ther_c expression above, in this case we have:

Thus, for each value oft, we find the values ofr_c andr_d = 100 − r_c; sincea is expressed in mm, the radii are also expressed in mm. We also have:θ_d = 0.91t and:

where angles are expressed in degrees.

This gives us all the data we need to draw the two cams with spiral profiles, all we have to do is select a reference point P. We notice that in the section we are interested in, from 0 to 30, the two cams do not make a full revolution: the driving cam makes 45.5° and the follower makes 8.99°. The radiusr_d of the drive cam varies from 67.8 mm to 72.7 mm, and the radiusr_c of the follower varies from 32.2 mm to 27.3 mm.

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