Here is a two-part Navy training film from 1953 that describes the inner workings of mechanical fire control computers. It covers seven mechanisms: shafts, gears, cams, differentials, component solvers, integrators, and multipliers, and does so in the well-executed fashion typical of the era.
Fire control systems depend on many factors that occur simultaneously, not the least of which are own ship’s speed and course, distance to a target, bearing, the target’s speed and course if not stationary, initial shell velocity, and wind speed and direction.
The mechanisms are introduced with a rack and pinion demonstration in two dimensions. Principally speaking, a shaft carries a value based on revolutions. From this, a system can be geared at different ratios.
Cams take this idea further, transferring a regular motion such as rotation to an irregular motion. They do so using a working surface as input and a follower as output. We are shown how cams change rotary motion to linear motion. While the simplest example is limited to a single revolution, additional revolutions can be obtained by extending the working surface. This is usually done with a ball in a groove.
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