Ball-and-disk integrator

The ball-and-disk integrator is a key component of many advanced mechanical computers. Through simple mechanical means, it performs continual integration of the value of an input. Typical uses were the measurement of area or volume of material in industrial settings, range-keeping systems on ships, and tachometric bombsights. The addition of the torque amplifier by Vannevar Bush led to the differential analysers of the 1930s and 1940s.

The basic concept of the ball-and-disk integrator was first described by James Thomson, brother of William Thomson, 1st Baron Kelvin. William used the concept to build the Harmonic Analyser in 1886. This system was used to calculate the coefficients of a Fourier series representing inputs dialled in as the positions of the balls. The inputs were set to measured tide heights from any port being studied. The output was then fed into a similar machine, the Harmonic Synthesiser, which spun several wheels to represent the phase of the contribution from the sun and moon. A wire running along the top of the wheels took the maximum value, which represented the tide in the port at a given time. Thomson mentioned the possibility of using the same system as a way to solve differential equations, but realized that the output torque from the integrator was too low to drive the required downstream systems of pointers.

A number of similar systems followed, notably those of Leonardo Torres y Quevedo, a Spanish physicist who built several machines for solving real and complex roots of polynomials; and Michelson and Stratton, whose Harmonic Analyser performed Fourier analysis, but using an array of 80 springs rather than Kelvin integrators. This work led to the mathematical understanding of the Gibbs phenomenon of overshoot in Fourier representation near discontinuities.

By the turn of the 20th century, naval ships were starting to mount guns with over-the-horizon range. At these sorts of distances, spotters in the towers could not accurately estimate range by eye, leading to the introduction of ever more complex range finding systems. Additionally, the gunners could no longer directly spot the fall of their own shot, relying on the spotters to do this and relay this information to them. At the same time the speed of the ships was increasing, consistently breaking the 20 knot barrier en masse around the time of the introduction of the Dreadnought in 1906. Centralized fire control followed in order to manage the information flow and calculations, but calculating the firing proved to be very complex and error prone.

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