Design Con 2015

Evolution of differential equation tools

-November 30, 2012

We all know how differential equations are a crucial part of engineering design and analysis. We have some powerful software tools for this today. Well, here is what led up to our modern analysis tools.

The Rutherford Journal has a nice “Chronology of Analog Computing” by Charles Care.

Back in 1878, Lord Kelvin developed a harmonic analyzer. The machine was Invented by William Thomson, Lord Kelvin, (1824-1907), a pioneering Irish physicist. The harmonic analyzer was designed to analyze graphical records of daily changes in atmospheric temperature and pressure. It was brought into use by the Meteorological Office in 1878. The tracing point is taken along the curve to be analyzed, and its movement causes the seven discs to rotate. Rolling spheres communicate this motion to the recording rollers.

Courtesy of the Science Museum

In 1927, Vannevar Bush, an engineer in the US, designed and built the differential analyzer (DA) which was in effect an analog computer with wheel-and-disk components that would solve differential equations with several independent variables by integral methods.

The principal usage of a differential analyzer is shown here in a You Tube video. Two integrators are connected to each other to perform the differential equation ydotdot = - y.
One solution is y = sin(t) and ydot = cos (t) which forms a perfect circle. This example was used to test the precision of differential analyzers.

 

Differential Analyzer built under Mergler in Instrument Research. The technician
is preparing a data report. This equipment is located at the Lewis Flight Propulsion
Laboratory (Credit: NASA)

Shown here is a wheel and disk illustration

Notation for an integrator using the wheel and disk (Courtesy of The future of things)

Engineers at Northrop Aircraft recognized in 1949 that analog computations could be done digitally by replacing a differential analyzer’s mechanical wheel-and-disk integrators with binary adders.

These “Digital Differential Analyzers,” or DDAs, were digital computers that acted like analog computers, but were more precise. They were used for real-time control applications, as well as simulations of airplane and missile designs—often paired with slower general-purpose digital computers.

A dozen Northrop employees formed Computer Research Corporation to build small airborne DDAs. The Polaris missile used one of the first digital computers for on-board guidance.

A Digital Differential Analyzer

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Mechanical analog computers in WWII naval gun directors

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