chalkdust This is where c0 and c1 come into play: these are the initial conditions for the integrators. Let us assume that c0 = 1 and c1 = 0, ie the lemost integrator starts with the value 1 at its output, which feeds into the second integrator, which in turn feeds the sign changing summer, which then feeds the first integrator. This will result in a cosine signal at the output of the first integrator and a minus sine function at the output of the second one, perfectly matching the analytic solution of (2). Such initial conditions are normally shown as being fed into the top of the rectangular part of an integrator symbol, but we have omied this in our diagrams. So if we have some computing elements, we have seen that we can arrange them to create an abstract model of a diﬀerential equation, giving us some form of specialised computer: an analogue computer! The implementation of these computing elements could be done in diﬀerent ways: time integration, for example, could be done by using the integrand to control the flow of water into a bole, or to charge a capacitor, or we could build some other intricate mechanical system. Some of the most important observations to make are the following:

Setup for the predator-prey simulation.

• Analogue computers are programmed not in an algorithmic fashion but by actually interconnecting their individual computing elements in a suitable way. Thus they do not need any program memory; in fact, there is no “memory” in the traditional sense at all. • What makes an analogue computer “analogue” is the fact that it is set up to be an analogy of some problem readily described by diﬀerential equations or systems of them. Even digital circuits qualify as analogue computers and are known as Digital Diﬀerential Analysers (DDA). • Programming an analogue computer is quite simple (although there are some pitfalls that are beyond the scope of this article). One just pretends that the highest derivative in an equation is known and generates all the other terms from this highest derivative by applying integration, summation, multiplication, etc until the right-hand side of the equation being studied is obtained, with the result then fed into the first integrator. As a remark it should be noted that Kelvin’s feedback technique, as it is known, can also be applied to traditional stored-program digital computers.

Examples of analogue computers Analogue computers were the workhorses of computing from the 1940s to the mid-1980s when they were finally superseded by cheap and (somewhat) powerful stored-program digital computers. Thus without them, the incredible advances in aviation, space flight, engineering and industrial processes aer the Second World War would have been impossible. A typical analogue computer of the 1960s was the Telefunken RA 770, shown on the next page. chalkdustmagazine.com

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Chalkdust, Issue 03

Popular mathematics magazine from UCL

Chalkdust, Issue 03

Popular mathematics magazine from UCL