Electronic Analog Computers: Networks Versus Tanks

The analogy between an electrical system and a mechanical system is well known. For example, a mass-spring damper system is equivalent to a circuit of resistors, capacitors and inductors, governed by the same second order ordinary differential equation and can be solved on an analog computer using summation and integration elements with continuous real time simulation and low power requirement. Systems of two degrees can be simulated via solving two coupled differential equations and coupling partial circuits. Accuracy is limited however, due to the noise in the circuits.

Analog computers were used in aeronautical industry in the fifties and sixties (before digital computers took over). Evolutionary partial differential equations can be reduced to systems of ordinary differential equations (continuous in time) via method of lines, where space derivatives are approximated with finite differences. Examples of heat and wave equations can be solved using analog computers with real time simulation. Stiff and nonlinear equations can also be treated using hybrid computers (see Ref. [19]). New ideas in this field are promising, for example the neural network systems [20].

13.4 Concluding Remarks

In this chapter, we discussed flow analogies including the Hele-Shaw cell, hydraulic analogy and electric analogy. Analog computer is also briefly covered. Other analo­gies exist. For example flowing foam, (a foam is a mixture of gas with a liquid at constant mass ratio).

Let p be the mass concentration of the gas and 1 – p that of the liquid, then the specific volume of the foam is

Подпись: 1 P (13.85)

where p and о are the densities of the gas and the liquid component, respectively.

Assuming the expansion of the gas in the bubbles to be isentropic and p > p/о, hence

Подпись:Electronic Analog Computers: Networks Versus Tanks(13.86)

Подпись: 2 2 Подпись: p dp = P P0 P Подпись: 2 Electronic Analog Computers: Networks Versus Tanks

The speed of sound c2 = dp/dp = pdp/dp = pc2, and from Bernoulli’s law, the velocity is given by

Подпись: c Подпись: q = M c Подпись: (13.88)

Hence,

Flowing foams and cavitating liquids are closely related. For more details see Oswatitsch [9].

In this regard, it is of historical interest that the analog solution of the torsion problem of a bar was proposed by L. Prandtl, and used by G. I. Taylor, two giants of fluid mechanics. An opening in the plane that has the same shape as the cross­section of a torsion bar covered with homogeneous elastic membrane, such as a soap film, under pressure on one side will have a lateral displacement satisfying the same equation as the Prandtl torsion stress function. Prandtl’s membrane analogy is discussed in many strength of materials books, see for example [21].

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