The solution of partial differential equations by means of electrical networks

Abstract

The material in this thesis is the result of a year's experience in the solution of problems on the Caltech Electric Analog Computer. Although much work has been done elsewhere, the solution of partial differential equations is a relatively new field for the Caltech Computer. It is natural that such an undertaking should initiate points of view and techniques that differ from those of other investigators., This thesis contains the development of certain ideas that have been useful to the author in the solution of problems on the computer. In Part I, finite difference methods are treated with reference to problems with one space variable. Techniques are developed for the representation of differential operators by means of electrical networks and the question of unequal lumping is discussed. The solutions of the fourth order differential equations of a beam are treated in Part II. Such practical considerations as the effects of parasitic impedances and cell size are investigated. Solutions are presented for the normal modes of a cantilever beam, the transient vibration of a cantilever beam, and the coupled modes of vibration of an airplane wing. Problems involving the scalar Laplacian operator are treated in Part III. A general asymmetric network is developed that is useful for problems with irregular boundaries and for problems where it is desired to have variable cell size. These techniques are illustrated with respect to a cavity resonator problem and an electromagnetic radiation problem. Elastic plate problems are treated in Part IV. The analogy for the elastic plate is an extension of the beam analogy to two dimensions. Here the difficult problems are those relating to the representation of boundary conditions, particularly of boundary conditions along an irregular edge. Some conclusions regarding the construction of a network analyzer designed specifically for the solution of partial differential equations are given in Part V. The chief conclusion is that such a computer must contain a much larger number of electrical elements than are at present available in the Caltech Computer.