Rozwiązanie zagadnienia błony metodą różnic skończonych z użyciem specjalnego liczydła

Solution of the membrane problem by means of the method of finite differences with the use of a special computation device

The membrane equation, (1), where u denotes the displacement and p the load, is replaced by the system of equations, (2), using the well known method of finite differences. This equation was established for a square net (Fig. 1), and expresses the relation between the load P, corresponding to the given nodal point or the net, and the displacements of the neighbouring points, u_s.

The solution is obtained by means of the method of successive approximations called by R. V. Southwell the “relaxation method”. Instead of computations arranged in table form, a kind of “chess board” (fig. 4) in the form of a net is used.

On each nodal point of the net a certain number of “stones” is laid, corresponding to the nodal force P (table 1).

The process of relieving the nodal points consists in shifting the «stones» to the neighbouring points (fig. 6). At the beginning of the computation higher values are assumed for the “stones” and the nodal forces are suppressed roughly. The values assumed in subsequent stages are progressively lower, the suppressing of forces becoming more and more accurate. This procedure is regulated in the form of a kind of “play rules”, which are simple enough to permit the solution of the problem to be passed over to unqualified personnel.

This method in numerical examples proved to be more rapid than other methods and at the same time less tedious for the calculator.