Machine Computation of Trigonometric Expansions

A new technique for the computer solution of problems in solid mechanics is introduced. The total potential energy of the system is expressed in terms of displacements that are expanded into trigonometric series, each of whose terms satisfies the boundary and symmetry conditions. Powers of these series from one to four occur in the potential energy expression, and it is necessary to multiply series by series before minimizing the energy with respect to the undetermined coefficients to find the equilibrium equations. Techniques are developed to perform this multiplication on the digital computer and to store the results on tape for use by a subroutine in which the energy is minimized. This technique, in which the exponents and subscripts of the undetermined coefficients and the arguments of the trigonometric terms are coded in integer or bit form, is not limited to applications involving minimization of the potential energy, but can also be used to aid in the solution of nonlinear differential equations with variable coefficients, where the dependent variables are expanded in trigonometric series. The method has been applied successfully to problems involving one and two independent variables.