Uniform convergence series to solve nonlinear partial differential equations: Application to beam dynamic

Abstract

Extended trigonometric series of uniform convergence are proposed as a method to solve the nonlinear dynamic problems governed by partial differential equations. In particular, the method is applied to the solution of a uniform beam supported at its ends with nonlinear rotational springs and subjected to dynamic loads. The beam is assumed to be both material and geometrically linear and the end springs are the Duffing type. The action may be a continuous load q = q(x) within a certain range and/or concentrated dynamic moments at the boundaries. The adopted solution satisfies the differential equation, the initial conditions, and the nonlinear boundary conditions. It has been previously demonstrated that, due to the uniform convergence of the series, the method yields arbitrary precision results. An illustration example shows that efficiency of the method.