A Trigonometric Recurrence Algorithm for Solving Nonlinear Problems in Structural Dynamics

Abstract

An analytical-numerical methodology for solving nonlinear problems governed by partial differential equations (PDE) is presented. The authors have previously used a method named WEM that consists essentially of the statement of extended trigonometric series of uniform convergence (UC). Theorems, demonstrated previously, ensure the UC of the essential functions and the exactness of the eigenvalues. WEM has been applied to nonlinear dynamic problems in two different ways: As a direct variational method and as a solution in the classical sense. The present tool is applied in the second fashion and starts from the statement of the extended trigonometric series for all the unknown functions and derivatives involved in the PDE. The nonlinearities are treated in the same way. The application of consistence conditions leads to recurrence relationships among the coefficients of the extended series. For the sake of comparison an initial conditions problem (the well-known Duffing oscillator) is numerically solved. Then a beam example is solved in detail: A linear (both material and geometrical) supported beam, with end springs of nonlinear analytical response, under the action of a dynamic distributed load and/or prescribed initial conditions, is studied. Two numerical examples are included.