Analysis Techniques for the Various forms of the Duffing Equation

Abstract

In this Chapter some analytical asymptotic techniques which can be used to solve different forms of the Duffing equation are presented. Two groups of analytical methods are shown: non-perturbation and perturbation techniques. The following asymptotic methods are considered: (1) the straightforward expansion method, (2) the parameter-expanding method (the elliptic Lindstedt-Poincaré method), (3) the generalized averaging method, (4) the parameter perturbation method (elliptic Krylov-Bogolubov method), (5) the elliptic harmonic balance method, (6) the elliptic Galerkin method (the weighted residual method), (7) the homotopy perturbation method and (8) the homotopy analysis method. For all of the methods presented the common factor is the generating solutions of the differential equations which describe the free or harmonically forced oscillations of the Duffing oscillator. These are based on Jacobi elliptic functions. To illustrate the use of these methods, some examples are given. To assess the accuracy of the approximate analytical solutions, they are compared with numerical solutions. It is shown that the analytical results obtained are in good agreement with the solutions from numerical integration even for the cases when the nonlinearity and/or the excitation force are not small. © 2011 John Wiley & Sons, Ltd. All rights reserved.