Improvement to the averaging method using the Jacobian elliptic function

Abstract

The averaging method was improved by using the Jacobian elliptic sine (sn), cosine (cn) and delta (dn) functions as generating solutions in order to obtain a highly accurate periodic solution for a strongly nonlinear dynamical system. The proposed method can be applied to a relatively general nonlinear system based on the single degree-of-freedom Duffing equation. Two methods of stability analysis were applied and evaluated for the approximate solutions obtained by the proposed method. The approximate solutions obtained by the proposed method for practical examples were compared to the solutions obtained by the shooting method. The results confirmed that the proposed method provides a more accurate solution and more accurate stability analysis results than those obtained by the conventional averaging method that uses trigonometric functions as the generating solution.