Improvement to the Elliptic Averaging Method Using the Sum of Jacobian Elliptic Cosine and Sine Functions as Generating Solution

Abstract

The present paper describes improved version of the elliptic averaging methods that provides a highly accurate approximate solution of a strongly nonlinear system based on the Duffing oscillator with hardening, softening or snap through spring. In the proposed method, the sum of the Jacobian elliptic cosine (cn) and sine (sn) function is incorporated as the generating solution of the averaging method. The proposed method can be used to obtain the odd-order solution, which includes odd-order harmonic component. The generation solution of the proposed method has six unknown constants, amplitude Ai , phase angle θi and modulus ki , (i=1,2) . Since the generating solution of the proposed method has multiplicity to satisfy the condition to determine six unknown constants, the proposed method has a many-valued property whereby several approximate solutions for one steady-state solution exist. In order to select the most accurate solutions, the method to evaluate the accuracy of all the solutions for one steady-state solution is described. The stability of the solution is analyzed by obtaining the characteristic multipliers of the variational equation. The method's validity is verified by comparing the results obtained by applying the proposed method to two nonlinear oscillators with the very accurate numerical solutions computed by the shooting method. The numerical results confirmed that the proposed method provides a more accurate solution than that obtained by the former elliptic averaging methods.