One of the most efficient ways of solving the weakly nonlinear - undamped- free duffing equation is Perturbations method. In this method, the response of the system (u) consists of two parts for first order expansion: response due to the linear case (u0) and a correction response due to the nonlinear case (u1). When, applying the Lindstedt - Poincare' method or Multiple scales method a solution which comprises both of the mentioned solutions could be obtained. On the other hand, in the Averaging Method, based on the assumptions of Method of Variation of Parameters, the correction response (u1) could not be emerged in the final solution. Are the results of Averaging method as accurate as Multiple Scales Method?? Why there is no such correction term?? Could it be imported??
I would be appreciated by your responses
It would be nice to see your calculations in detail. However, although not seen, I think that if the nonlinear dependency of the oscillator frequency on the system nonlinearity (via elliptic integrals) is properly taken care of, corrective terms in the final solution would appear. The details of the involved steps are presented in the following book:
“Introduction to Perturbation Techniques”
By ALI HASAN NAYFEH.
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