Some Considerations on Higher Order Approximation of Duffing Equation in the Case of Primary Resonance

Here, the higher order approximation of forced Duffing equation is studied. First, using the renormalization group method, the modulation equations of Duffing equation in the case of primary resonance is determined. The resulting modulation equations are identical with those previously obtained by method of multiple scales and generalized method of averaging. Second, the periodic steady state behavior of the solutions and the problem of spurious solutions in higher order approximation are considered. It is shown that depending on the truncation method of original phase and amplitude modulation equations, two types of frequency response equation may be obtained. One possesses spurious solutions for the case of softening nonlinearity and the other for the case of hardening nonlinearity. Furthermore, it is shown that the truncation of thefrequency response equation do not necessarily lead to more accurate results. Finally, by application of root classification of polynomials and Descartes'' rule of signs, a criterion is presented to detect the existence of spurious solutions in any point of frequency response equation without solving it. This method is also applicable to other nonlinear systems.