Some considerations on higher order approximation of Dung equation in the case of primary resonance

Here, the higher order approximation of forced dung equation is studied.first, using the renormalization group method, the modulation equations of dungequation in the case of primary resonance is determined. the resulting modulationequations are identical with those previously obtained by the method of multiple scales andgeneralized method of averaging. second, the periodic steady state behavior of the solutionsand the problem of spurious solutions in higher order approximation are considered.it is shown that depending on the truncation method of original phase and amplitudemodulation equations, two types of frequency response equation may be obtained. onepossesses spurious solutions for the case of softening nonlinearity, and the other for the caseof hardening nonlinearity. furthermore, it is shown that the truncation of the frequencyresponse equation do not necessarily lead to more accurate results. finally, by applicationof root classication of polynomials and descartes' rule of signs, a criterion is presentedto detect the existence of spurious solutions in any point of frequency response equationwithout solving it. this method is also applicable to other nonlinear systems.