On methods for continuous systems with quadratic, cubic and quantic nonlinearities

Methods for study of weakly nonlinear continuous systems are discussed. The method of multiple scales is used to analyze the nonlinear response of a relief valve under combined static and dynamic loadings. We determine a second-order approximation to the response of the system for the case of primary resonance. Second, we derive a second-order nonlinear ordinary differential equation that describes the time evolution of a single-mode, the so-called single-mode discretization. Then, we use the multiple scales method to determine second-order approximate solutions of this equation, thereby obtaining the equations describe the modulations of the amplitude and phase of the response. We show that the results of the second approach are erroneous.