In this paper three-mode interaction in harmonically excited system with cubic nonlinearities to harmonic excitations in the presence of auto-parametric resonances ω2≅3ω1 and ω3≅3ω2 is investigated in two cases. First case, when the amplitudes of external excitation are of order ε. Second case, when the amplitudes of external excitation are of order one. The method of multiple scales is used to determine a first-order nonlinear ordinary differential (average) equations governing the modulation of the amplitudes and phases of three modes. Numerical solutions are conducted to obtain the frequency curves of the three modes. The effect of the different parameters on both response and stability of the system is investigated. The obtained results are discussed followed by the main conclusions.

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Three-mode interaction in harmonically excited system with cubic nonlinearities - ResearchGate. Available from: http://www.researchgate.net/publication/242791270_Three-mode_interaction_in_harmonically_excited_system_with_cubic_nonlinearities [accessed Jul 27, 2015].