Abstract
An investigation is presented of the response ofthree-degree-of-freedom system with cubic non-linearities andauto-parametric resonance ω3~=3ω2 andω2~=3ω1 to a harmonic excitation ofthe third mode, where the ωn are the linearnatural frequencies of the system. The method of multiple scale isapplied to determine six first-order non-linear ordinary differentialequations that govern the time variation of the amplitudes and phasesof the interacting modes. The fixed points of the three equations areobtained and their stability are determined. Numerical solutions areconducted to obtain the response of the three modes. The effects ofthe different parameters on both response and stability of the systemare investigated. The obtained results are discussed followed by themain conclusions.
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