Three-to-one internal resonances in a general cubic non-linear continuous system

A general continuous system with an arbitrary cubic non-linearity is considered. The non-linearity is expressed in terms of an arbitrary cubic operator. Three-to-one internal resonance case is considered. A general approximate solution is presented for the system. Amplitude and phase modulation equations are derived. Steady state solutions and their stability are discussed in the general sense. The sufficiency condition for such resonances to occur is derived. Finally the algorithm is applied to a beam resting on a non-linear elastic foundation.