Three-to-one internal resonances in a curved beam resting on an elastic foundation

Transverse vibrations of curved beams are investigated. Different curvatures are considered. The beam is simply supported and resting on a nonlinear elastic foundation. The method of multiple scales is used in the analysis. A three-to-one internal resonance case is studied. It is possible when one of the mode numbers is three times the other mode. Amplitude and phase modulation equations are obtained. Steady state solutions and stability are discussed. A three-to-one internal resonance is possible for all types of curvatures under the assumptions in the article.