Three-to-one internal resonance of inclined marine riser

In this paper, we investigate the three-to-one internal resonance of an inclined marine riser. The model accounts for the flexural rigidity of the riser, variable axial load, nonlinear geometry, and initial static deflection due to self-weight. The applied tension with the static deflection are tuned such that the ratio between the fifth and first natural frequencies is three. Then, the multiple time scales (MTS) perturbation method is applied to study the internal resonance interactions when the structure is harmonically excited. The system equation is solved using a multi-mode Galerkin model and its results are compared to the perturbation results showing good agreement. Moreover, the frequency response curves of the fifth and first modes amplitudes exhibit Hopf and saddle node bifurcations. In addition, the interaction of the fifth mode with the first mode during internal resonance results into new emerging solutions and states. These phenomena are well observed in the force response curves and confirmed by the time history of the response of the structure, which can lead to complex dynamics that hinder the life of the riser by fatigue failure.