Multi-Mode Motion Dynamics of a Top Tensioned Riser Excited by Vortices and Varying Tension

A coupled model of a top tensioned riser (TTR) under combined excitations from vortices and time-varying tension is proposed. And the van der Pol oscillator is introduced to simulate the time-varying characteristics of vortex shedding. The first three mode approximations as a six degree-of-freedom (DOF) system of vibrations of the fluid-structure interaction system are derived. Two cases of resonances are analyzed by using the numerical method. One case is on the vortex induced vibration (VIV) of the six DOF system, where the 1-to-1 internal resonance (also named "lock-in" phenomenon) between frequencies of the first modes of structure and the wake oscillator are investigated. Bifurcation curves between dynamic responses in terms of the shedding frequency are derived. Comparing to the vibrations of the first order mode, the "lock-in" phenomenon for vibrations of the higher order modes can occur in the wider parameter range with large values. The second case is on the combined resonance between the 1-to-1 internal and 1-to-2 parametric resonance. Response curves versus the shedding frequency under different amplitude ratios are calculated by the Runge-Kutta method. Effects of system parameters including the amplitude ratio of the parametric excitation, shedding frequency on the response curves are investigated. Comparing to the case when only considering the 1-to-1 internal resonance in the VIV, results of the combined resonance show those large-amplitude responses of the structure and wake oscillator can be induced. Responses will increase rapidly as the amplitude of the varying tension increases. Unstable responses can occur within a large frequency band of nondimensional shedding frequency. Furthermore, the dynamic characteristics of the responses of the six DOF system including the bifurcation diagrams, time histories, phase-plane projection and Poincare sections are computed for different amplitude ratios. Results indicate that complex dynamics such as the quasi-periodic and chaotic solutions can occur for different amplitude ratios. The analysis can be helpful for the parameter design of deep-water risers.