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A coupling model between turbine blades with a varying rotating speed and oncoming vortices is constructed, where the coupling of the structure and the fluid is simulated by the van der Pol oscillation. Partial differential governing equations of motions for the coupled system are obtained and discretized by using the Galerkin method. The 1:2 subharmonic resonance and the 1:1 internal resonance are investigated with the multiple scale method and first-order averaged equations are then derived. Nonlinear responses and bifurcation characteristics are studied by a numerical integration method. Stability of bifurcation curves is determined by utilizing the Routh-Hurwitz criterion. The effect of system parameters including the detuning parameter, steady-state rotating speed, amplitude of periodic perturbation for the rotating speed and freestream velocity on vibration responses are investigated.

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A low dimensional model of a top tensioned riser (TTR) under excitations from vortices and timevarying tension is proposed, where the van der Pol wake oscillator is used to simulate the loading caused by the vortex shedding. The governing partial differential equations describing the fluid-structure interactions are formulated and multi-mode approximations are obtained using the Galerkin projection method. The one mode approximation is applied in this study and two different resonances are investigated by employing the method of multiple scales. They are the 1:1 internal resonance between the structure and wake oscillator (also known as 'lock-in' phenomenon) and the combined 1:1 internal and 1:2 parametric resonances. Bifurcations under the varying nondimensional shedding frequency for different mass-damping parameters are investigated and the results of multiple scale analysis are compared with direct numerical simulations. Analytical responses are calculated using the continuation method and their stability is determined by examining the eigenvalues of the corresponding characteristic equations. Effects of the system parameters including the amplitude of the tension variation, vortex shedding frequency and mass-damping parameter on the system bifurcations have been investigated. The analytical approach has allowed to probe bifurcations occurring in the system and to identify stable and unstable responses. It is shown that the combined resonances can induce large-

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