Nonlinear oscillations of floating offshore wind turbines under wave excitation

An approximate planar two-Degrees-Of-Freedom (DOF) analytical model is developed to investigate the nonlinear dynamic response of a Floating Offshore Wind Turbine (FOWT), mounted on a spar platform, under parametric excitations induced by the periodic variation of the buoyancy force due to sea waves. The equations of motion, studied under the main parametric resonance (PR) condition, account for the linear and nonlinear hydrostatic stiffnesses and their squares as explicit time-dependent functions of the free-surface elevation. The main novelties are: (i) a rigorous derivation of the wave-elevation-dependent linear and nonlinear hydrostatic stiffnesses, retaining the associated higher-harmonic parametric components; (ii) a fifth-order Multiple Time Scales Method (MTSM) analysis of the primary 2:1 parametric resonance, predicting small-amplitude resonant period-2 oscillations emerging from subcritical and supercritical Hopf bifurcations of the rest position; (iii) a complementary Harmonic Balance Method (HBM) study that reveals, beyond standard low-order formulations, a detached and non-resonant family of large-amplitude period-1 solutions; and (iv) a basin-based assessment of practical pitch stability. Numerical continuation, direct time-domain integrations, and basin-of-attraction computations confirm the unexpected coexistence between the classical small-amplitude resonant period-2 response and the detached non-resonant period-1 oscillation at finite and large amplitude. Remarkably, the period-1 attractor can exhibit wide basins of attraction even when the rest state is locally stable and within damping, wave-amplitude, and frequency ranges commonly considered acceptable in design practice, thus turning a seemingly safe operating point into a practically vulnerable one. These results demonstrate that conventional linear (and low-order nonlinear) analyses may systematically underestimate the risk of wave-induced large-amplitude pitch motions in spar-type FOWTs, and they provide a lightweight analytical framework together with basin-based robustness indicators for early-stage design and control screening.