Numerical study on 2:1 nonlinear parametric resonant responses of a Spar buoy in waves

Abstract

This paper investigates the 2:1 parametric resonant behaviors of a Spar buoy in waves from the perspective of nonlinear dynamics. A coupled dynamic model of an experimental Spar buoy is established, incorporating its mooring system and nonlinear hydrostatic stiffness. The 2:1 parametric resonant behaviors of the Spar buoy are simulated in the time domain. To elucidate the mechanical mechanisms, the governing equation of pitch motion is simplified to a damped Mathieu-Duffing equation. The first-order analytical solution of the damped Mathieu-Duffing equation is derived using the perturbation method when the incident wave frequency approaches twice the pitch natural frequency of Spar buoys. A refined stability chart is generated in the parameter plane, alongside a time-efficient quantitative evaluation method for parametric resonant responses. The nonlinear pitch motions of the Spar buoy are predicted using this simplified approach and validated against the coupled dynamic model under both regular and irregular wave conditions, the bifurcation and jumping phenomenon are simulated. The findings indicate that high-order nonlinear stiffness can trigger a steady-state non-zero solution for the parametric resonant amplitude, leading to significant pitch motion during Mathieu instability. Furthermore, wave elevation can induce pitch motion resonance even when heave motion is non-resonant. The irregular waves can also excite a relatively moderate parametric resonance of pitch motion for the Spar buoy. This proposed methodology may assist designers in assessing parametric instability during the preliminary design stage for Spar buoys.