Mathieu instability of Arctic Spar by nonlinear time-domain simulations

Abstract Arctic Spar typically has a sloped hull surface near the waterline to break ice through bending. However, the rapid variation of its waterplane area by the heave motion may cause unexpected nonlinear behaviors, which can result in instability. In the current study, the heave-to-pitch/roll and heave-to-heave Mathieu instability (MI) of Arctic Spar are investigated by using a nonlinear time-domain simulation program. The time-dependent nonlinear hydrostatic-restoring coefficient (HC) and nonlinear Froude-Krylov (FK) force are considered for the instantaneous wetted body underneath the instantaneous wave elevation. The effects of the two nonlinear terms for the generation of heave-to-pitch and heave-to-heave MI are systematically compared and discussed by employing four different approaches including conventional linear theory. Analytical equations of heave and pitch are also used with Hill's infinite determinants method to determine the boundaries of Mathieu's equation parameters where instability occurs. Through a series of numerical simulations, the occurrences of both heave-to-pitch and heave-to-heave MI have been clearly demonstrated both in regular and irregular waves. The simulated results agree well with the theoretical instability diagram. The importance of the combined effects of nonlinear FK forces and nonlinear HC in triggering pronounced heave-to-heave MI is underscored especially in the case of 100-year storm condition.