Modelling and simulation of periodic dynamics of submarine cables with bending stiffness using time-domain and multi-harmonic balance methods

Abstract This paper presents a comprehensive investigation of the nonlinear dynamic response of a submarine cable subjected to a harmonic fairlead excitation, explicitly incorporating bending stiffness, geometric extensibility and hydrodynamic drag. The governing equations form a coupled, nonlinear system of partial differential equations (PDEs) derived in a Lagrangian reference frame, resolved along the local tangential and normal directions of the cable. A formal asymptotic dispersion analysis establishes the existence of spatially periodic solutions under varying axial tension, providing theoretical justification for sustained travelling waves along the cable. To simulate the system response, two complementary numerical methods are developed: a high-resolution explicit Runge–Kutta time-domain (TD) integrator and a frequency-domain multi-harmonic balance method (MHBM). The dynamic behaviour is visualized through animated reconstructions of the cable profile and the corresponding velocity field in physical coordinates. A comparison of the results demonstrates excellent agreement between both solvers in terms of fairlead tension. Parametric investigations show that low flexural rigidity promotes long-range wave propagation, whereas increasing stiffness leads to motion localization near the excitation point. In addition, background ocean current, introduced through relative velocity in the drag formulation, significantly modifies the amplitude and symmetry of the dynamic tension response.