Control of nonlinear oscillations of a buoyant pendulum subjected to surface gravity waves via time-delay feedback

This study investigates the effectiveness of time-delayed feedback control (TDFC) in controlling the oscillatory behavior of a fully submerged buoyant pendulum subjected to surface gravity waves. A theoretical model is developed, incorporating a nonlinear ordinary differential equation to describe the transverse motion of the pendulum, driven by a linearized drag force induced by small-slope water waves. The dynamic response of the pendulum is analyzed using time-series data and frequency spectrum analysis, highlighting primary resonance and subharmonic response features. Analytical solutions are derived using the method of multiple scales and validated through numerical simulations, capturing the nonlinear characteristics of the system. The stability of the analytical stationary solutions is assessed via eigenvalue analysis of the Jacobian matrices. The influence of state feedback control with time delay on the system response is systematically examined. For fixed position and velocity gains, varying the time delay reveals that oscillations can be suppressed or amplified, depending on the chosen delay. This demonstrates the potential for optimizing control parameters to achieve specific outcomes, such as vibration suppression for onshore structures or enhancement for wave energy converters. The results confirm that TDFC provides an effective strategy for modulating the response of the pendulum, offering a robust framework for managing its dynamic behavior under hydrodynamic forcing.