Closed-form solution to multi-mode aerodynamic damping of monopile-supported offshore wind turbines

This paper presents an explicit solution to multi-mode aerodynamic damping of monopile-supported offshore wind turbines (OWTs), focusing on the first three fore-aft and side-side modes of the tower. The analytically derived aerodynamic damping matrix enables rapid quantification of aerodynamic coupling effects and modal damping of the OWT system at any operational point. A 14-degree-of-freedom OWT model is developed, incorporating essential features such as aeroelasticity, pitch control, and mechanical coupling. Second and third tower vibration modes in each direction are introduced to reveal multi-mode aerodynamic coupling effects. Comparative analyses with nonlinear OWT simulations confirm that the model employing linearized aerodynamic loads maintains high fidelity and robustness. Leveraging the linearized model, the wind speed dependence of multi-mode aerodynamic coupling and damping ratios of the OWT system is rigorously investigated. Findings highlight that tower top rotation plays a decisive role in generating multi-mode aerodynamic coupling. Excluding blade flexibility, tower top rotation, or aerodynamic coupling terms results in varying degrees of inaccuracy in evaluating multi-mode damping ratios, whereas the omission of hydrodynamic added mass matrix is of minimal consequence. In both the fore-aft and side-side directions, the aerodynamic damping of the first tower mode is significantly higher than that of the second and third modes. The third side- side mode exhibits negative aerodynamic damping ratios throughout the entire operational range, indicating a potential instability issue.