Linearization and Modal Problems in Multibody System Dynamics for Modern Wind Turbines

Wind energy, as a primary source of renewable energy, plays a crucial role in achieving global carbon neutrality goals. To enhance cost-efficiency, modern wind turbines are being scaled to larger sizes. However, this trend introduces significant challenges related to aeroelastic instability, which is increasingly evident in turbines with slender blades. The eigenvalue analysis method, based on linear aeroelastic models, is widely used to investigate the modal dynamics of wind turbines. While custom simulation tools have been developed to predict aeroelastic stability, discrepancies in aeroelastic damping ratios across different tools persist. Additionally, field validation using full-scale wind turbines under normal operational conditions remains either insufficient or lacking. This study investigates the problem of aeroelastic stability assessment in modern wind turbines using the eigenvalue approach. A linearization and modal analysis framework is developed within the context of multibody system dynamics, with a particular focus on its application to the modal dynamics of wind turbines. A comprehensive verification and validation procedure is proposed to demonstrate the effectiveness of the developed method. The slender blades of wind turbines are discretized using a generic tapered Timoshenko beam element, combined with the corotational formulation to capture geometric nonlinearities during large deflections. Rotor aerodynamics are computed using the classical blade element momentum method. The equations of motion for the wind turbine are derived using Newton-Euler equations, with mechanical joints modeled as algebraic constraints based on quaternion parametrization. Analytical linearization of the inertial, elastic, viscous, and algebraic terms is performed in the local reference frames at the rotor’s rotating center, while numerical differentiation is used to linearize the steady aerodynamic forces in the quasi-static equilibrium configuration. The various terms are assembled in consistent coordinate systems, yielding a set of linearized differential-algebraic equations. Multi-blade coordinate transformations are applied to both generalized coordinates and Lagrange multipliers, resulting in linear time-invariant systems that are suitable for eigenvalue analysis. An efficient sparse-preserving Krylov-Schur eigenvalue solver is employed to solve the eigenvalue problem, enabling the subsequent investigation of the wind turbine’s modal dynamics. Aeroelastic damping predictions are verified through numerical simulations using a series of virtual models, which are generated by scaling the blade torsional stiffness. A strong correlation between the damping ratio and the blade edgewise vibration level is observed in the numerical experiments. Field validation is performed on two prototype wind turbines, where the unstable wind speed range observed in the blade edgewise vibration closely aligns with the region of predicted negative damping. The verification and validation analyses confirm the applicability of the developed linearization and modal analysis framework for assessing the aeroelastic stability of wind turbines in engineering practice. A unified analytical expression for the tangent stiffness matrix of holonomic constraints is derived for the first time. Numerical examples highlight the importance of this matrix in ensuring proper convergence of static equilibrium analysis and in achieving accurate eigenvalue results for multibody systems with free-motion joints under stressed equilibrium conditions. As a complementary effort, a modal reduction procedure is developed for rotating subsystems comprising rigid bodies, finite elements, and mechanical joints, within the context of multibody system dynamics. The accuracy and effectiveness of this procedure are verified through a series of academic numerical examples. This work establishes an effective framework for linearization and modal analysis in multibody system dynamics, advancing the aeroelastic stability assessment of wind turbines and providing a foundation for future engineering applications and research.