Nonlinear Flapwise Modes Characteristics of a Long Rotating Blade

The rapid development of large-scale horizontal-axis wind turbines (HAWTs) has increased the demand for accurate prediction of flapwise vibration under complex aerodynamic conditions. This study establishes a nonlinear flapwise dynamic model incorporating geometric nonlinearity, centrifugal stiffening, and aerodynamic damping. The governing equations are derived using Hamilton’s principle and solved via the Galerkin method, with mass and stiffness orthogonality verified. Model accuracy is demonstrated through good agreement between assumed-modes and finite-element results. Using the multi-scale method, nonlinear modal characteristics under different damping conditions are investigated, revealing frequency shifts, nonlinear hardening, and aerodynamic energy dissipation. Numerical analyses show that rotational speed raises natural frequencies through centrifugal stiffening, geometric nonlinearity further enhances effective bending stiffness, and aerodynamic damping leads to frequency softening and vibration attenuation. The proposed framework provides a reliable theoretical basis for predicting nonlinear vibration and optimizing the structural design of large rotating blades.