In-Plane Nonlinear Dynamics of Wind Turbine Blades

Abstract

The partial differential equation that governs the in-plane motion of a wind turbine blade subject to gravitational loading and which accommodates for aerodynamic loading is developed using the extended Hamilton principle. This partial differential equation includes nonlinear terms due to nonlinear curvature and nonlinear foreshortening, as well as parametric and direct excitation at the frequency of rotation. The equation is reduced using an assumed cantilevered beam mode to produce a single second-order ordinary differential equation (ODE) as an approximation for the case of constant rotation rate. Embedded in this ODE are terms of a nonlinear forced Mathieu equation. The forced Mathieu equation is analyzed for resonances by using the method of multiple scales. Superharmonic and subharmonic resonances occur. The effect of various parameters on the response of the system is demonstrated using the amplitude-frequency curve. A superharmonic resonance persists for the linear system as well.