Analysis of Wind Turbine Blade Motion: Linear Formulation Including All Aeroelastic Load Couplings

A wind turbine blade similar to a helicopter rotor blade has the structure of a pretwisted beam of a variable airfoil asymmetrical cross-section. A number of approximate theories have been developed by different researchers to study the dynamic behavior of the blade of a horizontal axis wind turbine. Some researchers include warping, but they do not include the blade’s pretwisting. Others include the axial and torsional loadings and the coupling among these loadings but they ignore the bending loading. The new contribution in this study is the consideration of all the extensional, torsional and flexural loadings with their couplings, variable airfoil cross sections with warping effects, shear deflection, rotary inertia and with or without blade’s pretwist to obtain a more accurate dynamic model. To the best knowledge of the authors the simultaneous inclusion of all these factors has not been done before. The aerodynamic loadings (lift, drag and pitch moment) were calculated at each time step for a 14m blade that has a linear decreasing NACA4415 airfoil cross section utilizing a time dependent set of parameters such as angle of attack, material and air density, wind and blade speed, flow angle, yaw, pitch angles. Assuming that deformation is small, the total strain energy and total kinetic energy and external work due to the aerodynamic loading acting on the blade were calculated and used in the Lagrange equations of motion where we obtained the stiffness, mass and damping matrices of the linear dynamic equations of motion. Then the unknown displacements and rotations u, v and w in the directions of x, y and z axes respectively, the bending rotations θ₁, θ₂ about the y and z axes respectively and the torsional rotation ϕ about the x axis, were solved using the Newmark implicit iteration scheme.