Linear and non-linear deformations of a wind turbine blade considering warping and all aeroelastic load couplings

摘要

The structural dynamics behavior of the blade of a horizontal axis wind turbine that reacts to the different components of the aerodynamic loading were studied by many researchers using different approaches and assumptions. In the present research, the author considered all the extensional, torsional and flexural loadings acting on the blade with their couplings, variable airfoil cross sections with warping effects, shear deflection, rotary inertia and with or without blade's pretwist for both the linear small deformation case and the nonlinear large deformation case. To the best knowledge of the author the simultaneous inclusion of all these factors has not been done before. The "assumed modes method" was used, in which displacements are assumed to be an expansion of products of time-step dependent constants and polynomial functions of x (where x is the coordinate along the length of the blade) that satisfy the boundary conditions at the fixed end where x=0 (hub of the blade) and at the free end where x=L (tip of the blade). The mass matrix, linear and nonlinear stiffness matrices and the load vector (function of time step) of the dynamic equations of motion are deduced from the Lagrange equations of motion that were derived step by step. The steps of the linear and nonlinear Newmark implicit iteration schemes used for solving the linear and nonlinear dynamic equations of motion respectively were explained in detail. Numerical implementation examples for both linear and nonlinear cases were demonstrated for a 14m long blade with and without pretwisting that has specific material and geometrical properties and a decreasing NACA4415 airfoil cross section from hub to tip. For both of the linear and nonlinear examples, the aerodynamic loadings (lift, drag and pitch moment) and the nonlinear stiffness matrices were computed at each time step utilizing a time dependent set of parameters such as angle of attack, material and air density, wind and blade speed, flow angle, yaw and pitch angles. Then the unknown displacements u, v and w in the directions of x, y and z axes respectively, the bending rotations theta 1 and theta 2 about the y and z axes respectively and the torsional rotation phi about the x axis, were solved using the linear and nonlinear Newmark implicit iteration schemes. The linear case displacement result plots are shown to agree with the work of Younsi et al. The nonlinear case displacement result plots are shown to agree with the Ls-Dyna code.