Determination of natural frequencies and mode shapes of a wind turbine rotor blade using Timoshenko beam elements

When simulating a wind turbine, the lowest eigenmodes of the rotor blades are usually used to describe their elastic deformation in the frame of a multi-body system. In this paper, a finite element beam model for the rotor blades is proposed which is based on the transfer matrix method. Both static and kinetic field matrices for the 3-D Timoshenko beam element are derived by the numerical integration of the differential equations of motion using a Runge–Kutta fourth-order procedure. In the general case, the beam reference axis is at an arbitrary location in the cross section. The inertia term in the motion differential equation is expressed using appropriate shape functions for the Timoshenko beam. The kinetic field matrix is built by numerical integration applied on the approximated inertia term. The beam element stiffness and mass matrices are calculated by simple matrix operations from both field matrices. The system stiffness and mass matrices of the rotor blade model are assembled in the usual finite element manner in a global coordinate system accounting for the structural twist angle and possible pre-bending. The program developed for the above-mentioned calculations and the final solution of the eigenvalue problem is accomplished using MuPAD, a symbolic math toolbox in MATLAB®. The natural frequencies calculated using generic rotor blade data are compared with the results proposed from the FAST and ADAMS software.