Abstract
Abstract. 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.
Highlights
20 Vibration of an elastic system refers to a limited reciprocating motion of a particle or an object of the system
When the excitation frequency of the vibrating system is near any natural frequency, the undesirable resonant state occurs with large amplitudes, which may lead to damage or even collapse of the wind turbine or 25 its components
Both static and kinetic field matrices for the beam element are derived by applying in a special way a RUNGE KUTTA 4th order numerical integration procedure on the differential equations of motion
Summary
20 Vibration of an elastic system refers to a limited reciprocating motion of a particle or an object of the system. Wind turbines operate in an unsteady environment which results in a vibrating response (Manwell, McGowan, & Rogers, 2009) They consist of long slender structures (rotor blades and tower), of which resonant frequencies should be taken into account during the initial design and construction. The eigenmodes associated to the lowest natural frequencies are employed as shape functions to describe the elastic deformation of the rotor blade beam model in the frame of the usual simulation of the wind turbine as a multi-body system. Using the above standard relations and appropriate shape functions for the Euler-Bernoulli beam and Timoshenko beam, the stiffness matrix and consistent mass matrix for the finite beam element can be derived. The system stiffness and mass matrices for the rotor blade are assembled in a global coordinate basis in the usual finite element manner. Γy (4a-c) where, γz and γy are the constant shear strains which are not neglected in Timoshenko beam theory
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