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

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.

Flexural vibration of multistep rotating Timoshenko shafts using hybrid modeling and optimization techniques

“…However, no general analytical and exact solution is provided for the Timoshenko beam equation, except for very restricted cases and limited boundary conditions (Timoshenko et al, 1974; Rao, 2018). The investigation of existing literature indicates that the dynamic problems of rotor-bearing systems are mainly resolved by either the step-by-step integration process (Stanoev and Chandrashekhara, 2019), the analytical method (Wu et al, 2014; Rao, 2018), the transfer matrix method (TMM) (Deepthikumar et al, 2014), the assumed mode method (Masoud and Alsaid, 2009), the frequency-dependent TMM (Torabi et al, 2017), or the finite element method (FEM) (Ganguly et al, 2018). For example, the 3-D solid finite element method including bending, torsional and axial motions is developed by Yu et al (1999) to analyze the dynamic responses of shafts.…”

Nonlinear Flap-Wise Vibration Characteristics of Wind Turbine Blades Based on Multi-Scale Analysis Method

“…The blade is simplified as an Euler beam model, which can be solved by using the finite element method, the vibration pattern superposition method, and the direct integration method, but the vibration pattern superposition method for modal analysis of the kinetic equations is only applicable to linear calculations [4]. Hamilton's principle is used to represent the nonlinear kinematics of the blade; under the internal resonance condition, the equations may be analytically solved by the multiscale technique, and the accuracy of the perturbation method and the numerical solution can be compared [5][6][7].…”