Key words Modal analysis in structural dynamics, modal decomposition of the equations of motion, non-modal damping matrix, non-proportional damping, complex natural modes, complex left and right eigenvectors, quadratic eigenvalue problem.A general method for the modal decomposition of the equations of motion of damped multi-degree-of-freedom-systems is presented. The first variant of the presented method in earlier publications of the author, including the complex right eigenvectors, is briefly reviewed first. The second presented variant is also based on the corresponding eigenvalue problem of the damped structure including the complex left and right eigenvectors. After initial partitioning of the equations of motion a real modal transformation matrix is built by a combination of two complex transformations. For the general case of damped structures with non-modal symmetric damping matrix a modal analysis can be performed in real arithmetic. Two numerical examples with 3 and 10 DOF's demonstrate the accuracy and the advantages of the presented modal solution method.

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.

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Keywords: modal decomposition of the equations of motion, modal transformation method, left and right complex eigenvectors, complex eigenvalue problem, proportional/nonproportional damping. Abstract. In the general case of non-proportionally damped

Abstract. In the simulation of a wind turbine, the lowest eigenmodes of the rotor blades are usually used to describe their elastic deformation in the frame of a multibody system. In this paper, a finite element beam model for the rotor blades based on the transfer matrix method is proposed. Both static and kinetic field matrices for the 3D Timoshenko beam element are derived by numerical integration of the differential equations of motion using RUNGE KUTTA 4th order procedure. The beam reference axis in the general case 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 with the accounting for structural twist angle and possibly pre-bending. The program developed for the above calculations and the final solution of the eigenvalue problem is accomplished using MuPAD, a symbolic math toolbox of MATLAB®. The calculated natural frequencies using generic rotor blade data are compared with the results proposed from FAST and ADAMS software.