Articles | Volume 1, issue 2
Wind Energ. Sci., 1, 271–296, 2016
Wind Energ. Sci., 1, 271–296, 2016

Research article 30 Nov 2016

Research article | 30 Nov 2016

Modal dynamics of structures with bladed isotropic rotors and its complexity for two-bladed rotors

Morten Hartvig Hansen Morten Hartvig Hansen
  • Dept. of Wind Energy, Technical University of Denmark, Frederiksborgvej 399, 4000 Roskilde, Denmark

Abstract. The modal dynamics of structures with bladed isotropic rotors is analyzed using Hill's method. First, analytical derivation of the periodic system matrix shows that isotropic rotors with more than two blades can be represented by an exact Fourier series with 3/rev (three per rotor revolution) as the highest order. For two-bladed rotors, the inverse mass matrix has an infinite Fourier series with harmonic components of decreasing norm; thus, the system matrix can be approximated by a truncated Fourier series of predictable accuracy. Second, a novel method for automatically identifying the principal solutions of Hill's eigenvalue problem is introduced. The corresponding periodic eigenvectors can be used to compute symmetric and antisymmetric components of the two-bladed rotor motion, as well as the additional forward and backward whirling components for rotors with more than two blades. To illustrate the use of these generic methods, a simple wind turbine model is set up with three degrees of freedom for each blade and seven degrees of freedom for the nacelle and drivetrain. First, the model parameters are tuned such that the low-order modal dynamics of a three-bladed 10 MW turbine from previous studies is recaptured. Second, one blade is removed, leading to larger and higher harmonic terms in the system matrix. These harmonic terms lead to modal couplings for the two-bladed turbine that do not exist for the three-bladed turbine. A single mode of a two-bladed turbine will also have several resonance frequencies in both the ground-fixed and rotating frames of reference, which complicates the interpretation of simulated or measured turbine responses.

Short summary
The modal dynamics of wind turbines are the fingerprints of their responses under the stochastic excitation from the wind field. Commercial wind turbines have typically three-bladed rotors, and their modal dynamics are well understood. Two-bladed turbines are still commercially less successful, and this work also shows that their modal dynamics are significantly more complex than that of turbines with three or more blades.