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

Abstract

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.

Qualitative yaw stability analysis of free-yawing downwind turbines

“…This restoring yaw moment can be increased with an increasing cone angle, as the combination of cone and yaw angles creates a favourable wind speed projection and therefore increases the yaw stiffness as predicted by Eggleston and Stoddard (1987). This result confirms the observations of the measurements from, for example, Verelst and Larsen (2010) and Kress et al (2015).…”

“…To evaluate the dynamic stability of the free-yaw mode, a simple 2-DOF model is set up in Maple software (MapleSoft, version 2016.2). The 2 degrees of freedom (2-DOF) model is based on an existing 15-DOF model without cone angle, described by Hansen (2003) and Hansen (2016). The 2 degrees of freedom are the tower side-side motion (u x (t)) and the free-yaw motion (θ (t)).…”

“…Since the blade was passively unloaded at higher wind speeds and the inflow condition due to the position of the blade segments differed along the blade due to deformation, the different blade segments were subject to stall at different wind speeds. Kress et al (2015) used a scaled model of a commercial 2 MW downwind turbine to compare the restorative yaw moment in a water tunnel in a downwind and upwind configuration. They compared the influence of different cone angles, different yaw angles, and different tip speed ratios close to optimum tip speed ratio.…”

Fatigue and extreme load reduction on two‐bladed wind turbines using the flexible hub connection

“…Regarding time‐variance, it was pointed out by Skjoldan and Hansen as well as Cacciola that due to the rotation of the rotor with respect to a flexible but stationary support structure, every principal eigenmode possesses an infinite number of associated “fan” modes shifted in frequency by . It was shown for two‐bladed turbines by numerical analysis (Hansen and Kim et al) that these fan modes can have a nonneglectable contribution compared with the principal mode. The identification of the principal and the fan modes of time‐variant systems requires the use of periodic analysis methods, such as the implicit Floquet analysis or Hill's method, which can be computationally expensive for high‐fidelity models.…”

“…It was shown for two‐bladed turbines by numerical analysis (Hansen and Kim et al) that these fan modes can have a nonneglectable contribution compared with the principal mode. The identification of the principal and the fan modes of time‐variant systems requires the use of periodic analysis methods, such as the implicit Floquet analysis or Hill's method, which can be computationally expensive for high‐fidelity models. As the effects described above only lead to mild modifications of the dynamic system properties within the operational range of the turbine, the standstill eigenfrequencies are a valid indicator to analyze the modal sensitivity and the time‐variance of the system properties.…”