Six-Degree-of-Freedom Rotor-Load Cross-Spectral Density from Rotor-Plane Turbulence Cross-Spectra

Abstract

A frequency-domain response analysis of a wind turbine rotor requires the complete hub-load spectrum, but the customary scalar thrust and torque spectra retain neither the four in-plane components nor the coupling between them. This paper derives the full six-degree-of-freedom (6-DOF) rotor-load cross-spectral-density (CSD) matrix from the IEC Kaimal turbulence model. By decoupling the spatial order of inflow coherence from the temporal Doppler shift, the formulation introduces a two-index rotational-sampling kernel. Rotor symmetry and azimuthal averaging constrain the observed Doppler carrier strictly to blade-passing harmonics $pB{f}_{\mathrm{rot}}$, while the azimuthal aerodynamic transfer shifts the spatial coherence order to $pB+s$. A convergence study of the retained azimuthal order $s$ shows that $|s|\le 4$ suffices, the assembled CSD then deviating from the full-band result by less than $10^{-5}$ in relative norm. The formulation is compared with OpenFAST simulations of a rigid floating wind turbine over nine wind speeds: the six auto-spectral densities agree closely below 1~Hz, and the present theory over-predicts the spectra near the 6P and 9P rotational harmonics.