The dynamics of wind turbine rotor blades are generally expressed in rotating frames attached to the individual blades. The rotor, however, responds as a whole to excitations such as aerodynamic gusts, control inputs, and tower-nacelle motion-all of which occur in a nonrotating frame. Similarly, the tower-nacelle subsystem sees the combined effect of all rotor blades, not the individual blades. Multi-blade coordinate transformation (MBC) helps integrate the dynamics of individual blades and express them in a fixed (nonrotating) frame. MBC involves two steps: transformation of the rotating degrees of freedom, and transformation of the equations of motion. This paper details the MBC operation. A new MBC scheme is developed that is applicable to variable-speed turbines, which may also have dissimilar blades. The scheme also covers control, disturbance, output, and feed-forward matrices. Depending on the analysis objective, wind turbine researchers may generate system matrices either in the first-order (state-space) form or the second-order (physicaldomain) form. We develop MBC relations for both these forms. MBC is particularly essential for modal and stability analyses. Commonly, wind turbine researchers first compute the periodic state-space matrix, time-average it over the rotor rotational period, and then apply conventional eigenanalysis to compute modal and stability characteristics. Direct averaging, however, eliminates all periodic terms that contribute to system dynamics, thereby producing errors. While averaging itself is not always a bad approach, it must follow MBC. Sample results are presented to illustrate this point and also to show the application of MBC to the modal and stability analysis of a 5-MW turbine.

Offshore turbines are gaining attention as means to capture the immense and relatively calm wind resources available over deep waters. This paper examines the aeroelastic stability of a three-bladed 5MW conceptual wind turbine mounted atop a floating barge with catenary moorings. The barge platform was chosen from the possible floating platform concepts, because it is simple in design and easy to deploy. Aeroelastic instabilities are distinct from resonances and vibrations and are potentially more destructive. Future turbine designs will likely be stability-driven in contrast to the current loads-driven designs. Reasons include more flexible designs, especially the torsionally-flexible rotor blades, material and geometric couplings associated with smart structures, and hydrodynamic interactions brought on by the ocean currents and surface waves. Following a brief description of the stability concept and stability analysis approach, this paper presents results for both onshore and offshore configurations over a range of operating conditions. Results show that, unless special attention is paid, parked (idling) conditions can lead to instabilities involving side-to-side motion of the tower, edgewise motion of the rotor blades, and yawing of the platform.

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