A model for yawing dynamic optimization of a wind turbine structure

A novel design optimization model for placing frequencies of a wind turbine tower/nacelle/rotor structure in free yawing motion is developed and discussed. The main aim is to avoid large amplitudes caused by the yawing-induced vibrations in the case of horizontal-axis wind turbines or rotational motion of the blades about the tower axis in case of vertical-axis wind turbines. This can be a major cause of fatigue failure and might severely damage the whole tower/nacelle/rotor structure. The mathematical formulation considers a single pole tower configuration having thin-walled circular cross section with constant taper along the tower height. The nacelle/rotor combination is modeled as a rigid mass elastically supported at the top of the tower by the torsional spring of the yawing mechanism. Adequate scaling and non-dimensionalization of the various parameters and variables are given in order to make the model valid for a variety of wind turbine configurations and types of the material of construction. The resulting governing differential equation of motion is solved analytically by transforming it into a standard form of Bessel's equation, which leads to the necessary exact solutions for the frequencies and mode shapes. Several cases of study are examined for different values of the yawing stiffness and inertia parameters by considering both conditions of locked and unlocked yawing mechanism. Useful design charts are developed for placing the frequencies at their needed target values with no penalty of increasing the total structural weight of the system. In all, the developed model guarantees full separation of the system frequencies from the critical exciting yawing frequencies by proper choice of the optimization design parameters.