Frequency optimization of a wind turbine blade in pitching motion

Abstract

This article presents a mathematical model for optimizing frequencies of a wind turbine blade in pitching motion. Related design problems include the excessive blade torsional vibrations induced by continuous pitching, which is necessary to limit the power output and protect generator from damages in severe wind conditions. This, in turn, can have adverse effects on the fatigue life of the blades and might cause catastrophic failures near the resonant frequencies. The study considers a blade tapered linearly towards the tip representing the effective aerodynamical surface for producing the needed power. The blade inboard portion is modelled by an equivalent torsional spring simulating the torsional stiffness near the root. The model provides exact solutions of the resulting governing differential equation by utilizing analytical Bessel's functions of the first kind, where the functional behaviour of the torsional frequency has been thoroughly examined. The associated optimization problem is formulated by considering two forms of the objective function. The first one is represented by a direct maximization of the fundamental frequency, while the second one considers minimization of the square of the difference between the fundamental frequency and its target or desired value. In both strategies, an equality constraint is imposed on the total structural mass in order not to violate other economic and performance requirements. Design variables encompass the blade tapering ratio, chord, and shear wall thickness distributions, which are expressed in dimensionless form, making the formulation valid for a variety of blade configurations. In fact, the mathematical procedure implemented can be regarded as a beneficial design tool, in which significant increase in the overall stiffness/mass ratio level of the blade structure combined with full separation of the frequency from the undesired range that resonates with the pitching frequencies can be achieved.