**Appendix**


*L* effective blade length

*q(t)* time dependence of blade pitch angle.

*t* time variable

**7. Conclusions**

222 Advances in Wind Power

muli-criteria optimization technique.

*C* chord length of the airfoil section

*G* shear modulus of blade material

chord length at blade tip

*Co* chord length at blade root

*h* skin thickness of the blade

*ho* skin thickness at blade root *I* second polar moment of area

*B(x,t)* pitch angle about blade elastic axis: *Β*(*x*, *t*)=*β*(*x*).*q*(*t*),

**Appendix**

*Ct*

Efficient model for optimizing frequencies of a wind turbine blade in pitching motion has been presented in this chapter. The mathematical formulation is given with dimensionless quantities so as to make the model valid for a real-world wind turbine blade of any size and configuration. It provides exact solutions to the vibration modes of the blade structure in free pitching motion, against which the efficiency of other numerical methods, such as the finite element method, may be judged. Design variables include the chord length of the air‐ foil section, shear wall thickness and blade tapering ratio. Useful design charts for either maximizing the natural frequency or placing it at its desired (target) value has been devel‐ oped for a prescribed total structural mass, and known torsional rigidity near blade root. The fundamental frequency can be shifted sufficiently from the range which resonates with the excitation frequencies. In fact the developed frequency charts given in the paper reveal very clearly how one can place the frequency at its proper value without the penalty of in‐ creasing the total structural mass. Each point inside the chart corresponds to different mass and stiffness distribution along the span of constant mass blade structure. The given ap‐ proach is also implemented to maximize the frequency under equality mass constraint. If it happened that the obtained maximum frequency violates frequency windows, another val‐ ue of the frequency can be taken near the optimum point, and an inverse approach can be applied by solving the frequency equation for any one of the unknown design variables in‐ stead. Other factors under study by the author include the use of material grading concept to enhance the dynamic performance of a wind turbine blade. Exciting frequencies due to the turbulent nature of the wind, especially in large wind turbines with different types of boundary conditions, are also under considerations. Another extension of this work is to op‐ timize the aerodynamic and structural efficiencies of the blade by simultaneously maximiz‐ ing the power coefficient and minimizing vibration level under mass constraint using a

*X* design variables vector.

*x* distance along blade span measured from chord at root

*α = (1 – Δ)*

*β(x)* amplitude of the pitch angle

*ω* circular frequency of pitching motion

*ω* ^ normalized frequency

$$\mathcal{Y} \qquad \qquad \left(\stackrel{\wedge}{=} \stackrel{\wedge}{\omega}/\alpha\right)$$

$$\delta \qquad \qquad \qquad (=\gamma \Delta)$$

*ρ* mass density of blade material

*Δ* blade taper ratio (*Ct /Co*)

*θ (*=*αh* ^ *oC* ^ *o* <sup>3</sup> / *K* ^ *s)*
