**2. Structural dynamic analysis**

The isolated blade structure to be analyzed herein is illustrated in figure 4. The inboard pan‐ el having ignored length relative to the outboard one is considered as a flexible segment modeled by an equivalent torsion spring. The blade has a polar moment of area *I* spinning about its longitudinal axis, x, at an angular displacement *B(x,t)* relative to the pitch bearing at the rotor hub. The blade is analyzed considering the state of free torsional vibration about its elastic axis. The pitching mechanism and the short segment near the hub are assumed to have a linear torsional spring with stiffness *Ks*. Applying the classical theory of torsion (Rao, 1994), the governing equation of the motion is cast in the following:

$$\frac{\partial}{\partial \mathbf{x}} \mathsf{L} \mathsf{G} / (\mathbf{x}) \frac{\partial B(\mathbf{x}, t)}{\partial \mathbf{x}} \mathsf{J} = \rho I(\mathbf{x}) \frac{\partial^2 B(\mathbf{x}, t)}{\partial t^2} \tag{1}$$

which must be satisfied over the interval *0<x<L.*

The associated boundary conditions are described as follows:

**Case (I):** Pitch is active

**Figure 3.** Typical blade failure of a three-bladed, 2 MW wind turbine

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The scope of this chapter is not just to apply optimization techniques and find an opti‐ mum solution for the problem under study. The main aim, however, is to first; perform the necessary exact dynamical analysis of a pitch-regulated wind turbine blade by solving the exact governing differential equation using analytical Bessel's functions. Secondly, the behavior of the pitching fundamental frequency augmented with the mass equality con‐ straint will be investigated in detail to see how it changes with the selected design varia‐ bles. 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 fundamen‐ tal frequency, while the second considers minimization of the square of the difference be‐ tween 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 tapering ratio, blade chord and skin thickness distributions, which are expressed in dimensionless form, making the formulation valid for a variety of blade configurations. The torsional stiffness simulating the flexibility of the inboard panel near the rotor hub is also included in the whole set of design variables. Case studies include the locked and unlocked conditions of the pitching mechanism, in which the functional behavior of the frequency has been thor‐ oughly examined. The developed exact mathematical model guarantees full separation of the frequency from the undesired range which resonates with the pitching frequencies. In fact, the mathematical procedure implemented, combined with exact Bessel's function sol‐ utions, can be beneficial tool, against which the efficiency of approximate methods, such as finite elements, may be judged. Finally, it is demonstrated that global optimality can be

$$\begin{aligned} \text{at blade root (x=0)} & \text{G/} \frac{\partial B}{\partial \mathbf{x}} \Big|\_{\mathbf{x=0}} = 0 & \mathbf{a} \\ \text{at blade tip (x=L)} & \text{G/} \frac{\partial B}{\partial \mathbf{x}} \Big|\_{\mathbf{x=L}} = 0 & \mathbf{b} \end{aligned} \tag{2}$$

**Case (II):** Pitch is inactive

$$\begin{aligned} \text{at blade root (x=0)} & \overset{\partial B}{\partial x} \Big|\_{x=0} = K\_s B(0, t) & \mathbf{a} \\ \text{at blade tip (x=L)} \big|\_{x=L} = 0 & \mathbf{b} \end{aligned} \tag{3}$$

where *GJ(x)* and *ρI(x)* represent the torsional stiffness and the mass polar moment of inertia per unit length, respectively. The twisting angle *B(x,t)* is assumed to be separable in space and time, *Β*(*x*, *t*)=*β*(*x*).*q*(*t*), where the time dependence *q(t)* is harmonic with circular fre‐ quency *ω*. Substituting for *<sup>d</sup>* <sup>2</sup> *q dt* <sup>2</sup> <sup>=</sup> <sup>−</sup>*<sup>ω</sup>* <sup>2</sup> *q*, the associated eigenvalue problem can be written di‐ rectly in the form

$$\frac{d}{d\mathbf{x}}\mathbb{E}\mathbf{G}(\mathbf{x})\frac{d\boldsymbol{\beta}}{d\mathbf{x}}\mathbf{J} + \rho\mathbf{I}(\mathbf{x})\boldsymbol{\omega}^2\boldsymbol{\beta}(\mathbf{x}) = \mathbf{0} \tag{4}$$

**Figure 4.** Isolated blade in pitching motion.

The boundary conditions can be obtained from Eqs. (2) and (3). Considering a tapered blade with thin-walled airfoil section (refer to Figures1 & 4), the torsional constant and the second polar moment of area are directly proportional to *h* and *C3 ,* which are assumed to have the same linear distribution described by the expressions:

$$\begin{array}{ccc} \mathbf{C} = \mathbf{C}\_o (1 - \alpha \stackrel{\wedge}{\mathbf{x}}) & & \mathbf{a} \\ \mathbf{h} = h\_o (1 - \alpha \stackrel{\wedge}{\mathbf{x}}) & & \mathbf{b} \end{array} \tag{5}$$

*x* ^ and *<sup>α</sup>* are dimensionless parameters defined as:

$$\stackrel{\circ}{\alpha} = \frac{\alpha}{L} \text{ : } \alpha = (1 - \Delta) , \ \Delta = \text{C}\_t \Big|\ \text{C}\_0 \tag{6}$$

where *Δ* is the taper ratio of the wind turbine blade.

#### **3. Solution procedures**

For thin-walled, cellular blade construction, the total structural mass *M*, the torsional con‐ stant *J(x)*, and the polar moment of area *I*(x) can be determined from the expressions:

$$\begin{aligned} \mathbf{M} &= f\_1 \Big\| \mathbf{C} h \text{ (x)} \mathbf{dx} & \mathbf{a} \\ \mathbf{J} \{ \mathbf{x} \} &= f\_2 \mathbf{C}^{\text{\text{\textquotedblleft}1}{\text{\textquotedblright}}} \mathbf{b} \\ \mathbf{J} \{ \mathbf{x} \} &= f\_3 \mathbf{C}^{\text{\textquotedblleft}3} h \text{ (x)} \\ \mathbf{J} \{ \mathbf{x} \} &= f\_3 \mathbf{C}^{\text{\textquotedblleft}3} h \text{ (x)} \end{aligned} \tag{7}$$

where *f1*, *f2* and *f3* are shape factors depend upon the shape of the airfoil section, number of interior cells and the ratios between the shear web thicknesses and the main wall thickness *h(x)*. It is convenient first to normalize all variables and parameters with respect to a refer‐ ence design having uniform stiffness and mass distributions with the same material proper‐ ties, airfoil section, and type of construction as well (see Table 1). The dimensionless expressions for the total mass, torsional constant and polar moment of area are, respectively given by:

$$\begin{array}{ll}\text{Mass} & \stackrel{\text{\textdegree}}{M} = \stackrel{\text{\textdegree}}{\text{Ch}} \stackrel{\text{\textdegree}}{d} \text{x} & \text{a} \\ & \stackrel{\text{\textdegree}}{\text{Torsion constant}} & \stackrel{\text{\textdegree}}{J} = \stackrel{\text{\textdegree}}{\text{C}^3} \hbar & \text{b} \\ \text{Polar moment of area} & \stackrel{\text{\textdegree}}{I} = \stackrel{\text{\textdegree}}{\text{C}^3} \hbar & \text{c} \end{array} \tag{8}$$

Therefore, dividing by the corresponding reference design parameters, the governing differ‐ ential equation takes the following dimensionless form:

$$
\beta'' - \frac{4\alpha}{(1 - \alpha x)} \beta' + \stackrel{\frown}{\omega}^2 \beta = 0; \quad 0 \le \le 1\tag{9}
$$


Reference parameters: *Mr*=structural mass, *Jr*= torsion constant, *Ir*=2nd polar moment of area, where *Cr*=Chord length, *hr*=wall thickness, blade taper Δ=1.

**Table 1.** Definition of dimensionless quantities

**Figure 4.** Isolated blade in pitching motion.

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*x*

polar moment of area are directly proportional to *h* and *C3*

*x* ^ <sup>=</sup> *<sup>x</sup>*

*C* =*Co*(1−*αx*

*h* =*ho*(1−*αx*

same linear distribution described by the expressions:

^ and *<sup>α</sup>* are dimensionless parameters defined as:

where *Δ* is the taper ratio of the wind turbine blade.

**3. Solution procedures**

The boundary conditions can be obtained from Eqs. (2) and (3). Considering a tapered blade with thin-walled airfoil section (refer to Figures1 & 4), the torsional constant and the second

^) a

For thin-walled, cellular blade construction, the total structural mass *M*, the torsional con‐

stant *J(x)*, and the polar moment of area *I*(x) can be determined from the expressions:

*,* which are assumed to have the

^) b (5)

*<sup>L</sup>* , *<sup>α</sup>* =(1−*Δ*), *<sup>Δ</sup>* <sup>=</sup>*Ct* / *<sup>C</sup>*<sup>0</sup> (6)

The boundary conditions to be satisfied are *β* ′ =0 at both blade root and tip for the unlocked pitching condition and *β* ′ =( *K* ^ *s J* ^ 0 )*β* at root, *β* ′ =0 at tip for the locked condition, where the prime denotes here differentiation with respect to *x* ^. Using the transformation *<sup>x</sup>* ^ <sup>=</sup> <sup>1</sup> *<sup>α</sup>* <sup>−</sup> <sup>1</sup> *ω* ^ *y* (*α* ≠0), Eq. (9) takes the form:

$$\frac{d\,^2\beta}{dy\,^2} + \frac{4}{y}\frac{d\beta}{dy} + \beta = 0; \quad \delta \le y \le \gamma \tag{10}$$

which can be further transformed to the standard form of Bessel's equation by setting *β* =*ψ* / *y* <sup>3</sup> , to get

$$(y^2\frac{d^2\psi}{dy^2} + y\frac{d\psi}{dy} + (y^2 - \frac{9}{4})\psi = 0\tag{11}$$

This has the solution

$$
\psi(y) = \mathbb{C}\_1 \mathbb{I}\_{3/2} + \mathbb{C}\_2 \mathbb{I}\_{-3/2} \tag{12}
$$

where *C1* and *C2* are constants of integration and *J3/2* and *J-3/2* are Bessel's functions of order *k= ±3/2*, given by (Edwards & Penney, 2004):

$$\begin{aligned} \mathbf{J}\_{3/2}(y) &= \sqrt{\frac{2}{\pi y}} (\sin y - y \cos y) & \mathbf{a} \\ \mathbf{J}\_{-3/2}(y) &= -\sqrt{\frac{2}{\pi y}} (\cos y + y \sin y) & \mathbf{b} \end{aligned} \tag{13}$$

The exact analytical solution of the associated eigenvalue problem is:

$$\beta(y) = A \mathbb{I} \frac{y \cos y - \sin y}{y^3} \mathbf{I} + B \mathbb{I} \frac{y \sin y + \cos y}{y^3} \mathbf{I} \tag{14}$$

where *A* and *B* are constants depend on the imposed boundary conditions. Applying the boundary conditions, given in Eqs. (2) and (3), and considering only nontrivial solution the frequency equation can be directly obtained. The final derived exact frequency equations for both active and inactive pitching motion in appropriate compacted closed forms are sum‐ marized in the following:

$$\text{Baseline design with rectangular planform } \begin{array}{c} \text{(D=1)} \quad \stackrel{\frown}{\omega} \tan \stackrel{\frown}{\omega} = \stackrel{\frown}{K}\_s / \langle \stackrel{\frown}{h}\_o \stackrel{\frown}{\mathbb{C}}\_o \rangle \end{array} \tag{15}$$

$$\text{Active pitching } \tan\hat{\omega} = \frac{\hat{\mathbf{3}\hat{\omega}}(\mathbf{3} + \mathbf{y}\boldsymbol{\delta})}{\left(\mathbf{y}\,\boldsymbol{\delta}\right)^2 - \mathbf{3}\,\boldsymbol{\gamma}^2(\mathbf{1} + \boldsymbol{\Delta}^2) + \mathbf{9}(1 + \mathbf{y}\boldsymbol{\delta})} \tag{16}$$

$$\text{Locked pitching mechanism} \cdot \tan \hat{\omega} = \frac{(1 - 3\theta)(3\hat{\omega}^{\prime} - \gamma \cdot \delta^{\prime}) - 3\gamma^{2} \delta \theta}{3\gamma \delta (1 - 3\theta) + (3 - \delta^{2})(1 - 3\theta + \theta \gamma^{2})} \tag{17}$$

The definition of the various quantities in Eqs. (15), (16) and (17) is given in Table (1) and the appendix of nomenclature.

## **4. Optimization problem formulation**

The boundary conditions to be satisfied are *β* ′

=( *K* ^ *s J* ^ 0

prime denotes here differentiation with respect to *x*

*d* 2 *β d y* <sup>2</sup> <sup>+</sup>

*<sup>y</sup>* <sup>2</sup> *<sup>d</sup>* <sup>2</sup> *ψ d y* <sup>2</sup> <sup>+</sup> *<sup>y</sup>*

4 *y dβ*

which can be further transformed to the standard form of Bessel's equation by setting

*dy* + (*<sup>y</sup>* <sup>2</sup> <sup>−</sup> <sup>9</sup>

where *C1* and *C2* are constants of integration and *J3/2* and *J-3/2* are Bessel's functions of order *k=*

*<sup>π</sup><sup>y</sup>* <sup>3</sup> (sin*<sup>y</sup>* <sup>−</sup> *<sup>y</sup>*cos*y*) <sup>a</sup>

*<sup>π</sup><sup>y</sup>* <sup>3</sup> (cos*<sup>y</sup>* <sup>+</sup> *<sup>y</sup>*sin*y*) b

*<sup>y</sup>* <sup>3</sup> <sup>+</sup> *<sup>B</sup> <sup>y</sup>*sin*<sup>y</sup>* + cos*<sup>y</sup>*

where *A* and *B* are constants depend on the imposed boundary conditions. Applying the boundary conditions, given in Eqs. (2) and (3), and considering only nontrivial solution the frequency equation can be directly obtained. The final derived exact frequency equations for both active and inactive pitching motion in appropriate compacted closed forms are sum‐

*dψ*

)*β* at root, *β* ′

pitching condition and *β* ′

212 Advances in Wind Power

(*α* ≠0), Eq. (9) takes the form:

, to get

This has the solution

marized in the following:

*±3/2*, given by (Edwards & Penney, 2004):

*J*3/2(*y*)=

*<sup>J</sup>*−3/2(*y*)= <sup>−</sup> <sup>2</sup>

2

The exact analytical solution of the associated eigenvalue problem is:

*<sup>β</sup>*(*y*)= *<sup>A</sup> <sup>y</sup>*cos*<sup>y</sup>* <sup>−</sup>sin*<sup>y</sup>*

Baseline design with rectangular planform (D=1) *ω*

*β* =*ψ* / *y* <sup>3</sup>

=0 at both blade root and tip for the unlocked

=0 at tip for the locked condition, where the

<sup>4</sup> )*ψ*=0 (11)

*<sup>y</sup>* <sup>3</sup> (14)

^ <sup>=</sup> <sup>1</sup> *<sup>α</sup>* <sup>−</sup> <sup>1</sup> *ω* ^ *y*

(13)

) (15)

^. Using the transformation *<sup>x</sup>*

*dy* <sup>+</sup> *<sup>β</sup>* =0; *<sup>δ</sup>* <sup>≤</sup> <sup>y</sup> <sup>≤</sup> *<sup>γ</sup>* (10)

*ψ*(*y*)=*C*1*J*3/2 + *C*2*J*−3/2 (12)

^ tan*<sup>ω</sup>* ^ <sup>=</sup> *<sup>K</sup>* ^ *<sup>s</sup>* / (*h* ^ *oC* ^ *o* 3 Attractive goals of designing efficient structures of wind generators include minimization of structural weight, maximization of the fundamental frequencies (Maalawi & EL-Chazly, 2002; Maalawi & Negm, 2002; Maalawi & Badr, 2010), minimization of total cost per energy produced, and maximization of output power (Maalawi & Badr, 2003). Another important consideration is the reduction or control of the vibration level. Vibration can greatly influ‐ ence the commercial acceptance of a wind turbine because of its adverse effects on perform‐ ance, cost, stability, fatigue life and noise. The reduction of vibration can be attained either by a direct maximization of the natural frequencies or by separating the natural frequencies of the blade structure from the harmonics of the exciting torque applied from the pitching mechanism at the hub. This would avoid resonance and large amplitudes of vibration, which may cause severe damage of the blade. Direct maximization of the natural frequen‐ cies can ensure a simultaneous balanced improvement in both of the overall stiffness level and the total structural mass. The mass and stiffness distributions are to be tailored in such a way to maximize the overall stiffness/mass ratio of the vibrating blade. The associated opti‐ mization problems are usually cast in nonlinear mathematical programming form (Vander‐ plaats, 1999). The objective is to minimize a function *F(X)* of a vector *X* of design variables, subject to certain number of constraints *Gj (X) ≤ 0, j=1,2,…m*.

In the present optimization problem, two alternatives of the objective function form are im‐ plemented and examined. The first one is represented by a direct maximization of the fun‐ damental frequency, which is expressed mathematically as follows:

$$\text{Maximize } F(\underline{X}\_1) = \cdot \stackrel{\frown}{\omega}\_1 \tag{18}$$

where *ω* ^ 1 is the normalized fundamental frequency (see Table1) and *X =(C* ^ *<sup>o</sup>*, *h* ^ *<sup>o</sup>*, *Δ)* is the chosen design variable vector. The second alternative is to minimize the square of the differ‐ ence between the fundamental frequency *ω* ^ 1 and its target or desired value *ω* ^ \* , i.e.

$$\text{Minimize } F(\underline{\mathbf{x}}) = (\stackrel{\wedge}{\omega} \cdot \stackrel{\wedge}{\omega}^\*)^2 \tag{19}$$

Both objectives are subject to the constraints:

$$\text{Mass constraint: } \stackrel{\frown}{M} = \stackrel{\frown}{C} \stackrel{\frown}{h} \stackrel{\frown}{d} \stackrel{\frown}{\alpha} = 1 \tag{20}$$

$$\text{Side constraints:} \underline{\underline{X}}\_{\mathcal{L}} \underline{\underline{X}} \le \underline{\underline{X}}\_{\mathcal{u}} \tag{21}$$

where *XL* and *XU* are the lower and upper limiting values imposed on the design variables vec‐ tor *X* in order not to obtain unrealistic odd-shaped designs in the final optimum solutions. Ap‐ proximate values of the target frequencies are usually chosen to be within close ranges; sometimes called frequency – windows; of those corresponding to an initial baseline design, which are adjusted to be far away from the critical exciting pitching frequencies. Several com‐ puter program packages are available now for solving the above design optimization model, which can be coded to interact with structural and eigenvalue analyses software. Extensive computer implementation of the models described by Eqs. (18-21) have revealed the fact that maximization of the fundamental frequency is a much better design criterion. If it happened that the maximum frequency violates frequency windows, which was found to be a rare situation, another value of the frequency can be chosen near the global optima, and the frequency equa‐ tions (15-17) can be solved for any one of the unknown design variables instead. Considering the frequency-placement criterion, it was found that convergence towards the optimum solution, which is also too sensitive to the selected target frequency, is very slow.

## **5. Optimization techniques**

The above optimization problem described by Eqs.(18-21) may be thought of as a search in an 3-dimensional space for a point corresponding to the minimum value of the objective function and such that it lie within the region bounded by the subspaces representing the constraint functions. Iterative techniques are usually used for solving such optimization problems in which a series of directed design changes (moves) are made between successive points in the design space. The new design *Xi+1* is obtained from the old one *Xi* as follows:

$$\mathbf{X}\_{i+1} = \mathbf{X}\_i + a\_i \mathbf{S}\_i \tag{22}$$

$$\text{Such that } F\{X\_{i+1}\} \le F\{X\_i\} \tag{23}$$

where the vector *Si* defines the direction of the move and the scalar quantity *α<sup>i</sup>* gives the step length such that *Xi+1* does not violate the imposed constraints, *Gj (X)*. Several optimization techniques are classified according to the way of selecting the search direction *Si* . In general, there are two distinct formulations (Vanderplaats, 1999): the constrained formulation and the unconstrained formulation. In the former, the constraints are considered as a limiting subspace. The method of feasible directions is one of the most powerful methods in this cat‐ egory. In the unconstrained formulation, the constraints are taken into account indirectly by transforming the original problem into a series of unconstrained problems. A method, which has a wide applicability in engineering applications, is the penalty function method.

The *MATLAB* optimization toolbox is a powerful tool that includes many routines for differ‐ ent types of optimization encompassing both unconstrained and constrained minimization algorithms (Vekataraman, 2009). One of its useful routines is named *"fmincon"* which imple‐ ments the method of feasible directions in finding the constrained minimum of an objective function of several variables starting at an initial design. The search direction *Sj* must satisfy the two conditions *Sj .∇F< 0* and *Sj .∇G<sup>j</sup> < 0*, where *∇F* and *∇G<sup>j</sup>* are the gradient vectors of the objective and constraint functions, respectively. For checking the constrained minima, the Kuhn-Tucker test (Vanderplaats, 1999) is applied at the design point *XD*, which lies on one or more set of active constraints. The Kuhn-Tucker equations are necessary conditions for optimality for a constrained optimization problem and their solution forms the basis to the method of feasible directions.
