INTRODUCTION
Renewable resources, such as wind, are replaced continuously by nature. Electrical energy is a form of secondary energy and is derived from primary energy by using wind power. Wind velocities change with time and this inconstancy causes the difficulties to utilize wind power efficiently. Spee et al. (1995) presented a novel control strategy for a bruless doubly fed machine applied to variable speed wind power generation systems. Song and Dhinakaran (1999) explored the nonlinear wind turbine control method. In the design of wind turbine generator, fiber reinforced composites materials have been used. The wind turbine generator composite material in blade applications has stimulated a considerable amount of study. Storti and Aboelnaga (1987) studied the transverse deflections of a straight tapered symmetrical beam attached to a rotating hub as a model for the bending vibration of blades in turbomachinery. Wagner (1967) calculated the forced vibration response of subsystems attached to a foundation with finite stiffness or mass. Schultz and Tsai (1968) presented data concerning the dynamic material behavior in a glassfilament reinforced epoxy composite. Abarcar and Cunniff (1972) presented the fiber orientation effect on the vibration mode in cantilever beams made of unidirectional fiber reinforced composite materials. Shiau et al. (1996) investigated the vibration and optimum design of a rotating laminated blade subject to constraints on the dynamic behavior. Rand and Barkai (1997) presented a nonlinear formulation for the structural behavior of initially twisted solid and thin walled composite blades. Surace et al. (1997) presented the analysis of the model characteristics of straight, pretwisted nonuniform blades. Lekhnitskii (1963) solved the static response of an orthotropic composite beam subjected to the simultaneous action of an applied bending moment and an applied torque.
The dynamic characteristics of wind turbine generator composite blades are important in highspeed wind turbine generator designs. The determination of the dynamic characteristics for wind composite blades is one of the great importances. In this study the differential quadrature method is employed to formulate the eigenvalue problem of wind turbine generator composite blade in matrix form. The integrity and computational efficiency of the differential quadrature method in this problem will be demonstrated through a series of case studies.
MAXIMUM POWER OF WIND GENERATORS
Figure 1 shows the geometry of the wind generator. The wind turbine blades are manufactured from carbon fiber laminated composite. The measurements are acquired analog through multichannel analog to digital circuit. The measured data are stored digitally in the computer. The shaft torque is measured via a strain gauge transducer. The wind generator has three wind composite blades. The wind mechanical power P_{w} is expressed by

Fig. 1: 
Geometry of a wind generator 

Fig. 2: 
The output power of the wind generator for various wind speeds 
where C_{wl} is the wind energy loss constant, J is the turbine moment of inertia, ω is the turbine angular speed and P_{m} is the turbine mechanical power. The turbine mechanical power P_{m} is expressed by

Fig. 3: 
The energy losses of the wind generator for various wind speeds 

Fig. 4: 
The power coefficients of the wind generator for various wind
speeds 
where T is the torque, C_{ml} is mechanical energy loss constant. The wind turbine generator power P_{e} is expressed by
where ρ_{a} is the air density, C_{p} is the rotor efficiency
of the turbine, v_{w} is wind speed and r is wind turbine radius.

Fig. 5: 
The energy loss constants of the wind generator for various
wind speeds 
The air density dependents on the pressure and temperature of the air. Figure
2 shows the measured power of the wind generator for various wind speeds.
The results show that the wind blade speeds significantly affect the measured
output power. It is needed to keep the wind blade tip speed to obtain maximum
power. Figure 3 displays the measured energy losses of the
wind generator for various wind speeds. The results show that the wind blade
shaft speeds significantly affect the energy losses. The energy losses depend
on the rotation speeds of the wind generator. Figure 4 illustrates
the measured value of the power coefficient C_{p}. The results show
that the wind blade speeds significantly affect the measured value of the power
coefficient C_{p}. The power coefficient C_{p} is a function
of the wind blade speed. It is necessary to keep the wind blade tip speed to
obtain the measured value of the maximum power coefficient C_{p}. Figure
5 reveals the energy loss constants of the wind generator. There is a local
minimum energy loss constant for the wind generator under each wind speed.
DYNAMIC BEHAVIOR OF WIND GENERATORS
The wind blades are attached to a rigid hub. A model for the wind turbine generator composite blade with a rigid hub is assembled. Axes 1, 2 and 3 are the global axes and axes x, y and z are the local blade axes. θ_{f} is the fiber orientation and θ_{i} is the inclined angle of the blade. The kinetic energy of the rotating wind composite blades can be derived as
The corresponding strain energy of the rotating wind blades is
where u_{1}(z,t) is the displacement of the first blade and φ_{1}(z,t) is the twist angle of the first blade. u_{2}(z,t) is the displacement of the second blade and φ_{2}(z,t) is the twist angle of the second blade. u_{3}(z,t) is the displacement of the third blade and φ_{3}(z,t) is the twist angle of the third blade. For the wind turbine generator composite blades, the parameters used in the above equations are defined as
where b(z) and h(z) denote respectively the breadth and thickness of the blade. E_{11} and E_{33} are Young<’s moduli in 1 and 3 axes, v_{31} is the Poisson<’s ratio, G_{13}, G_{12} and G_{23} are shear moduli in 13, 12 and 23 planes, respectively. With considering the internal and external damping effects in the wind turbine generator composite blade, the virtual work δW in the blade can be derived as:
with
where Ω is the rotation speed of the hub, J_{z} is the polar moment
of inertia, ρ is the density of the material, r_{0} is the radius
of the hub. C_{0} and C_{1} are the external and the internal
damping coefficients of the wind composite blades, respectively. Substituting
Eqs. (4), (5) and (13)
into Hamilton equation
This leads to the equations of motion of the wind turbine generator composite blades as
The corresponding boundary conditions are
Consider u_{1}(z,t) = U_{1} (z)e^{λt}, φ_{1} (z,t) = φ_{1} (z)e^{λt}, u_{2} (z,t) = U_{2} (z)e^{λt}, φ_{2} (z,t) = φ_{2} (z)e^{λt}, u_{3} (z,t) = U_{3} (z)e^{λt }and φ_{3} (z,t) = φ_{3} (z)e^{λt}, the governing equations for the wind turbine generator blades can be rewritten as
The corresponding boundary conditions are
THE DIFFERENTIAL QUADRATURE METHOD
One of the fields among which one can find extensive differential quadrature method applications is structural mechanics. Han and Liew (1999) analyzed annular plates using the differential quadrature method. Striz et al. (1994) used the quadrature element method for static analysis. Du et al. (1996) used the differential quadrature method for buckling analysis. Quan and Chang et al. (1989a, 1989b) derived the weighting coefficients in a more explicit way. The basic concept of the differential quadrature method is that the derivative of a function at a given point can be approximated as a weighted linear sum of the functional values at all of the sample points in the domain of that variable. Using this approximation, the differential equation is then reduced into a set of algebraic equations. The number of equations depends on the selected number of sample points within the domain. As for any polynomial approach, the accuracy of the solution in this method may be improved by increasing the number of sample points. Possible oscillations in the numerical results arising from higher order polynomials can be avoided by using numerical interpolation methods. Suppose that there are N sample points in the domain. For a function f(z), differential quadrature method approximation for the mth order derivative at the sample point z_{i} is given by
where f(z_{i}) is the functional value at the sample point z_{i} and D_{ij}^{(m)} are the weighting coefficients of the mth order differentiation attached to these functional values.
In order to determine the weighting coefficients D_{ij}^{(m)} and to overcome the numerical illconditions in determining the weighting coefficients D_{ij}^{(m)}, a Lagrangian interpolation polynomial was introduced by Quan and Chang (1989a, 1989b), which is
where
Substituting Eq. (67) into Eq. (66) leads
to
Once the sample points are selected, the coefficients of the weighting matrix
can be obtained from equations (68) and (69).
The most convenient method is to peak up the sample points in an equal space sample point distribution. The selection of sample points always plays an important role (Bert et al., 1994; Malik and Bert, 1996) in the solution accuracy of the differential quadrature method. Poor accuracy results were obtained using this equally spaced distribution. An unequally spaced sample point distribution, i.e., ChebyshevGaussLobatto distribution (Bert et al., 1994; Malik and Bert, 1996), has been used to improve the calculation accuracy. The unequally spaced inner points of each blade using the ChebyshevGaussLobatto distribution in the present computation are distributed as
The boundary points of the clampedfree wind turbine generator composite blades in this computation are defined as
The differential quadrature weighting coefficients of second, third and fourthorder derivatives in the differential quadrature method, D_{ij}^{(2)}, D_{ij}^{(3)} and D_{ij}^{(4)} may be computed by (Bert et al., 1994; Malik and Bert, 1996).
By employing the differential quadrature method, Eq. (66)
is substituted into equations (42)(47).
The equations of motion for the wind turbine generator composite blades can
be rearranged in matrix form as
According to the differential quadrature method, boundary condition equations
(48)(65) take the following discrete forms:

Fig. 6: 
The frequencies of the wind generator for various wind speeds 

Fig. 7: 
The frequencies of the wind generator for various rotation
speeds 

Fig. 8: 
The frequencies of the wind generator for various inclined
angles 
Figure 6 illustrates the frequencies of the wind generator for various wind speeds. The material properties and the geometric dimensions of the wind composite blade are given as: G_{12} = 2.542 GPa, E_{11}/G_{12} = 3.7, G_{23} = 4.305 GPa, E_{33}/G_{23} = 30, G_{13} = 5.158 GPa, v_{31} = 0.3, r_{0} = 5.5 cm, θ_{f} = 0.0, L = 52.5 cm and ρ = 1550 kg m^{3}. The results reveal that the frequencies of the wind generator are strongly dependent on the blade shaft speed. The frequencies of the wind generator keep increasing with increasing the rotation speeds. Results indicate that the wind speeds are not the key factors to affect the natural frequencies of the wind generator. Figure 7 displays the frequencies of the wind generator for various rotation speeds. Numerical results in this example reveal that the frequencies of the wind generator might be affected by the rotation speed. Figure 8 shows the frequencies of the wind generator for various inclined angles θ_{i}. The numerical results show that the inclined angle θ_{i} has not great effect on the frequencies of the wind generator.
CONCLUSIONS
Due to the coupling effect of torsion and flexure in the wind turbine composite
blades, the dynamic characteristics of the wind turbine generator composite
blades are much more complicated than the wind turbine isotropic blades. The
wind blade speeds significantly affect the measured output power. It is possible
to maintain optimal energy generation by tracking the wind speed changes and
wind blade shaft speeds. The wind blade shaft speeds significantly affect the
energy losses and the power coefficients. Numerical results in this study show
that the rotation speed could affect the frequencies of the wind generator seriously.
Numerical results in this study showed that the inclined angle could not affect
the frequencies of the wind generators gravely. The formulation presented in
this article reveals that the differential quadrature method is convenient for
solving problems governed by fourth or higher order differential equations.
The simplicity of this formulation makes its own good candidate for modeling
more complicated applications.