1 Smart Active Vibration Control System of a Wind Turbine Blade Using Piezoelectric Material 5 Fig. 1.1 Anticipated figure of the beam with piezoelectric patches discarded in this research. Moreover, the piezoelectric patches are assumed to be set along the transversal direction of the structure. The length of each piezoelectric patch Lp became the following: Lp =x2bj −x1bj (1.1) where j shows the number of sensors, b indicates the beam, and 2 and 1 are the beginning and the end of the patch, respectively. x2bj and x1bj show the distance of the end tip and the beginning of the nth sensor-actuator from the base, respectively. In this research, Lp is considered as 10 cm for each sensor. Furthermore, the width of each piezoelectric patch is the same as the width of the beamb, and it is located at the end of each section on the beam. 1.2.2 Analytical Modeling of the Euler-Bernoulli Beam Includes PZT Patches Shearing deformations are neglected in the Euler-Bernoulli theory; therefore, shearing tension and strain are not included. In this theory, the shear force is derived by this equation: V = dM dx (1.2) The energy method and Lagrange equations are used to obtain the governing equation for the Euler-Bernoulli beam with PZT patches. Then the assumed mode method is employed to solve acquired obtained equation. The total strain energy of the Euler-Bernoulli beam is provided as the following: U = 1 2 L 0 A (σxxεx)dAdx (1.3) By inserting the associated equations for stress and strain of the Euler-Bernoulli beam, Eq. 1.3 can be represented as: U = 1 2 L 0 Dxx ∂uz (x, t) ∂x 2 dx (1.4) where Dxx is defined as: Dxx =EI(x) (1.5) σxx andεx are the normal stress and normal strain inx direction, respectively. Eis the Young modulus, andI(x) is the second moment of area in x direction. To derive the related strain energy equation for the beam, Eq. 1.5 needs to be substituted into Eq. 1.4, as the following: U = 1 2 L 0 EI(x) ∂uz(x.t) ∂x 2 dx (1.6) where uz(x) describes the transverse deflection of the beam at some positionx.
RkJQdWJsaXNoZXIy MTMzNzEzMQ==