1 Smart Active Vibration Control System of a Wind Turbine Blade Using Piezoelectric Material 7 be applied for the kinetic energy of the piezoelectric patches and the Euler-Bernoulli beam, respectively. As a result, the cumulative strain energy and kinetic energy of the beam including PZT patches are obtained as follows: U = 1 2 L 0 EI(x) ∂uz(x.t) ∂x 2 dx + 1 2 x2a.j x1a.j cE 11Ip ∂ 2 uz(x.t) ∂x2 2 +JpV(t) ∂ 2 uz(x.t) ∂x2 2 dxj=1:n, (1.16) T = 1 2 L 0 ρA(x) ∂uz(x.t) ∂t 2 dx + x2a.j x1a.j ρahab 1 2 ∂uz(x.t) ∂x 2 dxj=1:n (1.17) where j indicates the number of piezoelectric patches. 1.2.3 Assumed Mode Method To solve the governing equations of the beam and the piezoelectric patch, the assumed mode approximation is used. The transverse deflection under actuation of a surface-bonded PZT patch may be represented as: (1.18) whereÒk shows the admissible function which satisfies geometrical boundaries andqk describes the corresponding unknown. By replacing Eq. 1.18, into Eqs. 1.16 and 1.17, these equations can be represented as: (1.19) (1.20) The Lagrange equation is presented as: d dt ∂T ∂ ˙qj − ∂T ∂qj + ∂U ∂qj = 0 (1.21) The simple matrix form of the governing equation of the Euler-Bernoulli beam with the PZT patch is derived by substituting Eqs. 1.19 and 1.20 into the Lagrange Eq. 1.21 as: M{¨q}+K{q}=−V(t)η (1.22) where M, K, and ηare presented as: (1.23) (1.24) (1.25)
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