6 A. Hashemi and J. Jang The total formulation of the Euler-Bernoulli beam’s kinetic energy is provided as: T = 1 2 L 0 A ρν 2 z dAdxdx (1.7) where νz is velocity and it can be presented as: νz = ∂uz(x.t) ∂t (1.8) To formulate the Euler-Bernoulli beam’s cumulative kinetic energy and simplify it, Eq. 1.8 could be replaced into Eq. 1.7, as: T = 1 2 L 0 ρA(x) ∂uz(x.t) ∂t 2 dx (1.9) By assuming the linear PZT constitutive relations and obtaining the governing equation of the vibration of the PZT patches, the stress and strain of these patches can be derived as: σ p xx =c E 11S1 −e31E3 (1.10) ε p xx =S1 =−y ∂ 2 uz (x, t) ∂x2 (1.11) where 1, 2, and 3 show the X, Y, and Z directions, respectively. cE 11, e31, and E3 are modules of elasticity of the PZT in constant electric field, piezoelectric stress constant, and the electric field across the electrodes of the PZT, respectively. The following is the relation between the electrical field and the voltage applied to the piezoelectric patch electrodes: E3 = V(t) ha (1.12) V(t) is applied harmonic voltage. The correlation between the piezoelectric stress constant and the associated strain constant, d31, is provided as: e31 =c E 11d31 (1.13) Thus, the equations of the strain energy and kinetic energy of a piezoelectric patch are formulated as: U p j = 1 2 x2a x1a cE 11Ip ∂ 2 uz(x.t) ∂x2 2 +JpV(t) ∂ 2 uz(x.t) ∂x2 2 dx (1.14) T p j = 1 2 x2a x1a ρahab 1 2 ∂uz(x.t) ∂x 2 dx (1.15) To obtain the overall strain of the Euler-Bernoulli beam with the piezoelectric sensor and actuator, the strain energy of the piezoelectric patches must be contributed to the strain energy of the Euler-Bernoulli beam. The above process must
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