# Studies on a Piezoelectric Cylindrical Transducer for Borehole Dipole Acoustic Measurements
## 2 Principles and Numerical Analyses  
### BENDER Bar and Its Vibration Principles  
In this section, we first introduce the transducer vibration principles, and also conducted numerical analyses of the actuator and the whole transducer. A piezoelectric bender bar is shown in Figure 1. A metal layer and two piezoelectric ceramic layers are bonded at the two interfaces. The \(z\)-axis is parallel to both the thickness direction of the bender bar and the polarization direction of both piezoelectric ceramic layers. The \(x\)-axis is parallel to the length direction of the bender bar and the bar is symmetric about the \(x\)-\(y\)-plane. The middle plane of the bender bar bisects its thickness. Its vibration principle is that when a voltage is applied to both surfaces of each piezoelectric element, one extends and the other one contracts, which produces the bending motion of the entire bender bar [13; 14]. In our previous work [15], we considered the theory of the small deflection of a thin plate. Because the surfaces of the bender bar are traction-free, it is assumed that the stresses in the volume vanish, except for stress \(T_{xx},T_{xy}\) and \(T_{yy}\), i.e., \(T_{xz}=T_{yz}=T_{zz}=0\). The displacement in the \(z\)-direction due to bending of the bar is denoted as \(w\). We divided the bender bar into two parts; one is composite three-layer bender with two piezoelectric ceramic layers and one metal layer, and the other one is the isotropic metal layer part. In the middle composite layer part (\(|x|<L_{p}/2\)), the piezoelectric equations are as follows:  
\[\left\{ \begin{array}{l}T_{1}=\frac{1}{s_{11}^{D}(1-\sigma_{p}^{D})}S_{1}+ \frac{\sigma_{p}}{s_{11}^{D}(1-\sigma_{p}^{D})}S_{2}-\frac{S_{31}}{s_{11}^{D}( 1-\sigma_{p})}D_{3},\\ T_{2}=\frac{1}{s_{11}^{D}(1-\sigma_{p}^{D})}S_{1}+\frac{1}{s_{11}^{D}(1-\sigma _{p}^{D})}S_{2}-\frac{1}{s_{11}^{D}(1-\sigma_{p}^{D})}D_{3},\\ T_{6}=\frac{S_{31}}{s_{11}^{D}(1+\sigma_{p})}S_{6},\\ E_{3}=-\frac{S_{31}}{s_{11}^{D}(1-\sigma_{p})}(S_{1}+S_{2})+\overline{\beta}_{ 33}D_{3},\end{array} \right. \tag{1}\]  
where \(T_{1}\), \(T_{2}\), \(T_{6}\) are the stress components; \(S_{1}\), \(S_{2}\), \(S_{6}\) are the strain components; \(\sigma_{p}=s_{12}^{D}/s_{11}^{D}\) is the Poisson's ratio of piezoelectric material when electric displacement is constant; \(s_{11}^{D}\), \(s_{12}^{D}\) is the compliance coefficient component; \(\overline{\beta}_{33}\) is the dielectric isolation rate component; \(S_{31}\) is the piezoelectric constant component; and \(E_{3}\), \(D_{3}\) are the electric field intensity component and electric displacement component, respectively. Figure 1: Geometry of a trilaminar bender bar. For the isotropic metal layer part (\(L_{p}/2<|x|<L_{m}/2\)), the stress and strain equations are shown as follows [15]:  
\[\left\{ \begin{array}{l}T_{1}^{\prime}=\frac{E_{m}}{1-\sigma_{m}^{2}}(S_{1}+ \sigma_{m}S_{2}),\\ T_{2}^{\prime}=\frac{E_{m}}{1-\sigma_{m}^{2}}(\sigma_{m}S_{1}+S_{2}),\\ T_{6}^{\prime}=\frac{E_{m}}{2(1+\sigma_{m})}S_{6},\end{array} \right. \tag{2}\]  
where \(E_{m}\) is the [PERSON]'s modulus of metal material, and \(\sigma_{m}\) is the [PERSON]'s ratio of metal material. By deriving the formula in our previous work, we obtained the vibration equation of the composite layer:  
\[D\
abla^{2}\
abla^{2}w+\rho H\frac{\partial^{2}w}{\partial t^{2}}=0, \tag{3}\]  
where \(\
abla^{2}=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y ^{2}}\); \(D\) is the stiffness coefficient of composite layer; \(H=2H_{p}+H_{m}\), with \(H_{p},H_{m}\) as the thickness of the piezoelectric material and the metal material, respectively; \(\rho=2\rho_{p}\frac{H_{p}}{H^{2}}+\rho_{m}\frac{H_{m}}{H^{2}}\) is the average density of the composite layer, with \(\rho_{p},\rho_{m}\) being the density of the piezoelectric material and the metal material, respectively.