# Piezoelectric Composite Vibrator with a Bilaminated Structure for Bending Vibration
## 3 Finite Element Simulation of the Piezoelectric Composite Bilaminated Vibrator  
The 2-2 and 1-3 piezoelectric composite were simulated under free boundary condition via the finite element method to select a bilaminated piezoelectric composite vibrator. The electrical and vibration  
Figure 1: Structure of the piezoelectric composite bilaminated vibrator. Figure 3: Structures of the 2-2 and 1-3 type piezoelectric composite. (a) 2-2 type. (b) 1-3 type. Figure 2: Schematic of the series and parallel arrangements of the piezoelectric layers. (a) Parallel arrangement. (b) Series arrangement. characteristics of the piezoelectric composites were analyzed using the finite element analysis software ANSYS. The size of the 2-2 and 1-3 piezoelectric composites was 49.5 mm in length, 9.5 mm in width, and 2 mm in height, and the volume percentage of the piezoelectric ceramics was 0.8. First, the finite element simulation models of the 2-2 and 1-3 piezoelectric composite bilaminated vibrators were established in accordance with the solid structure, as shown in Figure 4. Blue represents the piezoelectric phase and red represents the polymer phase. Second, material parameters of the piezoelectric ceramics and epoxy resins were shown in Table 1[35]. The anisotropic PZT-5 piezoelectric ceramics were selected as the piezoelectric phase and the solid element Solid5 was selected as the element type. The material parameters included density, stiffness matrix, piezoelectric constant matrix, and dielectric constant matrix. The polymer phase is made of high polymer epoxy resin. The unit type is Solid45. Solid45 is used for three-dimensional solid structure model. The grid size was set to 1/10 of the wavelength at the central frequency in the finite element model [36]. In the present study, 1 and 0 V were applied to the top and bottom of the model at the \(z\)-axis direction. No constraints are imposed. The frequency range was 4000-6600 Hz, which was divided into 260 substeps with a 10 Hz interval between each step. The damping coefficient is set as 0.02, and the admittance curve of the piezoelectric vibrator is obtained. The calculation results are shown in Figure 5. The series resonance frequency, parallel resonance frequency, and bandwidth of the vibrator were obtained from the admittance curve, as shown in Table 2.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **PZT-5A** & **Parameters** & **Epoxy Resins** \\ \hline \(\rho\) (kg/m\({}^{3}\)) & 7750 & \(\rho\) (kg/m\({}^{3}\)) & 1050 \\ \(c_{11}^{E}\) (10\({}^{10}\)N/m\({}^{2}\)) & 12.1 & \(c_{11}^{E}\) (10\({}^{10  
The effective electromechanical coupling coefficients of the 2-2 and 1-3 vibrators were calculated using Equation (1) [30; 37], where \(f_{s}\) and \(f_{p}\) refer to the series resonance frequency and the parallel resonance frequency, respectively. The results are listed in Table 2.  
\[k_{e}=\sqrt{\frac{f_{p}^{2}-f_{s}^{2}}{f_{p}^{2}}} \tag{1}\]  
As shown in Figure 5, the series resonance frequency of the 2-2 vibrator was 4630 Hz, whereas that of the 1-3 vibrator was 5320 Hz. Furthermore, the bandwidth of the 2-2 vibrator increased 1.8 times in comparison with that of the 1-3 vibrator. Figure 6 shows the vibration modes of the two kinds of composites, different colors represent different displacements. In Figure 6a, the largest displacement appeared at the center of the upper and lower surfaces of the 2-2 vibrator. The result is similar to that of the 1-3 vibrator in Figure 6b. In Figure 7, the maximum vibration displacement of the 2-2 vibrator at the resonance point was \(7.87\times 10^{-11}\) m, and that of the 1-3 vibrator was \(3.72\times 10^{-11}\) m, where the driving voltage was 1 V. Moreover, the maximum vibration displacement of the 2-2 vibrator was twice greater than that of the 1-3 vibrator, suggesting that the 2-2 composite is superior to the 1-3 composite in the preparation of the piezoelectric composite b