# Analysis and Experimental Research of a Multilayer Linear Piezoelectric Actuator
## 3 Design and Analysis  
To optimize performance of the actuator, the dimensions of the multilayer piezoelectric ceramic need to be designed to make resonant frequencies of the L1 and the B2 modes of the ceramic as close as possible. The L1 and B2 modes' resonance frequencies of a long bar, whose length is much larger than its width and thickness, can be approximately calculated by the following equations [21]:  
\[f_{\rm L1}=\frac{1}{2L}\sqrt{\frac{1}{\rho\rm s_{11}^{E}}} \tag{5}\]  
\[f_{\rm B2}=\frac{\alpha^{2}W}{2\pi\sqrt{12L^{2}}}\sqrt{\frac{1}{\rho\rm s_{11} ^{E}}} \tag{6}\]  
where \(L\) is the length, \(W\) is the width, \(\alpha\) = 7.853 is a constant, and \(\rho\) is the density of the material. To make \(f_{\rm L1}=f_{\rm B2}\), we can calculate that \(L/W\approx 5.7\). However, the PZT-81 is anisotropic and these two equations do not calculate the influence of the contactor either, so deviation would be brought in when the resonant frequencies were calculated with these two equations. From Equations (5) and (6), we can see that both frequencies of the first longitudinal and second bending modes would decrease along with the increase of \(L\), but \(f_{\rm B2}\) decreases faster. So it is possible to draw the curves of the resonant frequency vs. the length, to obtain the cross-point where the optimized length is. With the method of finite element analysis (FEA), the resonant frequencies at various dimensions can be easily obtained through modal analysis. In this study, the commercial FEA software (ANSYS 12.1, ANSYS Inc., Canonsburg, Pennsylvania, PA, USA, 2009) was used, the element type was solid 226 with option K1 set to be electro-elastic/piezoelectric. The boundary condition was set to be mechanically free. The width and thickness were set to be 7.5 mm and 3 mm, respectively, and the results are shown in Figure 4. The length at the cross-point is around 30 mm. The PZT-81 (obtained from Konghong Corporation Limited, Xi'an, China) is a kind of anisotropy material with a density \(\rho\) = 7600 kg/m\({}^{3}\). The parameter matrixes of PZT-81 polarized in the Z+ direction are listed as follows:  
(1) Elastic matrix in flexibility form \(S^{\rm E}\) (unit: \(10^{-12}\) m\({}^{2}\)/N):  
\[S^{\rm E}=\left[ \begin{array}{cccccc}11.5&-3.7&-4.8&0&0&0\\ -3.7&11.5&-4.8&0&0&0\\ -4.8&-4.8&13.5&0&0&0\\ 0&0&0&30.4&0&0\\ 0&0&0&0&31.9&0\\ 0&0&0&0&0&31.9\\ \end{array} \right] \tag{7}\]  
Figure 3: Working principle of the multilayer piezoelectric actuator. The matrix in \(d\) form used in ANSYS is (unit: pC/N):  
\[d=\left[ \begin{array}{ccc}0&0&-97\\ 0&0&-97\\ 0&0&230\\ 0&0&0\\ 0&330&0\\ 330&0&0\end{array} \right] \tag{8}\]  
Relative permittivity matrix (\(\varepsilon^{\mathrm{T}}/\varepsilon_{0}\)):  
\[\varepsilon^{\mathrm{T}}/\varepsilon_{0}=\left[ \begin{array}{ccc}1200&0&0\\ 0&1200&0\\ 0&0&900\end{array} \right] \tag{9}\]  
According to the FEA results, the final dimensions are decided to be 30 \(\times\) 7.5 \(\times\) 3 mm\({}^{3}\), as mentioned previously. The dimension of the alumina ceramic contactor is \(\Phi\) 3 \(\times\) 4 mm. A photo of the piezo-vibrator is shown in Figure 5.  
Before making a piezoelectric vibrator, the d\({}_{33}\) of the multilayer piezoelectric ceramic used in this study was obtained using the quasi-static d\({}_{33}\) meter (ZJ-4 NV, Beijing Oriental YUANTONG Science Technology Development Co., Ltd., Beijing, China), and the \(C_{0}\) was measured using the impedance analyzer (4294A, Agilent Technologies Inc., Santa Clara, CA, USA) as shown in Table 1. The number of the piezoelectric ceramic layers is 15, and the material parameter matrix \(d\) is listed in Equations (9). Figure 4: Modal frequencies at various lengths of PZT (piezoelectric lead zirconate titanate) block. Figure 5: Photo of the piezoelectric vibrator. The value of d\({}_{33}\) is 230 pC/N from Equations (9). In the ideal situation, the d\({}_{33}\) of the multilayer piezoelectric ceramic should be 230 \(\times\) 15 = 3450 pC/N, but the measured value is about 2290 pC/N, as shown in Table 1, 66% of the ideal value.