## 3 Results
After selecting the optimal materials, various finite element analyses were performed with ANSYS Mechanical Multiphysics. Among the possible analysis types for models containing piezoelectric elements using ANSYS Multiphysics, static, modal, and harmonic analyses were performed in this work. Static response analyses were performed on two individual piezoelectric plates and two different pairs of plates to determine the total deformation or free displacement due to externally applied DC voltage. The two piezoelectric plates were modeled as PMN-28% PT and PIN24%-PMN-PT plates, because PMN-28% PT and PIN24%-PMN-PT were the top two ranked materials. The two different modes of pairing the plates are: the two plates having the same size, but being made of PMN-28% PT and PIN24%-PMN-PT materials, and the two plates being made of the same material, but with different dimensions. Based on the total displacement results, the static response analyses helped to select the better-performing pairing of the two piezoelectric plates to be used as an actuator for the piezoelectric damper project. The dimension of the larger plate was 30 mm \(\times\) 30 mm \(\times\) 10 mm, while the smaller plate dimension was 18 mm \(\times\) 18 mm \(\times\) 8 mm. For comparison, additional static response analysis was performed on a pair of piezoelectric plates of the same material and dimensions. In this case, the material was determined by the results of the individual plates, i.e., the material that results in higher deformation was used in this pair of plates. In order to carry out the abovementioned types of analyses on models, the characterization of selected materials in terms of compatibility with ANSYS is essential and was done as shown below. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Material** & \(k_{33}\) & \(\begin{pmatrix}S^{E}{}_{33}\\ \left(p\frac{\pi^{2}}{N}\right)\end{pmatrix}\) & \(\epsilon^{T}{}_{33}\) & \(d_{33}\) & **FOM** & **TOPSIS** & **TOPSIS** \\  & & \(\left(\frac{\pi^{2}}{m}\right)\) & \(\left(\frac{\pi^{2}}{m}\right)\) & \(\left(\rho\frac{C}{N}\right)\) & **Rank** & **Index** & **Rank** \\ \hline PIN24\%-PMN-PT & 0.89 & 49.04 & 42.06 & 1285 & 1 & 0.6558 & 2 \\ PMN-28\% PT & 0.90 & 36.15 & 48.68 & 1190 & 2 & 0.6749 & 1 \\ PZT-5H & 0.75 & 21.00 & 33.63 & 650 & 3 & 0.4153 & 4 \\ PZT-5J & 0.74 & 18.80 & 18.59 & 500 & 4 & 0.4302 & 3 \\ PZT-5A & 0.72 & 18.60 & 15.93 & 390 & 5 & 0.3868 & 5 \\ PZT-4 & 0.71 & 17.00 & 14.60 & 372 & 6 & 0.3786 & 6 \\ PZT-8 & 0.64 & 13.50 & 9.80 & 275 & 7 & 0.3309 & 9 \\ ZnO film & 0.69 & 14.70 & 12.39 & 246 & 8 & 0.3439 & 8 \\ (PVDF-TrFE)75/25 & 0.23 & 300.00 & 0.07 & 33.5 & 9 & 0.0306 & 10 \\ PZT film (4 \(\upmu\)m on Si) & 0.23 & 300.00 & 0.07 & 33.5 & 10 & 0.030670 & 11 \\ PVDF & 0.18 & 472 & 0.0673 & 32.5 & 11 & 0.009976 & 12 \\ AlN film & 0.33 & 4.74 & 0.072 & 12 & 12 & 0.358883 & 7 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Ranking of the piezoelectric materials for actuators according to FOM and TOPSIS approaches. Property matrices of PMN-28% PT in Z-polarization:  
\[C^{E}(6\times 6)=\left[ \begin{array}{cccccc}11.58&10.23&9.31&0&0&0\\ &11.58&9.31&0&0&0\\ &&10.71&0&0&0\\ &&&6.4&0&0\\ sym.&&&6.4&0\\ &&&6.01\end{array} \right]\times 10^{10}Nm^{-2}, \tag{11}\]  
\[e(3\times 6)=\left[ \begin{array}{cccccc}0&0&0&0&0&8.69\\ 0&0&0&0&8.69&0\\ -13.12&-13.12&21.72&0&0&0\end{array} \right]Cm^{-2}, \tag{12}\]  
\[\varepsilon^{S}(3\times 3)=\left[ \begin{array}{cccc}1467&0&0\\ 0&1467&0\\ 0&0&895\end{array} \right]\times\varepsilon_{0}, \tag{13}\]  
where: \(C^{E}\) denotes the stiffness matrix at constant electric field, \(e\) is the piezoelectric stress matrix, \(\varepsilon^{S}\) is the dielectric at constant strain matrix, and \(\varepsilon_{0}\) is vacuum permittivity; its value is 8.85 \(p\frac{F}{m}\). Density of PMN-28% PT\(\rho=8100\)\(\frac{kg}{m^{2}}\)[42]. Property matrices of PIN24%-PMN-PT in Z-polarization:  
\[C^{E}(6\times 6)=\left[