## 4 Discussion  
It has been observed that the results of the FOM and TOPSIS analyses are almost consistent in terms of the ranking of the state-of-the-art categories of piezoelectric materials for actuators. The categories in order of decreasing actuator performance in both approaches are single-crystal composites, PZT soft ceramics, PZT hard ceramics, thin films, and polymers. Specifically, PMN-24% PT is the best piezoelectric material for actuators among the candidates according to TOPSIS analysis, while PIN24%-PMN-PT is the best according to the FOM approach. The differences in the results of the two material selection approaches  
Figure 16: Frequency vs. amplitude graph of the face.  
Figure 17: Frequency vs. phase angle graph of the face.  
are due to the fact that the FOM approach considered \(d_{33}\) to be the sole determining criteria, while the TOPSIS approach considered the relative weights of \(d_{33}\), \(k_{33}\), \(S^{E}\)\({}_{33}\), and \(T_{33}\) as determining factors. Both PMN-24% PT and PIN24%-PMN-PT are classes of single crystal composites and ranked at the top in both material selection approaches. In general, the presence of a morphotropic phase boundary in the crystals of both materials with the given composition of each material makes them exhibit excellent piezoelectric properties.  
After selecting the optimal materials for the actuator, various finite element analyses were performed with ANSYS Mechanical Multiphysics, as shown in Figure 18. The static analysis results are summarized in Table 8.  
Of the two-candidate pair of plates, the pair of larger PMN-24% PT and PIN24%-PMN-PT plates (Case 3) had greater deformation as shown in Table 8 and was found to be the optimal plate actuator for active noise cancellation (AVC). Therefore, modal and harmonic response analyses were performed on this pair of plates. Even though the pair of two larger PIN24%-PMN-PT plates (Case 5) had the greatest deformation compared to all cases, it was just considered as a reference for the maximum achievable free displacement, due to the high cost of the material.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**No.** & **Cases** & **Method** & **Deformation (\(\mu\)m)** \\ \hline \hline
1 & PMN-24\% PT plate & Method 1 & 1.39 \\  & Method 2 & 1.39 \\
2 & PIN24\%-PMN-PT plate & Method 1 & 1.60 \\
3 & Pair of larger PMN-24\% PT and PIN24\%-PMN-PT plates & 1.60 \\  & (Candidate 1) & Method 1 & 2.46 \\
4 & Pair of larger PIN24\%-PMN-PT and smaller PMN-24\% PT plates & \\  & (Candidate 2) & Method 1 & 2.21 \\
5 & Pair of two larger PIN24\%-PMN-PT plates (as reference) & Method 1 & 3.08 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Static response analysis results of all cases.  
Figure 18: The “project schematic” of all the analyses carried out.  
The first ten natural frequencies, the first three modal shapes, and the frequency response of the optimal actuator (Case 3) were discussed in the modal and harmonic response analyses sections. From these results, it was observed that the highest longitudinal (3-3) deformation amplitude of the third case is 187.96 \(\upmu\)m at 23 KHz frequency, which is very close to the second natural frequency.