# Modelling and Study of the Effect of Geometrical Parameters of Piezoelectric Plate and Stack
## 4 Discussion  
The identified parameter was found to be the best fit for the plate operated at the same frequency. The damping coefficient and the spring constants were the parameters identified as being used for stack modelling. The manufacturer's data sheet was used to determine the remaining parameters. The complex vibration generated by the upper and lower plates of single stack at different frequencies fit with the theoretical model, with a slightly higher percentage error than the same frequency actuated. Because the triggering voltage was somewhat lower than the transducers' resonance frequency, the response of displacement and frequency was supposed to be linear. Figure 6a-c depicts the response of the output displacement over the input voltage in the given wide variety of frequency. On the graph, the sensitivity of the material to the frequency is depicted and, according to the results, the thicker PZT plate responded at the lower frequency and more damping efficiency at that frequency. The displacement response was also higher for the thicker PZT plate, which could be practically predictable. Figure 7 illustrates the influence of numerous cross-sectional areas on a constant plate thickness. It was observed that the area determined the elongation rather than the sensitivity. The smaller the area, the greater the displacement response with respect to the applied load. Because the applied load (voltage) was parallel to the thickness axis, the plate resonance frequency was essentially dependent on the thickness. The analytical solution of the plate's impedance response over a large frequency range is shown in Figure 8. The analytical solution verifies that the plate's response was identical to the simulation result in Figure 6. The resonant frequency of a plate is affected by its thickness. In [19], the characteristics of a piezoelectric plate were determined and described using both a non-classical and a classical method. The conclusion of the basic dynamic equation solved and simulated in this study was largely consistent with the results of the non-classical approach. Figure 6: _Cont_. Figure 6: FEM analysis of harmonic response of (**a**) 2 mm, (**b**) 4 mm, (**c**) 8 mm thick piezo plates for the first six natural frequencies. Figure 7: Frequency domain response of different areas of piezoelectric plate at wide frequency band. The model, which was operated at the same frequency spectrum, agreed with the experimental data with a margin of error of 7.14% as depicted in Figure 9. The model operated at different frequency spectra, on the other hand, agreed with the experimental data, with a slightly larger percentage of inaccuracy. As a result, because the margin of error is tolerated, it is possible to characterize the model based on the mathematical modelling results. As the maximum displacement, the upper plate displacement of the piezoelectric stack was calibrated for the sinusoidal voltage inputs of each plate in the single stack. The maximum displacement from the same plate stack was 0.035 micrometer, while the maximum displacement from different plates stack was 0.025 micrometer, confirming that same-sized plates in piezoelectric stacks produce greater displacement than differently sized plates as shown in Figure 10. This was due to the fact that the electrode area exposed to the electrical field in the same size plate was greater than in the different size plate. Figures 11 and 12 shows a frequency and time domain comparison of a stack made up of the same and different size plates actuated at 10 kHz and 8 kHz frequencies for each  
Figure 8: Analytical frequency response of different piezoelectric plates based on thickness effect. Figure 9: Time domain response of single stack composed of two same-sized piezoelectric plates modulated at the same voltage frequency separately. plate in the stack, confirming that the same size plate produces more displacement than a differently composed stack. However, at 10 kHz, the displacement is about the same for both stacks.