# Modelling and Study of the Effect of Geometrical Parameters of Piezoelectric Plate and Stack
## 3 Results  
The experimental data from the three independent frequencies of the displacement results were used to identify parameters. Next, the average values of the damping coefficient and spring constant were obtained, and the results were also guaranteed to characterize the entire system. The displacement response obtained in this experiment was excited by sinusoidal voltage. The three sets of experimental data used to estimate the parameters were:  
* The experimental results of the displacement response of the stack composed of different-sized plates--big and small --were actuated at 10 kHz and 8 kHz frequencies of voltage, respectively.
* The experimental results of the displacement response of the stack composed of the same-sized plates were actuated at 10 kHz and 10 kHz frequencies of voltage, respectively.  
Harmonic response is used to analyze the dynamic response of the plate at the obtained natural frequency. The linear model of the piezoelectric plate was excited with harmonic loads at different frequencies to investigate the system response in amplitude. For the  
\begin{table}
\begin{tabular}{l c c} \hline \hline
**Notation** & **Value** & **Unit** \\ \hline Piezoelectric coefficient (\(d_{33}\)) & \(3.28\times 10^{-10}\) & C/N \\ Electrode area of plate 1 (\(A_{1}\)) & \(9\times 10^{-4}\) & m\({}^{2}\) \\ Thickness of plate 1 (\(t_{1}\)) & 0.01 & m \\ Electrode area of plate 2 (\(A_{2}\)) & \(3.24\times 10^{-4}\) & m\({}^{2}\) \\ Thickness of plate 2 (\(t_{2}\)) & 0.008 & m \\ Electromechanical conversion coefficient of plate 1 (\(N_{1}\)) & 3.3708 & N/V \\ Electromechanical conversion coefficient of plate 2 (\(N_{2}\)) & 1.5169 & N/V \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameter adopted from the dimensions and constants of piezo plate for generating graphs.  
Figure 5: Parameter identification methodology.  
obtained modal frequency, the harmonic response of the plate was conducted with help of ANSYS Multiphysics.  
The harmonic results obtained through the help of Ansys Multiphysics are depicted in the above figures. Three different plates with thicknesses of 8 mm, 4 mm, and 2 mm, were used as samples to investigate the thickness effect on a wide range of frequencies. The 8 mm thick plate responded more closely at lower frequency than both the 4 mm and 2 mm piezo plates for the given input excitation voltage. The results obtained in ANSYS Multiphysics were further used to plot and compare the thickness effects using MATLAB software.  
These results are very helpful to characterize piezoelectric material and enhance the optimum utilization of piezoelectric material in mechatronics systems. The data were analyzed for three different types of piezo plates, each with a different thickness and surface area. The findings were simulated for the expansion of piezo plates along the longitudinal axis, i.e., the assumption was the deformation of \(d_{33}\) (piezoelectric coefficient polarized along longitudinal axis). Within the modal frequency range, the plate was triggered at a harmonic voltage of 400 V pick. A time domain analysis of the system triggered at various voltage spectrum was carried out. The input alternating current voltage was a sinusoidal wave with an amplified voltage at varied kilohertz of upper and lower plate frequency. It is difficult to obtain complete information from the time domain. Using the FFT tool, the time domain results were plotted in the frequency domain, and the peak amplitude of the responses was detected at the forcing frequency.