## 3 Simulation
Figures 9 and 10 are the damping characteristic graphs of the ultrasonic system and piezo element. According to the conditions in Table 1, the characteristic equation, characteristic root, damping period, vibration period, and resonance frequency are calculated, and these are shown in the graphs of Figures 9 and 10. The red graph in Figure 9 shows the output stage of the ultrasonic system, including the piezo element and \(LC\) filter, and the blue graph shows the damping curve and envelope of the piezo element, excluding the parasitic capacitor \(C_{0}\). Figure 10 shows the damping curve and envelope of the piezo element, including the \(LC\) filter and parasitic capacitor \(C_{0}\), added to Figure 9. The green color is a damping graph for the \(LC\) filter only. If a simulation is performed with \(L_{1}\) 190, 240 mH and \(C_{1}\) 120, 140 pF, which are the minimum and maximum value ranges according to the pressure fluctuation of the piezo element, data can be obtained that can estimate the characteristics of the ultrasonic system resonance frequency and the mechanical resonance frequency of the piezo element. Figure 8: S-plane graph of pressure variations in piezo elements. Figure 10: Damping characteristics graph 2 of ultrasonic system and piezo elements. Figure 9: Damping characteristics graph 1 of ultrasonic system and piezo elements. Figure 11 is a graph of the relationship between the ultrasonic system resonance frequency and the piezo element mechanical resonance frequency. The data were obtained through the process of calculating the characteristic equation, calculating the characteristic root, calculating the damping period and the vibration period, and calculating each resonance frequency, and then this was displayed as a graph. The x-axis represents the entire system resonance frequency (\(f_{system}\)) of the ultrasonic system output stage, and the y-axis represents the piezo element mechanical resonance frequency (\(f_{piezo}\)). As can be seen in Figure 11, the ultrasonic system resonance frequency and the piezo element mechanical resonance frequency each have a nonlinear relationship, and it was verified that this can be analyzed by calculating the characteristic equation and the damping characteristic. The ultrasonic system resonance frequency can be seen to be divided into right and left sections based on 30 kHz due to nonlinear characteristics. In order to analyze the ultrasonic system resonance frequency, the right resonance frequency blue section was divided into 1, and the left resonance frequency pink section was divided into 2. In this study, we propose a method to estimate the optimal resonance frequency through regression analysis to maximize the mechanical output of the piezo element by measuring the frequency of the entire system combined with the input filter and the piezo element. Therefore, it is important to select a model suitable for the piezo element showing nonlinear characteristics, learn the data, and simulate them. MATLAB simulation results showed that linear regression analysis and multiple regression analysis among various machine learning models were not suitable for estimating the resonance frequency due to the nonlinear characteristics of the piezo element itself. Accordingly, we attempted to estimate the optimal resonance frequency through a nonlinear regression analysis model and LSTM regression analysis suitable for piezo elements with nonlinear characteristics. The input data for learning were the series \(R_{1}L_{1}C_{1}\) and \(L_{f}\) and \(C_{f}\) of the \(LC\) filter, which are responsible for the mechanical output of the piezo element. At this time, the relationship between the entire system frequency and the resonance frequency of the piezo element was estimated by inputting a monotonous increasing pattern of \(L_{1}\) 190 to 240 mH and \(C_{1}\) 120 to 140 pF, which affect the resonance frequency among the series \(R_{1}L_{1}C_{1}\) of the piezo element.