## 3 Numerical Investigation of the Actuator
This shows that superposition of the vibration modes can be obtained using several sets of variables. The design variable values of H =49 mm and \(D_{in}\) = 16.5 mm were chosen for further investigation because they provided the lowest difference between the natural frequencies. Frequency values of 51202 Hz and 51245 Hz were obtained for the first longitudinal and the second bending vibration modes, respectively. A view of the vibration modes is given in Figure 3.  
Figure 2: Dependence of the difference between the natural frequencies of the first longitudinal and second bending vibration modes on the diameter of the hole and height of the cylinder. The next step of the numerical study was to investigate the electrical and mechanical characteristics of the actuator. The displacement and velocity amplitudes of the contact point were analyzed. Moreover, the motion trajectories of the contact point (CP) were calculated. Also, analysis of the impedance-frequency and phase-frequency characteristics was performed. In order to obtain these characteristics, the same numerical model of the actuator was analyzed in the frequency and time domains. Firstly, the impedance-frequency and phase-frequency characteristics were calculated to obtain the electrical characteristics of the actuator. The results of the investigation are given in Figure 4. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Geometrical Parameters**} & \multicolumn{2}{c}{\
\begin{tabular}{c} **The First Longitudinal** \\ **Vibration Mode, Hz** \\ \end{tabular} } & \multicolumn{2}{c}{
\begin{tabular}{c} **The Second Bending** \\ **Vibration Mode, Hz** \\ \end{tabular} } & \multirow{2}{*}{\(\Delta f=|w_{L1}-w_{B2}|,Hz\)} \\ \hline H = 35 mm; \(\text{D}_{\text{in}}\) = 16.5 mm & 66,246 & 66,190 & 56 \\ H = 45 mm; \(\text{D}_{\text{in}}\) = 18 mm & 55,937 & 55,878 & 59 \\ H = 49 mm; \(\text{D}_{\text{in}}\) = 16.5 mm & 51,202 & 51,245 & 43 \\ H = 53 mm; \(\text{D}_{\text{in}}\) = 15 mm & 46,931 & 46,997 & 66 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The values of design variables and optimization criteria. Figure 3: The vibration modes of the waveguide; (a) the first longitudinal vibration mode at 51,202 Hz; (b) the second bending vibration mode at 51,245 Hz; CP, contact point. By analyzing the results, it can be noted that the impedance values of the actuator at the first longitudinal and second bending vibration modes were 2.35 k\(\Omega\) and 2.15 k\(\Omega\), respectively. Similar impedance values at the resonant frequency show that damping of the actuator is similar when both vibration modes are excited. Also, low values of impedance show that the actuator has high-efficiency electrical energy conversion. The amplitude-frequency and velocity-frequency characteristics of the contact point were studied as well. The amplitude of the excitation voltage was set to 200 V\({}_{\text{P-P}}\) while excitation signals were applied to the piezoceramic rings, as shown in Figure 1b. The frequency range from 49 kHz to 54 kHz with a frequency step of 5 Hz was analyzed. The displacement and velocity amplitudes of the contact point located on the top surface of the waveguide were analyzed. The graph related to the displacement amplitudes is given in Figure 5. An analysis of the velocity amplitudes is given below Figure 5.  
The graphs in Figure 5 show that the actuator generates displacements in the X, Y, and Z directions. Also, it can be pointed out that the first longitudinal and second bending modes of the actuator are excited simultaneously. The amplitude peak in the Z direction is slightly shifted down in the frequency  
Figure 4: Impedance–frequency and phase–frequency characteristics of the actuator: (**a**) the characteristics of the first longitudinal vibration mode; (**b**) the characteristics of the second bending vibration mode. domain compared to the other peaks. The difference between peaks is 42 Hz.