## 3 Numerical Investigation of the Actuator  
Numerical modeling of the actuator was performed to obtain suitable modal shapes, to analyze electrical and mechanical characteristics of the actuator, and to study vibrations of the contacting elements while a single electrode is affected by the excitation signal. Comsol Multiphysics 5.4 was used to build the numerical model. Geometric dimensions were set as shown in Table 1. Material properties used to build the model are shown in Table 3. C17200 beryllium bronze was used for the cylindrical part of the actuator, piezoelectric characteristics were set for the ring, and finally, alumina oxide properties were used for spherical contacting elements (Figure 1). Boundary conditions were set as follows: gravity force was included, the model was analyzed as a mechanically free system, and short circuit electric boundary conditions were used during modal-frequency analysis. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{
\begin{tabular}{c} **Motion** \\ **Direction** \\ \end{tabular} } & \multicolumn{3}{c}{**Rotary**} & \multicolumn{3}{c}{**Planar**} \\ \cline{2-7}  & **SW\({}_{1}\)** & **SW\({}_{2}\)** & **SW\({}_{3}\)** & **SW\({}_{4}\)** & **SW\({}_{5}\)** & **SW\({}_{6}\)** \\ \hline
0\({}^{\circ}\) & 0 & 0 & 1 & 0 & 1 & 0 \\
120\({}^{\circ}\) & 1 & 0 & 0 & 1 & 0 & 0 \\
240\({}^{\circ}\) & 0 & 1 & 0 & 0 & 0 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Switching between electrodes used for motion direction control. Figure 3: Excitation schematics of the actuator; 1—signal generator; 2—switching box used for angular motion control; 3—switching box used for linear motion control; 4—polarization direction of piezoceramic ring. Firstly, modal-frequency analysis of the actuator was performed to identify modal shapes that can be used for achieving planar and rotational motions. The spherical payload was substituted by the distributed load placed on the three spherical contacts located on the top of the piezo ceramic ring. Suitable vibration modes were found at the frequencies of 18.02 kHz and 46.199 kHz (Figure 4). It can be seen that vibrations of individual parts of the actuator occur as different vibration modes. Three supports have radial vibrations at the frequency of 18.02 kHz that can be used to induce planar motion of the actuator. The third radial vibration mode of the piezoceramic ring is obtained at the frequency of 46.199 kHz and can be used to rotate the payload. It must be noted that there is loose coupling between vibrations of the spherical contacts located at the top and bottom of the actuator when the actuator vibrates at these two modes. Therefore, it will be possible to excite planar and angular movements independently. The next step of the numerical investigation was the analysis of impedance-phase-frequency characteristics with the goal of indicating operation frequencies of the actuator. The calculations were performed at the following electrical boundary conditions: the electrode dedicated to angular or linear motion control was affected by excitation voltage of  
Figure 4: Modal shapes of the actuator: (**a**)—the first radial mode (18.02 kHz); (**b**)—the third radial mode of the piezoceramic ring (46.199 kHz).