## 3 Numerical Investigation of the Motor  
The goals of the numerical investigation were to validate the operating principle of the motor, to obtain optimal geometrical parameters of the stator and to calculate the mechanical and electrical characteristics of the motor. Numerical modeling was performed by employing Comsol Multiphysic 5.4 (Burlington, MA, USA). The material properties used to build the finite element model are shown in Table 2.  
Firstly, optimization of the stator geometrical parameters was performed. The sum of displacement amplitudes in X and Y directions was selected as an objective function while design variables were a length of the cut in the center of plate and length of rectangular cuts made at the center of the stator. Cuts are important elements of the stator design because it allows to reduce structural stiffness and increase the amplitude of the contact point vibrations. The optimization problem can be written as follows:  
\[\max_{u_{tot}}=(|u_{x}(L,L_{cut})|+|u_{y}(L,L_{cut})|); \tag{1}\] \begin{table}
\begin{tabular}{l l l} \hline \hline \multirow{2}{*}{**Material Property**} & **Beryllium Bronze** & **PI Ceramics** \\  & **DIN 2.1247** & **PIC151** \\ \hline Density, kg/m\({}^{3}\) & 8360 & 7800 \\ Young’s modulus, N/m\({}^{2}\) & \(10\times 10^{9}\) & \(7.6\times 10^{10}\) \\ Poisson’s coefficient & 0.34 & - \\ Isotropic structural loss factor & 0.02 & - \\ Relative permittivity & - & \(\varepsilon_{33}\)T\({}_{/60}\) = 1200 \\  & & \(\varepsilon_{11}\)T\({}_{/60}\) = 1500 \\ Elastic compliance coefficient [10\({}^{-12}\) m\({}^{2}\)/N] & - & \(\text{S}_{33}\)E = 19.00 \\ Elastic stiffness coefficient \(\text{c}_{33}\)\({}^{0}\), N/m\({}^{2}\) & - & \(1.6\times 10^{10}\) \\ Piezoelectric constant d\({}_{33}\) [10\({}^{-12}\) m/V] & - & 225 \\ Piezoelectric constant d\({}_{31}\) [10\({}^{-12}\) m/V] & - & \(-97\) \\ Piezoelectric constant d\({}_{15}\) [10\({}^{-12}\) m/V] & - & 330 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Properties of the materials. Figure 4: Vibration mode of the motor. subjected to  
\[L_{\min}\leq L\leq L_{\max}; \tag{2}\]  
\[L_{cut}^{\min}\leq L_{cut}\leq L_{cut}^{\max}; \tag{3}\]  
\[f_{\min}\leq f\leq f_{\max}. \tag{4}\]  
here, \(u_{tot}\) is a sum of displacement amplitudes in X and Y directions; \(u_{x}\) is displacement amplitude in X direction; \(u_{y}\) is displacement amplitude in Y direction; \(L\) is a length of rectangular cuts made at center of stator; \(L_{cut}\) is a length of rectangular cuts made at the center of each plate; \(L_{\min}\) and \(L_{\max}\) are minimum and maximum length values of rectangular cuts made at center of stator; \(L_{cut}^{\min}\) and \(L_{cut}^{\max}\) are minimum and maximum length values of rectangular cuts made at the center of each plate; \(f_{\min}\) and \(f_{\max}\) are minimal and maximum limits of analyzed frequency range, respectively; \(f\)--operation frequency of the motor. Other dimensions of the stator are shown in Table 1.  
Boundary conditions of the model used for the optimization problem were set as follows: four clamping places of the motor were fixed rigidly (Figure 1). The excitation scheme of the electrodes was used, as shown in Figure 3. The voltage amplitude of 50 V\({}_{\text{p-p}}\).was applied during simulation. The gravity force was included in the model. Frequency domain study and linear optimization algorithm were applied to solve the optimization problem. The frequency range was set from \(f_{\min}\)--12 kHz to \(f_{\max}\)--41 kHz with a step of 50 Hz. \(L_{\min}\) and \(L_{\max}\) were set to 0.25 mm and 2.5 mm, respectively. The increment step was set to 0.25 mm. \(L_{cut}^{\min}\) and \(L_{cut}^{\max}\) were set to 1 mm and 8 mm, respectively, while the increment step of 1 mm was used. Dependence of the objective function from design variables \(L\), and \(L_{cut}\) are shown in Figure 5. The color legend of the plot shows the values of the objective function in meters. Three candidate pairs of the L\({}_{\text{cut}}\) and L were found analyzing the results of calculations.