## 3 Numerical Investigation of the Motor
The design variable values and displacement amplitudes in X, Y, and the total sum of amplitudes are listed in Table 3.  
Figure 5: The plot of objective function dependence from design variables. It can be seen that the largest displacement amplitudes in X and Y directions are obtained when L\({}_{\text{cut}}\) and L have a value of 7 mm and 2 mm, respectively. While the motor is configured with these values, the displacement amplitude in the X direction is 7.4 \(\upmu\)m, and it is 40.5% higher value compared to the case No. 1 and 77% higher compared to the case No. 2. Displacement amplitude in Y direction reaches 9.8 \(\upmu\)m, and it is 29.5% higher compared to case No. 1 and 30.5% higher compared to the case No.2. Finally, the total sum of the displacement amplitudes in X and Y direction reaches 17.2 \(\upmu\)m, and it is 35.5% higher than case No. 1 and 50.6% higher compared to case No. 2. Therefore, the aforementioned L\({}_{\text{cut}}\) and L values were chosen for further numerical investigations. The next step of the numerical investigation was to perform a modal analysis of the motor. A numerical model with optimized stator dimensions was used. The mechanical boundary conditions were the same as in the optimization problem, while short circuit electric boundary conditions were applied. The proper modal shape of the stator was obtained at 15.4 kHz (Figure 4). It can be seen that the first in-plane bending mode of the plates is obtained. Furthermore, beams of the V-shaped type structure have the first in-plane vibration mode as well. Therefore, it can be concluded that motor vibration mode is suitable for rotary motor operation. The next step of the numerical study was dedicated to the investigation of mechanical and electrical characteristics of the motor. Firstly, impedance-frequency and phase-frequency characteristics were calculated in the frequency domain. The analysis was performed for one piezo ceramic plates group (\(Z_{1}\)), which was affected by a single harmonic excitation signal. Meanwhile other ceramics were set to open loop condition. The goal of the analysis was to indicate the resonance frequency of the motor. The results are given  
The lowest impedance of 605 \(\Omega\) was obtained at 15,721.75 Hz. The difference between modal and resonant frequencies of the motor is 303.25 Hz. This difference occurs due to the different electrical boundary conditions used for modal and frequency domain analyses. The effective coupling coefficient \(k_{\text{eff}}\) of the motor is 0.0483. The next step of the numerical investigation was dedicated to validating the operation principle of the motor. The time-dependent study was set up to simulate one time period (T) of motor operation that was equal to 64 \(\upmu\)s. An excitation voltage of 100 V\({}_{\text{P-P}}\) was applied. Boundary conditions and excitation scheme of the electrodes were the same as in the previous simulations. The operation sequence of the motor per time period of vibrations is shown in Figure 4:0  
It can be seen that the contact point CP makes a closed trajectory in a counterclockwise direction during the motor operation in one period. In order to change the motion direction, the phases of the excitation signals must be changed to the opposite. Moreover, it can be seen that beams of the V-shaped structure vibrate in the first in-plane vibration mode as well. Further, the numerical investigation was focused on the characteristics of contact point displacement amplitudes in X and Y directions at 200 V\({}_{\text{P-P}}\) excitation voltage amplitude. The frequency-domain study was employed, and the frequency range from 15.67 kHz to 15.78 kHz with a step of 0.05 Hz was analyzed. The results of the investigation are given in Figure 4:1  
Figure 4:2 The operation sequence of the motor during one period; CP—contact point. The highest displacement amplitudes in X and Y directions were obtained at a resonant frequency of 15.72 kHz.