## 3 Numerical Modeling of the Actuator
The aim of this study was to find out the modal shapes and natural frequencies of the actuator that can be used for rotor driving. Trajectories of the displacements were investigated, these trajectories are on the top contact element, shown in Figure 4.  
Firstly, a modal-frequency analysis of the piezoelectric transducer was performed. Frequency range from 20 to 110 kHz was studied. Results showed that several modes had dominating displacements in axial and radial directions of the ring-shaped transducer. Such type displacements are needed to generate an elliptical 3D motion of contact point. The modal shape of the piezoelectric transducer at resonance frequency 93.2 kHz is presented in Figure 5.  
Lastly, modal shapes of unimorph actuator were analyzed when the disc and triangle-shaped flanges were used. The needed modal shapes were determined at the frequencies of 94.8 kHz and 95.3 kHz for the actuators with a disc and triangle-shaped flanges, as shown in Figure 6. Contact elements are displaced in radial and axial directions; therefore, the contact point moves in the XZ plane and these modes of vibrations and can be used for rotor driving. Figure 4: A point on contact element which was used for trajectory calculations. Figure 5: The modal shape of the ring-shaped piezoelectric transducer at a frequency of 93.2 kHz. Figure 3: Numerical models which were used for simulations: (**a**) piezoelectric actuator, when flange is bonded using soft rubber spacers; (**b**) piezoelectric unimorph with ring-shaped flange; (**c**) piezoelectric unimorph with triangle-shaped flange. Harmonic response analysis of the actuator was performed as well. The aim of numerical simulations were to determinate the rebound of the contact element and find out its trajectory of motion while voltage is applied to one of the segmented electrodes. An amplitude of driven electric signal was 50 V and this signal was provided for one of the top electrodes while the bottom electrode was grounded. The analysis was performed in a frequency range of 85 kHz-105 kHz, while the solution step was 50 Hz. The piezoelectric actuator has a symmetrical structure. Hence only one segment of the top electrode was excited. Numerical simulation was performed when the actuator had a disc-shaped and triangle-shaped flange, respectively. The displacement amplitudes in X, Y, and Z directions were studied and compared. The results of the harmonic response analysis are presented in Figures 7 and 8. Figure 6: Modal shapes of vibration of the unimorph actuator: (**a**) with disc-shaped flange (94.8 kHz); (**b**) with triangle-shaped flange (95.3 kHz). Figure 7: Results of numerical analysis of harmonic response with the disc-shaped flange: (**a**) amplitude vs. frequency characteristic; (**b**) At resonance mode, 95.8 kHz. A comparison of the displacement amplitudes of the actuator with disc and triangle-shaped flanges are presented in Figure 2:0  
Figure 2:1 Results of numerical analysis of harmonic response with the triangle-shaped flange: (**a**) amplitude vs. frequency characteristic; (**b**) At resonance mode, \(96.2\,kHz\). Figure 2:2 Cont. From the graphs, it can be seen that dominant vibrations appear in radial and axial directions, i.e., X and Z accordingly, and the shape of the flange has a big influence on the displacement amplitude. The higher amplitude was determined in the X-axis direction for an actuator with a disc-shaped flange. The value of this displacement is 6.2 \(\upmu\)m. Axial displacements in the Z-axis direction is also for an actuator with a disc-shaped flange and equal to 0.92 \(\upmu\)m. It must be noted that using a triangle shape for the flange, vibration amplitude is significantly smaller compared with the disc-shaped actuator. The actuator with a disc-shaped flange reaches its resonant peak at 95.8 kHz, while the actuator with a triangle-shaped flange at a frequency of 96.2 kHz. Amplitudes in X and Z directions decrease accordingly with the higher resonant frequencies.