# Development of Piezoelectric Inertial Rotary Motor for Free-Space Optical Communication Systems
## 3 Numerical Investigation of Motor  
Numerical investigations were performed to confirm the operation principle and indicate the mechanical, electromechanical, and thermal characteristics of the motor. In addition, an optimization study was performed to obtain optimal geometric and mechanical characteristics of the motor. The Finite Element Model (FEM) was built using COMSOL  
Figure 4: Excitation schematic of the motor: (1)—signal generator; (2)—polarization direction of piezoceramic plates. Multiphysics 5.6 software with respect to the geometrical characteristics, which are represented in Figure 3 and Table 2. The boundary conditions were established as follows: the stator supports, at the free ends, were rigidly fixed. Electrical boundary conditions as well as polarization directions of the piezoceramics were established with respect to the schematic shown in Figure 4. Finally, the material properties were included in the model. The stator was set as made of C17200 beryllium bronze while the piezoelectric plates were set as made of hard NCE81 piezoelectric material, and finally, the contact elements were set as made of alumina oxide material. The characteristics of the materials used to build the model are given in Table 2.  
The first step of the numerical investigation was dedicated to finding the optimal geometric parameters of the stator. For this purpose, a frequency domain study was performed in the frequency range of 7 kHz to 20 kHz with a step of 10 Hz, while the excitation signal was set at 100 V\({}_  
Upon analysis of the results shown in Figure 5, it was found that the maximum displacement amplitude in the y-direction was achieved when h was equal to 7.5 mm and \(L\) was equal to 20 mm, reaching a peak of 15.76 \(\upmu\)m. On the other hand, there were other combinations of parameters that propose similar displacement amplitudes, such as \(h\) equal to 7 mm and \(L\) equal to 19.5 mm with displacement amplitude of 14.61 \(\upmu\)m and \(h\) equal to 6.5 mm with \(L\) equal to 19 mm displacement amplitude reaching 13.16 \(\upmu\)m. So, these results suggest that the optimal geometric parameters to maximize use of the motor have been identified and can be used for further investigations. The next phase of the numerical investigation focused on conducting a modal analysis of the motor. For this study, the geometric parameters and material properties established during the optimization process were applied to the numerical model, while other physical and geometric parameters were established as given in Tables 1 and 2. The primary objective was to determine the modal shape and its natural frequency suitable for the generation of angular motion using the inertial stick-slip principle. The results of the calculations are given in Figure 6.  
Figure 5: Displacement amplitudes of the motor while different geometrical parameters are used. Figure 6: The modal shape of the stator at 12.82 kHz. The results of the modal analysis, as shown in Figure 6, revealed that the modal shape of the stator suitable for motor operation via the inertial operation principle is 12.82 kHz. The bimorph plates at this natural frequency vibrate at the first bending mode which allows to obtain the inertial stick-slip principle while a sawtooth signal is applied to the plates. Moreover, Figure 6 shows that the stator supports are placed at the nodes of the vibration mode nodes. Such positioning of the supports allows to reduce structural damping of the whole stator. Moreover, as the figure shows, alumina oxide contacts are at the antinodes of the vibration mode. This allows the proper transfer of the vibration amplitudes of the stator to the rotor and, as a result, the proper performance of the whole motor. Overall, considering the results of the modal analysis, it can be stated that the operation of the motor in the first bending mode, as the well as the positions of the supports and contacts, will ensure proper operation of the motor.