# Development of Piezoelectric Inertial Rotary Motor for Free-Space Optical Communication Systems
## 3 Numerical Investigation of Motor
The next stage of the numerical investigation was dedicated to the calculation of the impedance and phase frequency characteristics of the motor at a temperature of 20 \({}^{\circ}\)C. For this purpose, a frequency domain study was conducted over a range from 12,100 to 13,200 Hz, with a step size of 5 Hz, while other boundary conditions were set as in the previous case. The results of the calculations are given in Figure 7.  
As can be observed in Figure 7, the resonance frequency of the motor was identified at 12,660 Hz while the anti-resonance frequency is at 12,800 Hz. A slight mismatch of 45 Hz or around 0.5% between the resonance and natural frequencies can be attributed to discrete frequency steps used in the frequency domain analysis. Furthermore, the effective electromechanical coupling coefficient \(k_{\mathit{eff}}\) of the motor was calculated and found to be 0.1084 (Equation (4)). \[\frac{k_{\mathit{eff}}^{2}}{1-k_{\mathit{eff}}^{2}}=\frac{f_{a}^{2}-f_{r}^{2} }{f_{r}^{2}} \tag{4}\]  
Here, \(k_{\mathit{eff}}\) is the electromechanical effective coupling; \(f_{a}\) is the antiresonance frequency; and \(f_{r}\) is the resonance frequency. The next stage of the numerical study aimed to investigate the displacement-frequency characteristics while various excitation signal amplitudes were applied to the stator. The analysis was carried out over a frequency range of 12,200 Hz to 13,200 Hz, with a step size of 5 Hz, while varying the excitation signal amplitude from 40 V\({}_{\text{P-p}}\) to 180 V\({}_{\text{P-p}}\) in increment steps of 20 V\({}_{\text{P-p}}\). The results of the calculations are given in Figure 8.  
Table 2.0 Impedance– and phase–frequency characteristics of the stator at 20 \({}^{\circ}\)C.  
The simulation results, shown in Figure 8, demonstrate that the displacement amplitudes of the motor are directly related to the amplitude of the excitation signal. At the lowest excitation voltage of 40 V\({}_{\text{P-P}}\), the displacement amplitude reached approximately 8.81 \(\upmu\)m or 220.21 nm/V\({}_{\text{P-P}}\). On the contrary, at the maximum excitation amplitude of 180 V\({}_{\text{P-P}}\), the displacement amplitude peaked at 18.96 \(\upmu\)m or 105.36 nm/V\({}_{\text{P-P}}\). These results indicate that the motor displacement response scales predictably with the applied voltage, ensuring stable and controllable motion across a wide range of excitation signal amplitudes. To fully represent the displacement characteristics of the stator, a summary of maximum displacement amplitudes as well as their ratio to the excitation signal was made. The summary is given in Table 2.1  
As can be found in Figure 9, the motors exhibit stable and well-predictable displacement characteristics over the whole range of excitation signal amplitudes. This discloses that motor control and driving can be implemented via simple and cost-effective excitation and driving systems, as well as confirms that the motor is capable of delivering consistent and reliable displacement outputs, which are critical for applications requiring precise and adjustable actuation. The steady increase in displacement amplitude with higher excitation voltages suggests that the motor can be effectively tuned to achieve the desired performance by adjusting the driving voltage, providing flexibility in practical space applications. The next step of the numerical investigation was dedicated to the indication of the motor operation sequence during one period of vibration. For this purpose, a time domain study was set up with the same boundary conditions as described above while the excitation signal amplitude was set to 180 V\({}_{\text{P-P}}\) and the calculation time was set to a period of resonance frequency which was equal to 78.98 \(\upmu\)s. The results of the calculations are given in Table 2.2  
Therefore, Figure 10a represents the waveform of the electrical signal used during the time domain study.