# Development of Piezoelectric Inertial Rotary Motor for Free-Space Optical Communication Systems
## 3 Numerical Investigation of Motor
On the contrary, at higher temperatures (30 \({}^{\circ}\)C and 40 \({}^{\circ}\)C), the impedance values drop, suggesting a decrease in electrical and structural resistance. The phase-frequency characteristics, shown in Figure 11b, further illustrate the effect of temperature on the electromechanical characteristics. The phase shift around the resonance frequency of 12,700 Hz is more pronounced at lower temperatures, with a steep phase transition observed at \(-20\)\({}^{\circ}\)C. As the temperature increases, the phase transition becomes less sharp, reflecting a change in the dynamic response of the motor. These results underscore the critical influence of temperature on motor performance, especially in environments such as LEO, where temperatures fluctuate widely. Finally, a summary of the calculated values is given in Table 2.8  
The resonant frequency decreases with increasing temperature, starting at 12,775 Hz at \(-20\)\({}^{\circ}\)C and dropping to 12,605 Hz at 40 \({}^{\circ}\)C. Similarly, the impedance at resonance decreases from 3602.97 \(\Omega\) at 20 \({}^{\circ}\)C to 2751.58 \(\Omega\) at 40 \({}^{\circ}\)C. The phase angle also shows a significant shift, from \(-75.81\)\({}^{\circ}\) at \(-20\)\({}^{\circ}\)C to \(-86.41\)\({}^{\circ}\) at 40 \({}^{\circ}\)C. The effective coupling coefficient \(k_{\mathit{eff}}\) varies slightly with temperature, ranging from 0.1078 at 20 \({}^{\circ}\)C to 0.1072 at 40 \({}^{\circ}\)C, suggesting a minimal impact of temperature on the energy conversion efficiency in the motor. These results demonstrate that higher temperatures lead to lower resonant frequencies, impedance, and a more negative phase angle, which affect the motor's performance in temperature-variable environments. Calculations of the frequency-displacement characteristics were performed in order to indicate displacements in relation to temperatures. For this purpose, the frequency domain study was set up with the same boundary conditions as in the case before, while the excitation signal amplitude was set to 100 V\({}_{\text{p-P}}\). The results of the calculations are shown in Table 2.9  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Temperature, \({}^{\circ}\)C** & \begin{tabular}{c} **Resonant** \\ **Frequency, Hz** \\ \end{tabular}  & **Impedance, \(\Omega\)** & **Phase, \({}^{\circ}\)** & \(k_{\mathit{eff}}\) \\ \hline \(-20\) & 12,775 & 3602.97 & \(-75.81\) & 0.1078 \\ \(-10\) & 12,745 & 3498.72 & \(-80.33\) & 0.1080 \\ \(0\) & 12,720 & 3108.87 & \(-82.69\) & 0.1081 \\ \(10\) & 12,690 & 3044.27 & \(-84.18\) & 0.1082 \\ \(The simulation results illustrate the effect of temperature variations on the displacement amplitudes and resonance frequency of the piezoelectric inertial motor. At \(-20\)\({}^{\circ}\)C, the resonance frequency is approximately 12,810 Hz with a displacement amplitude of 12.62 \(\upmu\)m (126.14 nm/V\({}_{\text{P-P}}\)). At 10 \({}^{\circ}\)C, the resonance frequency is about 12,725 Hz with a displacement amplitude of 13.01 \(\upmu\)m (130.05 nm/V\({}_{\text{P-P}}\)). At higher temperatures, the resonance frequency and displacement amplitude shift as follows: At 30 \({}^{\circ}\)C, the resonance frequency is approximately 12,670 Hz with a displacement amplitude of 13.25 \(\upmu\)m (132.51 nm/V\({}_{\text{P-P}}\)), and at 40 \({}^{\circ}\)C, the resonance frequency is about 12,640 Hz with a displacement amplitude of 13.31 \(\upmu\)m (133.14 nm/V\({}_{\text{P-P}}\)). As temperature increases from \(-20\)\({}^{\circ}\)C to 40 \({}^{\circ}\)C, the resonance frequency decreases from 12,810 Hz to 12,605 Hz, a total deviation of 170 Hz, indicating temperature-dependent reductions in stiffness affecting resonance behavior. Meanwhile, the displacement amplitude increases from 12.62 \(\upmu\)m at \(-20\)\({}^{\circ}\)C to 13.31 \(\upmu\)m at 40 \({}^{\circ}\)C, with a deviation of 0.70 \(\upmu\)m, suggesting improved mechanical response at higher temperatures.