# Development of Piezoelectric Inertial Rotary Motor for Free-Space Optical Communication Systems
## 3 Numerical Investigation of Motor
The graph includes six time points, marked from \(t_{0}\) to \(t_{5}\), indicating the time points when the shape of the view of the motor vibration was captured. So, during the analysis of Figure 10b, which represents the motor vibration sequence, it can be found that the vibration mode and excitation schematics are suitable for the generation of rotor angular motion through the inertial stick-slip principle i.e., the sequence shows that the motor operates in stick mode from \(t_{0}\) to \(t_{2}\), then from \(t_{2}\) to \(t_{3}\) slip mode is initiated, and from \(t_{3}\) to \(t_{5}\), again, stick mode is obtained. Continuous repetition of these stages will ensure angular motion generation of the rotor. Therefore, on the basis of these results, it can be stated that motor operation at the desired configuration and principle were numerically confirmed. Table 2.3 Displacement–frequency characteristics of the stator. The final step of the numerical investigation involved assessing the operation of the piezoelectric inertial motor under various temperature conditions, simulating the environment of low Earth orbit (LEO). Firstly, the numerical investigation was conducted to analyze the impedance- and phase-frequency characteristics of the motor under these temperature conditions. The study covered a temperature range from \(-20\)\({}^{\circ}\)C to \(40\)\({}^{\circ}\)C, with a temperature increment step of \(10\)\({}^{\circ}\)C. For each temperature setting, a frequency domain study was performed in the range of 12,200 Hz to 13,200 Hz with a step size of 5 Hz. The results of the calculations are provided in Table 2.4  
Table 2.5 Summary of motor displacement–frequency characteristics: (**a**)—maximum displacement values; (**b**)—maximum displacement values ratio to excitation signal amplitude. The results of the numerical analysis reveal that the impedance- and phase-frequency characteristics of the motor exhibit a clear temperature-dependent behavior. As observed  
Table 2.6 Impedance– and phase–frequency characteristics of the motor under different temperature conditions: (**a**)—impedance–frequency characteristics of the motor; (**b**)—phase–frequency characteristics of the motor. Table 2.7 Sequence of motor vibrations during one period: (**a**)—waveform of excitation signal; (**b**)—view of motor vibrations on defined time point. in the impedance characteristics (Figure 11a), the resonant frequency and the impedance value shifts with changes in temperature. At \(-20\)\({}^{\circ}\)C and \(-10\)\({}^{\circ}\)C, the resonant frequencies are observed to be higher, approximately 12,845 Hz and 12,805 Hz, respectively. This increase in resonant frequency can be attributed to the reduced thermal expansion of the motor materials at lower temperatures, leading to stiffer mechanical properties and, as a result, higher structural damping. As the temperature increases to 30 \({}^{\circ}\)C and 40 \({}^{\circ}\)C, the resonant frequency decreases to around 12,685 Hz and 12,640 Hz, respectively. This decrease is due to the thermal expansion of the materials at higher temperatures, which causes a reduction in stiffness, resulting in a lower resonance frequency. The impedance peaks also become less pronounced, and the phase transitions are less sharp, indicating a decrease in the electromechanical response of the motor. At temperatures around 0 \({}^{\circ}\)C to 20 \({}^{\circ}\)C, the resonant frequency and the corresponding impedance and phase characteristics present intermediate values, reflecting a balance between the effects of thermal contraction and expansion on the mechanical properties of the materials. As depicted in Figure 11a, the impedance characteristics of the motor show significant variations across the temperature range. At lower temperatures (\(-20\)\({}^{\circ}\)C and \(-10\)\({}^{\circ}\)C), the impedance values near the resonance frequency are higher, indicating an increase in the electrical and structural resistance of the motor.