## 3 Numerical Investigation of the Motor
The difference of 11.75 Hz is obtained compared to the resonant frequency indicated by impedance frequency characteristics. The maximum displacement amplitude in the X direction of 91.75 \(\upmu\)m, while the ratio value is 0.458 \(\upmu\)m/V. The highest displacement amplitude in Y direction reached 117.8 \(\upmu\)m, while the ratio is 0.589 \(\upmu\)m/V. The contact point trajectory was simulated in the XY plane at an excitation voltage of 200 V\({}_{\text{P-P}}\). The time depended study was used for this purpose. One time period (T \(\approx\) 64 \(\upmu\)s) of motor vibrations was analyzed. The excitation scheme of the motor was the same as in previous cases. The results of the simulation are given in Figure 4:3 It can be seen that the contact point trajectory in the XY plane has an elliptical shape. The length of the major axis of the contact point trajectory is 116 \(\upmu\)m, while the length of the minor axis is 89.2 \(\upmu\)m. The ratio between lengths is 1.3. Figure 4:4 Displacement amplitudes of the contact point at different excitation voltage; (**a**)—displacement amplitude in the X direction; (**b**)—displacement amplitude in the Y direction. and calculated impedance values is 574 \(\Omega\). The difference is mainly caused by the glue layer that was neglected during the numerical study. The average rotation speed of the piezoelectric motor under different preload forces was measured as well. The experimental setup was built for this purpose (Figure 12). It included a computer, four-channel function generator WW5064 (Tabor Electronics, Israel), power amplifier E-619 (PI Ceramic GmbH, Germany), four-channel oscilloscope Yokogawa DL2000 (Yokogawa, Japan) and non-contact tachometer UT 372 (Uni-T, China). Figure 4:5 Impedance-frequency characteristics of the motor. Figure 4:6 Experimental setup; 1—a computer with data acquisition software; 2—a four-channel function generator; 3—a four-channel power amplifier; 4—oscilloscope; 5—a non-contact tachometer; 6—a prototype of the motor. The results of measurement of an average rotation speed of the rotor under different voltages and preload forces is given in Figure 4:7 The amplitude of excitation voltage was changed from 20 V\({}_{\text{P-P}}\) to 200 V\({}_{\text{P-P}}\) while the frequency was fixed at 16.35 kHz. Analysis of the results shows that the rotor starts rotating at the voltage of 20 V\({}_{\text{P-P}}\) in all cases, except when the preload force is 30.12 mN. Moreover, the average rotation speed at this preload force exhibits the lowest rotation speed starting from 140 V\({}_{\text{P-P}}\). Therefore, the maximum preload force of the motor is 30.12 mN. The lowest average rotation speed was measured when the preload force of 7.64 mN was used. The lowest values were obtained when the excitation voltages were in the range from 20 V\({}_{\text{P-P}}\) to 120 V\({}_{\text{P-P}}\). It can be assumed that such kinds of results were achieved because of the low friction force between the stator and rotor. The highest average rotation speed of 756.45 RPM was obtained when the preload force of 22.65 mN and voltage of 200 V\({}_{\text{P-P}}\) was applied. The lowest rotation speed of 155.77 RPM was obtained at 20 V\({}_{\text{P-P}}\) with the same preload force. Therefore, an average increment of the rotation speed of 3.34 RPM/V is provided by the motor. It should be noted that the standard deviation of rotation speed values, in most cases, does not exceed 15%-25% of average values. In addition, the instantaneous maximum rotation speed of the motor was indicated. The highest instantaneous rotation speed of 972.62 RPM was obtained at 200 V\({}_{\text{P-P}}\) voltage and 22.65 mN preload force. Compared to the average rotation speed value at these conditions, the difference is 216.17 RPM, or instantaneous rotation speed is 22.23% higher. This value does not exceed the standard deviation of average rotation speed and can be taken as the maximum rotation speed of the motor for driving at a fixed frequency of 16.35 kHz.