## 4 Experimental Investigation of the Traveling Wave Actuator  
A prototype was made in order to perform experimental investigations and verify the operating principle of the actuator (Figure 7). The dimensions and material parameters of the prototype were the same as those shown in Tables 1-3. Figure 6: Displacement trajectories of the contact point: (**a**) in the XY plane; (**b**) in the XZ plane; (**c**) in the YZ plane. Figure 7: Prototype of the traveling wave actuator. Firstly, measurements of the impedance-frequency and phase-frequency characteristics were performed. A 4192A impedance analyzer (HP, Palo Alto, CA, USA) was used for this purpose. The results of the measurement are shown in Figure 8. Two resonance frequencies can be seen in the plot. The first longitudinal vibration mode has a frequency of 51.75 kHz, while the second bending vibration mode has a frequency of 52.35 kHz. The difference between the resonance frequencies is 600 Hz. Compared to the results of modal analysis, this difference is notably higher. It is mainly caused by manufacturing errors, as well as slight differences in the material properties and actuator clamping. Also, it can be noted that the impedance values at the resonance frequencies have a small difference. The impedance at the first longitudinal vibration mode is 1.1 k\(\Omega\), while that at the second bending vibration mode is 1.03 k\(\Omega\). The difference is 700 \(\Omega\). Comparing the impedance values obtained numerically and experimentally, a difference of 1 k\(\Omega\) was obtained. The mechanical characteristics of the actuator were measured as well. The experimental study was split into the following steps: experimental verification of the operating principle, measurements of the contact point displacement amplitudes in the XYZ directions, and rotation speed measurements. Firstly, experimental verification of the operating principle was performed. For this purpose, a PSV-500 scanning vibrometer was used (Polytec, Waldbronn, Germany). The goal of the measurements was to obtain trajectories of the contact point (CP) motion. The top surface displacements of the actuator were measured during one period of vibrations. The results of the measurements at 51.65 kHz are given in Figure 9.  
Figure 9 presents the motion of the contact point in \(2\frac{1}{2}\pi\) and presents a visualization of the vibrations on the top surface of the waveguide. It can be seen that the contact point vibration has a harmonic form. So, in accordance with the excitation scheme (Figure 1b), it can be concluded that a traveling wave is generated on the top surface of the waveguide. Also, the visualization shows that the traveling wave has additional motion in the longitudinal direction. Therefore, it shows that the first longitudinal vibration mode is excited as well. The contact point velocity and displacement amplitudes in the XYZ directions were measured as well. For this purpose, an experimental setup was built (Figure 10). It included a WW5064 four-channel function generator (Tabor Electronics, Israel), an E-619 power amplifier (PI Ceramic GmbH, Lederhose, Germany), a Yokogawa DL2000 four-channel oscilloscope (Yokogawa, Tokyo, Japan), a UT 372 noncontact tachometer (Uni-T, Dongguan, China), and a U3B proximity sensor (Lion  
Figure 8: Measured impedance–frequency and phase–frequency characteristics. Precision, Oakdale, MN, USA) that was used to measure velocity and displacement amplitudes of the waveguide in the XYZ directions. Finally, a computer was used to record and manage the data. The results of contact point displacement trajectory measurements in the XYZ directions under different excitation voltages are given in Figure 11. It must be noted that three different points were measured; the measured points were located on the outside surfaces of the actuator for measurement in the X and Y directions, while a point located on the top surface of the actuator was measured in the Z direction.