## 4 Experimental Results
### Mechanical Characteristics Measurement  
Next, the displacement and velocity of the prototype were measured by a grating sensor (RL32 BBE001D15F; Renishaw Plc., Gloucestershire, UK) with a resolution of 1 nm [25], which was effective for acquiring the displacement changes in the prototype. A data collector was designed to visualize the grating sensor data with the PC software based on LabVIEW. The maximum thrust force of the prototype was measured by a force sensor (LH-S09A3-5N; Liheng Sensor Technology Co., Ltd., Shanghai, China) using a spring as a cushion; the load characteristic of the prototype was measured by a fixed pulley and weights which supplied a stable load parallel to the direction of the prototype's movement, as shown in Figure 9b. To obtain the optimal driving frequency, the output velocity of the prototype was measured under different frequencies. As shown in Figure 10, the optimal working frequencies of the ILUM-S and ILUM-D are around 35 kHz and 37.5 kHz, while the applied voltage is 500 V\({}_{\text{p-p}}\) and the preload force is 8 N. It should be noted that the preload force was calculated based on the number of rubber rings, the elastic modulus, and deformation. During the mechanical characteristic measurement, the actual working frequency of the prototype was generally higher than the resonant frequency acquired in the vibration measurement as a result of the boundary condition changes due to the preload force. It can also be found that the ILUM-S had a practicable output performance while the excitation frequency exceeded the resonance frequency. The cause of this phenomenon could be that the higher bending mode of the SA is excited by the harmonic excitation of the sawtooth wave signal, and subsequent research on this is in progress. Figure 10: The output velocity versus the frequency of the applied sawtooth wave signal with 500 V\({}_{\text{p-p}}\) voltage. Figure 9: Experimental system for measuring mechanical characteristics: (**a**) setup and prototype; (**b**) schematic diagram of the experimental system. After the driving frequencies of the motors were determined, it was necessary to investigate the optimal applied voltage for the motor for subsequent further measurements. The relationship between the thrust force and the amplitude of the applied voltage was explored. As shown in Figure 11a, the thrust force of the ILUM-S can reach 0.644 N when the amplitude of the applied voltage is 420 V\({}_{\text{P-p}}\), compared to 1.654 N for the ILUM-D under 500 V\({}_{\text{P-p}}\). Meanwhile, the relationship between the velocity and the amplitude of the applied voltage is illustrated in Figure 11b; the output velocity of the ILUM-S also reaches 4.798 mm/s when the amplitude of the applied voltage is 420 V\({}_{\text{P-p}}\), and 3.785 mm/s under 500 V\({}_{\text{P-p}}\). For the ILUM-D, the output velocity can reach up to 8.746 mm/s when the applied voltage is 500 V\({}_{\text{P-p}}\). The output velocity results from the experimental measurements are lower than those from the simulation results because of the additional slider friction in the mechanical characteristic measurement. In addition, it can be seen that under the condition of an 8 N preload force and the applied voltage range of 300 V\({}_{\text{P-p}}\) to 500 V\({}_{\text{P-p}}\), the output performance of the ILUM-D is close to a linear positive correlation with the applied voltage, whereas that of the ILUM-S is not. It is important to emphasize here that the maximum applied voltage was set at 500 V\({}_{\text{P-p}}\) since a higher applied voltage could cause damage to the PZTs. The plot of the output velocity versus the carrying load is shown in Figure 12. The output velocities of the ILUM-S and the ILUM-D were measured under the optimal frequency and voltage amplitude which were obtained in the previous measurements.