## 4 Experimental Studies  
Experimental studies were accomplished. Three experimental prototypes of the piezoelectric actuator were designed and built for the result validation of the numerical study. One prototype was a ring-shaped transducer attached to the soft rubber base. The other two prototypes were of unimorph-type with the disc and triangle-shaped flanges and piezoelectric ring-shaped transducers bonded using epoxy resin. The piezoelectric material of PZT-4 was used for the transducer, the dimensions and properties of the materials are shown in Table 1. Laser scanning vibrometer Polytec PSV-500-3D-HV (Waldbronn, Germany) and driving voltage amplifier P200 (FLC Electronics AB, Partille, Sweden) were used for the experimental studies [21; 22]. The experimental setup is presented in Figure 11.  
An experimental investigation of the dynamic characteristics of the actuators using a Polytec scanner was executed with the following steps: first of all, planar surfaces of the actuators were scanned, and virtual meshes were created. Later, the driven electric signal was determined in a frequency range from 50 to 110 kHz. This signal was amplified by a voltage amplifier and applied for one of the segmented electrodes of the transducer. 3D scanning vibrometer allows a non-contact, optical measuring of the amplitudes of vibrations and defines the operational frequencies of the actuators. The deformations of the scanned actuators are shown in Figures 12-14. Firstly, the piezoelectric transducer without flanges was analyzed. This piezoelectric transducer was glued on soft rubber tape and attached to the steel plate. Soft rubber tape reduces constraints, and piezoelectric transducer can perform closer to its natural frequencies of vibration modes. The setup and vibration mode shape of the piezoelectric transducer is shown in Figure 12.  
Secondly, an actuator with triangle-shaped flange was measured. The results of measurements are shown in Figure 13 at the vibration frequency of 93.72 kHz. Lastly, an actuator with a disc-shaped flange was analyzed. The results of this analysis show that operational vibration modes are achieved at 94.4 kHz. The vibration mode shape of the actuator with the disc-shaped flange and measured amplitude of vibration versus frequency graph are presented in Figure 14.  
Figure 11: Measurement instruments used in experiments: Polytec Laser Doppler Scanner PSV-500-3D-HV. Figure 12: Measured vibrations of the piezoelectric transducer glued on the soft rubber tape: (**a**) prototype deformations and shape of the transducer driven with harmonic electric signal of the 92.48 kHz; (**b**) amplitude vs. frequency graph, peaks of oscillations at 61 and 92.48 kHz. Figure 13: Cont. It could be noticed that deformations of the actuators are similar to what was achieved in numerical simulations. Displacements in Z direction are relatively small compare to X and Y directions, when piezoelectric transducer operates without flange. When the flange is added, the amplitudes in Z direction increases and higher displacements are achieved. This is an important feature of the actuator design, and it crucial for the elliptical trajectories' generation of the contact point and proper operation of the actuator. Moreover, it can be noticed that results of displacement amplitudes of numerical calculations are similar to experimental studies, actuator with disc-shaped flange is ~1% and with triangle-shaped flange ~3% higher than experimental measurements. The errors can be induced by various causes: slightly different material properties, inaccurate prototypes' dimensions and others. Deformations of actuators and their oscillation modes during experimental studies are more pronounced and visible in excitation zones than numerical calculations. This could be a reason of additional mass of wire soldering points, not free fixtures, errors of measurements.