## 2 Design and Operation Principle of the Actuator  
The actuator presented herein consists of a cylindrical waveguide with a conical hole inside, three piezoceramic rings, and a clamping system (Figure 1). The waveguide is composed of two parts that are joined by a thread junction. Such a waveguide configuration is used to locate a piezoceramic ring at a node of longitudinal vibrations of the waveguide. The waveguide was made from aluminum alloy 6061, and PZT8 hard piezoceramic rings were used for the actuation. The two piezoceramic rings were located at the node of the first longitudinal and second bending vibration modes, while one ring was glued at the bottom of the waveguide. The clamping system was located at the bottom of the waveguide and consists of a supporting cylinder, a ball bearing, a ball bearing bed, a fixing bolt, and a fixing ring. A schematic of the actuator is shown in Figure 1. The main dimensions of the actuator are given in Table 1.  
The dimensions of the actuator were obtained by numerical modeling and are described in detail in the next section. The conical hole of the waveguide was designed to amplify displacement and velocity amplitudes at the top surface of the waveguide. Moreover, due to the nonuniform stiffness of the waveguide, the node of the first longitudinal and second bending vibration modes was shifted down from the center of the cylinder; as a result, displacement amplitudes are amplified additionally because of the increased lever arm. The clamping system of the actuator operates as a counter-loosening system for the clamping bolt and composes the supporting cylinder that is placed at the center of the waveguide bottom. The supporting cylinder is clamped to the waveguide using an M3 bolt, while the other end of the supporting cylinder is put to the inner face of the ball bearing. The outer ring of the ball bearing is clamped into the ball bearing bed rigidly. Such a clamping system prevents possible loosening of the clamping force applied to the M3 bolt during actuator operation. The operating principle of the actuator is based on the superposition of the second bending and the first longitudinal vibration modes. Excitation of the second bending vibration mode is performed by the piezoceramic ring which was glued at the bottom of the waveguide, while the first longitudinal vibration mode is excited by two piezoceramic rings which were placed at the node of the first longitudinal mode. (Figure 1a). Excitation of the second bending vibration mode by piezoceramic placed at the bottom of the waveguide ensures generation of the traveling wave. For this purpose, the electrode of the piezoceramic ring was divided in four equal sections which are effected by four harmonic signals with phase difference \(\pi\)/2 (Figure 1b). Such an arrangement of the excitation signals ensures that the waveguide sequentially bends in accordance with the phase of the excitation signals; in this way, an even traveling wave is generated in the clockwise direction on the top surface of the waveguide. In order to generate a traveling wave in the counterclockwise direction, the phases of the excitation signals must be changed by \(\pi\). Moreover, the conical hole which was made in the waveguide amplifies vibrations of the traveling wave due to the nonuniform stiffness of the waveguide, i.e., the stiffness at the top of the waveguide is lower compared to that at the bottom of the waveguide. To amplify displacements in the axial direction, the first longitudinal vibration mode of the waveguide is excited by two piezoceramic rings which were located at the node of the vibration mode. The first longitudinal vibration mode is excited by a single harmonic signal (Figure 1b). Therefore, it can be assumed that the traveling wave generated on the top surface of the waveguide has additional motion in a longitudinal direction, which improves the dynamic characteristics of the actuator.