## 2 Design and Operation Principle of the Motor  
The motor presented in this paper consists of a cross-shaped stator that contains four identical rectangular plates (Figure 1). The plates are rigidly connected to each other by an angle of 90 degrees and form the cross-shaped stator. Each plate has a rectangular cut in the centerline that was made to reduce the structural stiffness of the plates and to increase amplitudes of vibrations. In addition, four symmetrical cuts are made at the center part of the stator for the same reason. The circular hole is drilled in the center of the stator and is used for rotor positioning and driving. Sixteen piezo ceramic plates are glued on both sides of the stator and are used to excite the in-plane bending mode of vibrations. A double-sided hemispherical rotor is placed in the center of the stator. The rotor consists of two hemispheres, a shaft, a preload spring, and a clamper. Clamping of the motor is implemented by four V-shaped type beam structures that are used to mount motor using four M1.4 bolts. The V-shaped structure was designed so that the frequency of the first in-plane bending mode of the beams would coincide with the operating frequency of the motor. Such a design of the clamping allows reducing structural damping of the motor. The isometric view of the motor design is given in Figure 1.  
The geometrical parameters of the stator are shown in Table 1. The length of the cuts was obtained by solving an optimization problem. The optimization problem is described in Section 3, while schematics and dimensions of the stator are given in Figure 2 and Table 1. \begin{table}
\begin{tabular}{l l l} \hline \hline \multicolumn{1}{c}{**Parameter**} & \multicolumn{1}{c}{**Value**} & \multicolumn{1}{c}{**Description**} \\ \hline L\({}_{\text{Stator}}\) & 10 [mm] & Length of the stator plate \\ L\({}_{\text{cut}}\) & 7 [mm] & Length of the cut in the center of plate \\ L\({}_{\text{pzt}}\) & 6.5 [mm] & Length of piezo ceramic plates \\ L & 2 [mm] & Length of additional cut \\ D & 5 [mm] & Distance from the end of stator to the clamping \\ W\({}_{\text{stator}}\) & 6.5 [mm] & Width of stator plate \\ W\({}_{\text{pzt}}\) & 3 [mm] & Width of piezo ceramic plate \\ W\({}_{\text{cut}}\) & 0.5 [mm] & Width of the cut in the center of plate \\ W & 0.25 [mm] & Width of additional cut \\ t & 0.25 [mm] & Width of supporting beam \\ t\({}_{\text{pzt}}\) & 0.2 [mm] & Thickness of piezo ceramic plate \\ t\({}_{\text{stator}}\) & 0.6 [mm] & Thickness of passive part of the stator \\ t\({}_{\text{total}}\) & 1.2 [mm] & Total thickness of the stator \\ R & 12.6 [mm] & Distribution radius of stator clamping points \\ \(\alpha\) & 30 [\({}^{\circ}\)] & Inclination angle between two supporting beams \\ \(\alpha_{1}\) & 45 [\({}^{\circ}\)] & Inclination angle of additional cut \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometrical parameters of the stator. Figure 1: Design of the motor; 1—clamping bolts of preload spring clamp; 2—preload spring clamp; 3—preloading spring; 4—hemispherical rotors; 5—shaft; 6—piezo ceramic plates; 7—motor mounting plate; 8—clamping bolts; 9—assembled stator. Analyzing geometrical parameters of the motor, it can be noted that the total volume of the stator is 202.8 mm\({}^{3}\) while the mounting area needed for the motor is 169 mm\({}^{2}\). The approximate total weight of the motor is 5.5 g, including rotor and preload components. It must be pointed out that the design of the motor is scalable. Therefore, dimensions of the motor can be changed based on the requirements. Sixteen piezo ceramic plates are used to excite vibrations of the stator. Each top and bottom surfaces of the stator have eight piezo ceramic plates glued on them (Figure 3). The polarization direction of the plates is aligned along with the thickness. So, the plates located on the top and bottom surfaces have opposite polarization directions. Four harmonic signals, with a phase difference of \(\pi\)/2, are used for motor driving.