## 3 Numerical Modeling of the Actuator  
The 3D simplified model of the unimorph actuator was designed, and numerical simulations were executed. The aim of the simulations was to determine resonant frequencies and vibration modes using  
Figure 1: Structure of the piezoelectric 3 degrees of freedom (DOF) rotary ultrasonic motor (USM) with a spherical rotor (**a**); view of the piezoelectric unimorph actuator (**b**) and schematic representation of the USM drive and layout of segmented electrodes of the piezoelectric transducer (**c**): 1-spherical rotor; 2-piezoelectric transducer; 3-flange; 4-contact element; 5-nut; 6-bolt; 7-signal generator; 8-controller. three different designs of actuator's flange. Numerical modeling was performed when the piezoelectric actuator was analyzed as unimorph structure, i.e., a transducer and flange were bonded using epoxy resin, and the other piezoelectric transducer was fixed on the flange using soft rubber spacers. The 3D vibration amplitudes and trajectories of contact points of the actuator were analyzed. All simulations were performed in two steps: firstly, modal analysis was executed, and then, harmonic response studies were performed. USM actuator consists of a piezoelectric ring-shaped transducer, contact elements, and thin flange. All geometry was designed and prepared using Autodesk Inventor Professional 2018, and COMSOL Multiphysics 5.4 was utilized for the calculations. Preparing models for calculations, only specific components were designed i.e., piezoelectric transducer, contact elements, and thin flange. Other components such as the glue layer, soft rubber spacer, and electrodes, were not included. Piezoelectric ring-shaped transducer dimensions were as follows: outside diameter was 20 mm, inside diameter was 15 mm, and the thickness was 3 mm. PZT-4 hard type piezoceramics was used for the production of piezoelectric transducer. This piezoceramics is widely used for the USM motors [13; 14; 15]. The disc and triangle-shaped flange were investigated. The thin flanges were made from phosphor bronze C52100 and had the following dimensions: outer diameter was 35 mm, central hole was 15 mm, and thickness was 0.3 mm. Dimensions of the actuators and flanges are presented in Figure 2. Contact elements are made from friction-resistant material--Aluminum oxide (Al\({}_{2}\)O\({}_{3}\)). Dimensions of these contact elements were as follows, i.e., width: 1 mm; length: 2.2 mm, and height: 1.5 mm. The properties of used PZT-4 piezoceramic and dimensions of components are presented in Table 2 [11; 16; 17; 18; 19; 20]. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Property** & **Unit** & **Transducer** & **Flange** & **Contact Element** \\ \hline Material & & & PZT-4 & Phosphor bronze C52100 & Aluminum oxide (Al\({}_{2}\)O\({}_{3}\)). \\ Density & kg/m\({}^{3}\) & 7500 & 8800 & 39650 \\ Dimensions & mm & OD-20; ID-15; T-3 & OD-35, ID-34; T-0.3 & W-1; L-22; H-1.5 \\ Poisson coefficient & – & 0.31 & 0.34 & 0.21 \\ Young Modulus & GPa & 63 & 110 & 215 \\ Dielectric permittivity, 10\({}^{3}\) & F/m & \(c_{11}=148\), \(c_{22}=1.48\), \(c_{23}=1.3\) & - & - \\ Piezoelectric coupling matrix, 10\({}^{-12}\) & C/N & \(d_{31}=-123\), \(d_{32}=-123\), \(d_{33}=289\), \(d_{34}=496\), & - & - \\  & & & \(c_{11}=123\), \(c_{23}=-4.05\), \(c_{31}=-51\), \(c_{22}=123\), & & \\ Compliance matrix, 10\({}^{-12}\) & m\({}^{2}\)/N & \(c_{23}=-51\), \(c_{23}=155\), \(c_{31}=39\), \(c_{35}=39\), & - & - \\  & & & \(c_{66}=327\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Dimensions and material properties of the actuator. Figure 2: Dimensions of actuators and flanges: (**a**) actuator with disk-shaped flange; (**b**) actuator with triangle-shaped flange. Modal-frequency analysis was accomplished while the actuator was not clamped, and short circuit boundary condition was applied. Piezoelectric transducer and assembled unimorph actuator with two types of the flanges were analyzed (Figure 3).