# A Disc-Type High Speed Rotary Ultrasonic Motor with Internal Contact Tenth
## 3 Numerical Modeling and Results  
The first step of numerical modeling is to obtain the characteristic frequencies and modal shapes of the stator by finite element method (FEM). A numerical simulation of the stator was performed by using FEM software COMSOL Multiphysics. A finite element model of the stator is shown in Figure 5a. The model contains a titanium layer and two piezoelectric ceramic rings. An adhesive layer was neglected in the model. The initial inner and outer diameters of the titanium alloy plate were set to 8 mm and 20 mm, respectively. The length, width and angle of the contact teeth were set to 3.25 mm, 1.2 mm and 18\({}^{\circ}\) initially. The stator was meshed by three-dimensional standard tetrahedral finite elements. After several iterations, a complete finite element mesh was defined and it contained 57,905 domain elements, 21,244 boundary elements and 1132 edge elements. The material properties of titanium alloy and piezoelectric ceramics are listed in Table 1. As shown in Figure 5b, the B\({}_{03}\) modal shape of the initial dimensions is calculated at the frequency of 98.737 kHz. Figure 5b with color legend represents the value of total displacement, and it can be observed that the total displacement of the contact teeth fits well with the principal design requirement of this modal shape. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Material Property** & \
\begin{tabular}{c} **Titanium Alloy** \\ **(Ti6A4V) [12]** \\ \end{tabular} &
\begin{tabular}{c} **Piezoelectric Ceramics** \\ **PIC151 [16]** \\ \end{tabular} \\ \hline Density, kg/m\({}^{3}\) & 4450 & 7800 \\ Young’s modulus, N/m\({}^{2}\) & 110 \(\times\) 10\({}^{9}\) & 7.6 \(\times\) 10\({}^{10}\) \\ Poisson’s coefficient & 0.34 & - \\ Isotropic structural loss factor & 0.025 & - \\ Relative permittivity & - & \(\epsilon_{33}\)\({}^{\mathrm{T}}\)/\(\epsilon_{0}\) = 1200 \\  & & \(\epsilon_{11}\)\({}^{\mathrm{T}}\)/\(\epsilon_{0}\) = 1500 \\ Elastic compliance coefficient (10\({}^{-12}\) m\({}^{2}\)/N) & - & \(\epsilon_{31}\)\({}^{\mathrm{E}}\) = 15.00 \\  & & \(5_{33}\)\({}^{\mathrm{E}}\) = 19.00 \\ Elastic stiffness coefficient c\({}_{33}\)\({}^{\mathrm{D}}\), N/m\({}^{2}\) & - & 1.6 \(\times\) 10\({}^{10}\) \\ Piezoelectric constant d\({}_{33}\) (10\({}^{-12}\) m/V) & - & 225 \\ Piezoelectric constant d\({}_{31}\) (10\({}^{-12}\) m/V) & - & \(-\)97 \\ Piezoelectric constant d\({}_{15}\) (10\({}^{-12}\) m/V) & - & 330 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Properties of the materials. Figure 4: Actuating principle of the motor. Figure 5: (**a**) Finite element method (FEM) mesh of the stator; (**b**) B\({}_{03}\) mode shape of the stator. After confirming the working mode of the stator, the next step of optimization was to find the optimal dimension parameters of the stator, to allow the contact point \"x\" (Figure 6) of the stator to gain maximum amplitude in this working mode. Frequency domain analysis method was chosen for use in the process of optimization. Harmonic response analysis was performed to calculate amplitudes of the contact points. The amplitudes of the contact point depend on the inner and outer diameters of the stator, as well as the length, width and angle of the contact teeth. Because of the scalability of the USM, and considering the limitations of size and complexity of manufacturing, the inner and outer diameters of the stator were chosen to be 8 mm and 20 mm, the same as the ceramic ring. Thus, the length, width and angle of the contact teeth were optimized to find the maximum oscillation amplitude of the contact point.