# A Disc-Type High Speed Rotary Ultrasonic Motor with Internal Contact Tenth
## 3 Numerical Modeling and Results
The formulas of this optimizing problem are listed below:  
\[u_{tot}^{\max}=\sqrt{u^{2}{}_{x}(b,l,\alpha)+u^{2}{}_{y}(b,l,\alpha)}; \tag{2}\]  
Subjected to  
\[b_{\min}\leq b\leq b_{\max}; \tag{3}\]  
\[l_{\min}\leq l\leq l_{\max}; \tag{4}\]  
\[\alpha_{\min}\leq\alpha\leq\alpha_{\max}; \tag{5}\]  
\[f_{\min}\leq f\leq f_{\max}. \tag{6}\]  
where \(u_{tot}\) is oscillation amplitude of the contact point, \(b\), \(l\) and \(\alpha\) are the width, length and angle of the contact teeth, respectively, \(b_{\min}/b_{\max}\), \(l_{\min}/l_{\max}\) and \(\alpha_{\min}/\alpha_{\max}\) are the minimum/maximum limits of the width, length and angle, \(f\) is the resonant frequency of the stator and \(f_{\min}/f_{\max}\) are the minimum/maximum limits of the frequency. As is shown in Figure 6, the horizontal distance between the contact point and the central axis \(x\) is related to the three variables, i.e., \(b\), \(l\) and \(\alpha\). The similar triangle theorem is used to solve this plane analytic geometry problem, and the distance \(x\) is described as follows: Equation (7) is used to determine the position of the contact point when different length, width and angle of the contact teeth are applied:  
\[x=\frac{\big{(}l\cdot\sin\alpha-\frac{b}{2}\cdot\cos\alpha\big{)}\cdot\bigg{[} \sqrt{\big{(}l\cdot\sin\alpha-\frac{b}{2}\cdot\cos\alpha\big{)}^{2}+\big{(}4- l\cdot\cos\alpha-\frac{b}{2}\cdot\sin\alpha\big{)}^{2}-\frac{b}{2}}\bigg{]}}{ \sqrt{\big{(}l\cdot\sin\alpha-\frac{b}{2}\cdot\cos\alpha\big{)}^{2}+\big{(}4- l\cdot\cos\alpha-\frac{b}{2}\cdot\sin\alpha\big{)}^{2}}} \tag{7}\]  
To initiate the FEM analysis, the scan range should be determined, based on the dimension parameters of the contact teeth. The scanning range of each simulation parameter  
Figure 6: Contact point between stator and rotor. set is shown in Figure 4:0 Scanning interval of frequency was determined by calculating the characteristic frequency of the stator B\({}_{03}\) mode, which corresponds to the two groups of critical sizes (l\({}_{\text{min}}\) = 2.8 mm, b\({}_{\text{min}}\) = 0.8 mm, \(\alpha_{\text{min}}\) = 15; l\({}_{\text{max}}\) = 3.2 mm, b\({}_{\text{max}}\) = 1.2 mm, \(\alpha_{\text{max}}\) = 30\({}^{\circ}\)) in the dimension parameters value range. Characteristic frequencies of B\({}_{03}\) mode corresponding to the critical sizes were calculated at 98.5 kHz and 101.5 kHz, respectively. Thus, the scanning interval of frequency was set from 98 kHz to 102 kHz with the scanning step of 100 Hz. Frequency domain simulations were performed for the 375 combinations of the dimension parameter variables, which are listed in Figure 4:1  
Boundary conditions of the model used for the optimization were set. The mechanical boundary of the stator was set to be free. The excitation scheme of the electrodes was used, as shown in Figure 4:2 The voltage amplitude of 20 V\({}_{\text{PP}}\) was applied on the electrodes, and the frequency range was set from 98 kHz to 102 kHz with a frequency scan step of 0.1 kHz. 2D plots of the amplitude of contact points in B\({}_{03}\) mode vary with length, width, and angle of contact teeth are shown in Figure 4:3 The color legend of the plot shows the value of the amplitude in the order of micrometers. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Scanning Step** \\ \hline External diameter, R & 20 mm & - \\ Internal diameter, r & 8 mm & - \\ Contact teeth length, \(l\) & (2.80 mm, 3.20 mm) & 0.1 mm \\ Contact teeth width, \(b\) & (0.8 mm, 1.2 mm) & 0.1 mm \\ Contact teeth width, \(\alpha\) & (15\({}^{\circ}\), 30\({}^{\circ}\)) & 1\({}^{\circ}\) \\ Frequency of applied voltage & (98 kHz, 102 kHz) & 0.1 kHz \\ \hline \hline \end{tabular}
\end{table} 
Table 2: FEM parameter scan list. Figure 7: 2D plots of the amplitude of contact points in B\({}_{03}\) mode vary with length, width, and angle of contact teeth. It is shown that the largest displacement amplitude is realized when \(l\), \(b\) and \(\alpha\) have the values of 18 mm, 1.1 mm and 15\({}^{\circ}\), respectively.