## 4 Design and Analysis
### Modal Analysis and Sensitivity Analysis  
The resonance frequencies of Mode A and Mode B should be near enough as an indispensable qualification for the stimulus of the elliptical trajectory at the driving foot, as mentioned above. Hence, it is necessary to explore the relationship between the resonance frequencies of two modes and the structural parameters. By means of FEM software (ANSYS 10.0, ANSYS Inc, Canonsburg, PA, USA), the motor's finite element model has been built to perform the modal analysis. Afterwards, the resonance frequencies of the two vibration modes were tuned to be close by adjusting the stator's structural parameters. Figure 4a shows the section view and the main parameters of the stator, and the initial values of these parameters are listed in Table 2. According to the selected materials and the initial dimensions of the stator, a finite element model was built in ANSYS, as shown in Figure 4b. Duralumin alloy (\(\rho\) = 2810 kg/m\({}^{3}\), \(E\) = 7.2 \(\times\) 10\({}^{10}\) N/m\({}^{2}\), \(\sigma\) = 0.33) was selected to make the T-shaped horn part; meanwhile, steel (\(\rho\) = 7850 kg/m\({}^{3}\), \(E\) = 2.06 \(\times\) 10\({}^{11}\) N/m\({}^{2}\), \(\sigma\) = 0.269) is the material of the flange bolt and end caps, and the PZT ceramic material is PZT-4. The mechanical and physical parameters of the PZT ceramics are listed as follows:  
\[d=\left[ \begin{array}{cccccc}0&0&0&0&5&0\\ 0&0&0&5&0&0\\ -1.6&-1.6&3.3&0&0&0\end{array} \right]\times 10^{-10}\mathrm{C/N} \tag{2}\]  
\[C^{E}=\left[ \begin{array}{cccccc}15&8.4&6.8&0&0&0\\ 8.4&15&6.8&0&0&0\\ 6.8&6.8&12.9&0&0&0\\ 0&0&0&3.3&0&0\\ 0&0&0&0&2.8&0\\ 0&0&0&0&0&2.8\end{array} \right]\times 10^{10}\mathrm{N/m^{2}} \tag{3}\]  
\[\varepsilon^{T}=\left[ \begin{array}{cccc}8.1&0&0\\ 0&8.1&0\\ 0&0&6.7\end{array} \right]\times 10^{-9}\mathrm{F/m} \tag{4}\]  
Figure 4: The section view and finite element model of the stator: (**a**) The section view and the main parameters of the stator; (**b**) Finite element model of the stator. The exponential-shaped horn is defined by the classical theory of the half-wavelength horn whose length is  
\[l_{h}=\frac{c}{2f}\sqrt{1+\left(\frac{lnN}{\pi}\right)^{2}} \tag{5}\]  
where \(c,f,N\) are the longitudinal wave velocity, the resonant frequency and the amplitude amplification factor. In general, the longitudinal wave velocity is 5100 m/s in duralumin. Particularly, the horizontal and vertical contraction ratios \(N_{H},N_{V}\) are defined as follows:  
\[N_{H}=\sqrt{S_{H1}/S_{H2}} \tag{6}\]  
\[N_{V}=\sqrt{S_{V1}/S_{V2}} \tag{7}\]  
where \(S_{H1}\), \(S_{V1}\) represent the horizontal and vertical big end cross-sectional areas of the horns; \(S_{H2}\), \(S_{V2}\) represent their small cross-sectional areas close to the driving foot. As mentioned before, the horn structure shape and parameters impact the resonant frequency significantly. Besides, from Equation (5) we can obtain that its length is influenced obviously by \(N_{H}\) and \(N_{V}\). To reach the purpose of the frequency regulation of two working modes, the contraction ratio parameters, which are convenient for adjusting the horn configurations, are defined as follows:  
\[C_{1}=\frac{a}{b},C_{2}=\frac{c}{d},H_{1}=\frac{e}{e+f},N_{1}=\frac{S_{H1}}{S_ {H2}},N_{2}=\frac{S_{V1}}{S_{V2}} \tag{8}\]  
where \(a\), \(b\), \(c\), \(d\), \(e\), \(f\) are shown in Figure 4.  
Several parameters were defined as design parameters, as described in Equation (8), to investigate their contributions to the frequency variations of Mode A and Mode B, which are called sensitivities to the modal frequencies. The sensitivities of the design parameters were calculated in ANSYS and shown in Figure 5, where the \(x\) axis represents the corresponding variance ratio to their original values and the \(y\) axis represents the corresponding resonant frequencies of Mode A and Mode B. According to the sensitivity analysis results, the selected design parameters can be divided into two groups.