## 3 Simulation and Analyses  
### Modal Analyses  
First of all, the feasibility of the stator needs to be verified by using the commercial software ANSYS R18.0 to acquire the stator and assembly modal analyses results. It should be noted here that the modal analyses in this paper were conducted under free boundary conditions and the stators and the driving shaft were bonded. The material properties of the ILUM-D are listed in Table 1.  
The PZTs used in the proposed motor were four PZT-8 ceramic rings with an inner diameter of 6 mm, outer diameter of 10 mm, and thickness of 0.5 mm. The PZT-8 rings were used due to their higher mechanical quality and better stability at high drive levels [22]. For the PZT-8 ceramic rings used in this paper, a signal with high voltage amplitude was applied in order to achieve full performance. The density of each PZT-8 was 7500 kg/m\({}^{3}\). Their piezoelectric matrix, stiffness matrix, and dielectric matrix are as follows [23]:  
\[[e]=\left[\begin{array}{ccc}0&0&-7.209\\ 0&0&-7.209\\ 0&0&15.118\\ 0&0&0\\ 12.332&0&0\\ 0&12.332&0\\ \end{array}\right]\text{N/V\text{\textminus}m} \tag{1}\]  
Figure 2: Schematics and motion principle of the proposed ILUM-D: (a) applied sawtooth wave signal and the signal for reverse motion; (b) motion principle of the ILUM-D.  
\[[C^{E}]=\begin{bmatrix}1.206&0&0&0&0&0\\ 0.535&1.206&0&0&0&0\\ 0.515&0.515&1.045&0&0&0\\ 0&0&0&0.313&0\\ 0&0&0&0&0.313&0\\ 0&0&0&0&0.346\end{bmatrix}\times 10^{11}\,\mathrm{N/m^{2}} \tag{2}\]  
\[[\varepsilon^{T}]=\begin{bmatrix}1.53&&\\ &1.53&\\ &&1.50\end{bmatrix}\times 10^{-8}\,\mathrm{F/m} \tag{3}\]  
To determine the resonance frequency of the stator assembly, a finite element model was built and subsequent simulations were carried out. The feasibility of the stator was verified by acquiring the working vibration mode through the modal analyses shown in Figure 3. The isolation region and constrain region of the stator were not involved during the vibration, as shown in Figure 3a. The resonance frequency of the stator was calculated as 44.111 kHz. The modal analyses of the stator assembly, which consists of stator(s) and the driving shaft, were also conducted, as shown in Figure 3b,c. In particular, there was no stress or strain within the driving shaft, which merely transmits the stator vibrations during the operation process. The resonance frequencies of the single stator assembly (SA) and the double-stator assembly (DA) were calculated as 33.052 kHz and 36.049 kHz.  
The modal analyses results indicate that the addition of the driving shaft decreases the bending mode frequency of the SA, potentially because of the extra mass added to the stator. However, the bending mode frequency of the DA is higher than that of the SA, which could be due to the changed overall boundary conditions, especially for the driving shaft.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Component**} & \multirow{2}{*}{**Materials**} & \multirow{2}{*}{**Density (kg/m\({}^{3}\))**} & \multicolumn{2}{c}{**Young’s**} & \multirow{2}{*}{**Poisson’s Ratio**} \\  & & & & & \\  & & & & & Modulus (N/m\({}^{2}\)) \\ \hline \multirow{2}{*}{
\begin{tabular}{c} Stator \\ Moving element \\ Driving shaft \\ \end{tabular} } & Structural Steel & 7850 & 2.0 \(\times\) 10\({}^{11}\) & 0.3 \\  & Stainless Steel & 7760 & 2.1 \(\times\) 10\({}^{11}\) & 0.28 \\  & & & \(E_{1}\) = 1.6 \(\times\) 10\({}^{11}\) & \(\gamma_{12}\) = 0.33 \\  & Carbon Fiber & 1800 & \(E_{1}\) = 9.7 \(\times\) 10\({}^{9}\) & \(\gamma_{23}\) = 0.28 \\  & & & \(E_{1}\) = 9.7 \(\times\) 10\({}^{9}\) & \(\gamma_{13}\) = 0.33 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Material properties of the ILUM-D.  
Figure 3: Modal analyses results in ANSYS: (**a**) stator, 44111 Hz; (**b**) SA, 33052 Hz; (**c**) DA, 36049 Hz.