## 3 Simulation and Analyses
### Transient Analyses  
For further acquisition of the stator assembly vibration characteristics, dynamic simulations were conducted in ADINA based on the results of the modal analyses. It should be noted here that the transient analyses in this paper were conducted under fixed boundary conditions. The parameters of the voltage signal applied to the SA were set as 33 kHz and 500 V\({}_{\text{p-p}}\), and the parameters of the voltage signal applied to the DA were set as 36 kHz and 500 V\({}_{\text{p-p}}\). The Z-displacement and Z-acceleration of the driving shaft under the sawtooth wave signal with 500 V\({}_{\text{p-p}}\) applied voltage were obtained. As shown in Figure 4, a significant divergence exists between the minimum and maximum vibration amplitudes of the driving shaft due to an asymmetrical excite signal. Though the motion range of the DA shaft is narrower than the SA shaft, as shown in Figure 4a, the velocity and acceleration range of the DA shaft is wider than the SA shaft, which indicates that the double-stator structure is capable of higher carrying abilities, as shown in Figure 4b,c. Based on the results of the stator assembly dynamic simulations, the preload force can be set properly based on [PERSON]'s second law. Since the motor is functional on account of normal inertial motion, the contact model of the motor is proposed as shown in Figure 5a. The contact model consists of the driving shaft and moving element, and the whole model aims to verify the feasibility of the motor structure as well as to obtain usable transient results. The transient response of the moving element displacement in the axial direction under the sawtooth wave signal with 500 V\({}_{\text{p-p}}\) applied voltage is shown in Figure 5b, and \(R^{2}\) indicates the degree of fit of the fitting line to the original displacement curve; the closer to 1, the more representative of the original displacement data. It can be calculated that the output velocity of the ILUM-S under a 33 kHz excitation frequency is 4.351 mm/s and that of the ILUM-D under 36 kHz is 9.515 mm/s. The velocity curves of the moving element and the driving shaft were also obtained as shown in Figure 6. First of all, it is clear that there is almost no relative stationary motion between the moving element and the driving shaft during both motor operations. The velocity of the driving shaft is significantly decreased compared to that in Figure 4b as a consequence of friction. The forward displacement and the backward displacement of the moving element/driving shaft are expressed as \(\delta_{\text{F}}\), \(\delta_{\text{B}}\) and \(\epsilon_{\text{F}}\), \(\epsilon_{\text{B}}\), as shown in Figure 6. The velocity integration results of \(\delta_{\text{F}}\), \(\delta_{\text{B}}\) are obtained using MATLAB R2018a, as shown in Table 2, and \(\epsilon_{\text{F}}+\epsilon_{\text{B}}=0\), since the motion of the driving shaft is reciprocating during one operation period. Combined with the data in Table 2, it can be found that the \(\delta_{\text{F}}\) of the ILUM-D expressed as \(\delta_{\text{FD}}\) is larger than that of the ILUM-S expressed as \(\delta_{\text{FS}}\), while the \(\delta_{\text{B}}\) of the ILUM-D expressed as \(\delta_{\text{BD}}\) is smaller than that of the ILUM-S expressed as \(\delta_{\text{BS}}\), resulting in a larger effective step displacement in one operation period expressed as \(\Delta\delta\), as shown in Table 2. Furthermore, the proportion of the forward phases in one operation period is also calculated and expressed as \(\varphi_{F}/T\), as shown in Table 2, indicating there is a slight distinction between the motor's forward proportions. Figure 4: The driving shaft of the stator assembly time domain response curves under the 500 V\({}_{\text{p-p}}\) sawtooth wave voltage: (**a**) displacement in the axial direction; (**b**) velocity in the axial direction; (**c**) acceleration in the axial direction.