## 2 Configuration and Working Mechanism  
The proposed ILUM-D consists of four PZTs and electrodes, two stators with the same structure, a driving shaft, and the moving element, as illustrated in Figure 1b. The ring-shaped PZTs are attached to the inner and outer surfaces of the stators with identical polarization directions. The utilization of ring-shaped PZTs avoids a maximum strain region in the stator, which prevents the PZTs from fragmenting at high voltages. On account of the technical requirements of the vibration mode and the installation process, the stator is divided into the excite region, isolation region, and constrain region, as shown in Figure 1a. The symmetrical structure of the motor allows for easy installation and commissioning, which ensures that the driving shaft is positioned horizontally and vertically on the stator surface. The cylindrical driving shaft is fixed between the stators and transfers the axial displacement generated by the stators to the moving element; hence, the resonant frequency of the driving shaft should be verified to avoid the proposed ILUM-D working frequency.  
Generally, inertial movement is achieved by applying sawtooth wave signal voltage to the PZTs. The proposed stators are excited by the PZTs in an external bending vibration mode as shown in Figure 1c, while the driving shaft works in an up-down motion. The moving element is pressed into the driving shaft with a rubber band. To generate an axial displacement of each stator, only one driving signal source is necessary, as shown in Figure 1d.  
First of all, the sawtooth wave signal symmetry is used to illustrate the proportion (\(t_{1}/T\)) of rising time in a cycle as shown in Figure 2a. The sawtooth wave signals adopted in this paper are 100% or 0% symmetry, which means that the rising time (falling time) of the signal occupies an entire period, and the difference between them only determines the motion direction. The schematic of the motion principle is demonstrated under a sawtooth wave signal with a symmetry of 100%; the ILUM-D working period can be divided into slow forward, turning point, and fast backward based on the applied sawtooth signal. Precisely speaking, the asymmetric motion is reflected significantly on the driving shaft and the motion between the moving element and the driving shaft is almost slip-type, since the proposed ILUM-D works under a sawtooth signal with resonant frequency. As shown in Figure 2b, in the slow forward period, the position of the moving element can be described with the distance \(x_{\text{I}}\) at the moment I. During sections I-II-III, the applied voltage rises slowly and the value of the acceleration of the driving shaft also rises slowly, while the position of the moving element is \(x_{\text{II}}\) to \(x_{\text{III}}\) and the position of the driving shaft is \(D_{\text{II}}\) to \(D_{\text{III}}\). After the turning point III, the applied voltage drops sharply during sections III-IV-V.  
Figure 1: Schematics and working principle of the proposed ILUM-D: (**a**) main structure dimensions of the stator; (**b**) configuration of the ILUM-D; (**c**) vibration configuration of the stator; (**d**) PZT arrangement.  
In the fast backward period, the driving shaft moves back to the \(D_{\text{IV}}\) position while the moving element is still moving forward and keeps the \(x_{\text{IV}}\) position as a result of its inertia; then, the moving element will keep moving until reaching the \(x_{\text{V}}\) position, which leads to backward motion \(\mid x_{\text{V}}-x_{\text{IV}}\mid\). In this way, the relative movement between the moving element and the driving shaft is generated with the value of \(\mid x_{\text{V}}-x_{\text{I}}\mid\), and the motion direction of the moving element can be changed by applying a sawtooth wave signal with 0% symmetry.