# A Novel Dual-Rotor Ultrasonic Motor for Underwater Propulsion
## 3 Operating Principle Analysis  
The working principle of the dual-rotor ultrasonic motor is shown in Figure 4. When applying single-phased sinusoidal signal with resonant frequency to piezoelectric ceramic wafer, it excited the ultrasound vibration of the stator due to the inversed piezoelectric effect. The vibration of the driving feet at two ends of the fixed metal plate drove the driving disks pressed on it to rotate by the contact friction, thereby enabling the parallel shafts to rotate in a reversed direction at the same time. The vibration mode for the ultrasonic motor was the first longitudinal in-plane vibration mode. The vibration direction of the driving foot at each end was opposite to each other [27; 28]. The longitudinal vibration of the metal plate caused the driving foot to have a linear motion along the surface of the shaft and to push the shaft to rotate in one direction. Due to the central symmetry of the structural design and the engagement of the gear unit, rotary directions for the two rotors were opposite, that is, the left rotor rotated clockwise, the right rotor rotated counterclockwise. The appearance size of the spindle-shaped stator was 20 mm \(\times\) 35 mm \(\times\) 2 mm (width \(\times\) length \(\times\) thickness), and the driving feet at both ends was 5 mm in width. The piezoelectric wafer was \(\mathcal{O}\)20 mm \(\times\) 1 mm and polarized in the thickness direction. The ground electrode at the bottom face of the wafer was extended from the side to the top face for connection. The rotating shaft was 12 mm  
Figure 4: Operating principles of dual-rotor ultrasonic motor. Figure 3: Structure of pre-pressure modulating unit. in diameter and 3 mm in thickness, and the distance between the two parallel shafts was 43 mm. The stator was assembled to the bottom-center position between the rotating shafts with its dual driving feet contacting both shafts. In order to determine the stator's vibration mode and frequency, the finite element method-based simulation software COMSOL Multiphysics was utilized to perform the eigenfrequency analysis. Sequentially, frequency domain analysis was performed to obtain the vibration displacement at resonance. During the simulation, all material properties were assumed to be constant, which means the non-linear effect of materials (e.g., hysteresis of piezoelectric material) was not taken into consideration. Phosphor bronze and PZT-8H were selected for the metal plate and piezoelectric wafer, respectively. In particular, PZT-8H material has a piezoelectric constant \(d_{33}\) of \(200\times 10^{-12}\) C/N, electromechanical coupling factor \(k_{33}\) of 0.60, mechanical quality factor \(Q_{\text{m}}\) of 800, dielectric dissipation factor tan\(\delta\) of 0.5%, density of 7450 kg/m\({}^{3}\), and Curie temperature \(T_{\text{c}}\) of 300 \({}^{\circ}\)C [29; 30]. For this numerical simulation, the three-dimensional model of the stator was meshed into 8385 free tetrahedral elements and all physical boundaries of the stator were set to be free. Calculation results from the eigenfrequency analysis are plotted in Figure 5a. It was observed that the metal plate vibrated in the first longitudinal mode, which demonstrated the working principle of the stator. The resonance frequency of this working mode was 61.4 kHz. With the voltage of 150 V (peak-to-peak) at resonance, the calculated results from the frequency domain analysis reveal that the first longitudinal vibration amplitude of the stator was about 1.4 microns. When the stator extended, the driving foot pushed driving disks to rotate shafts from the initial position (top figure) to the new position (left figure). After that, the stator returned to the original state (bottom figure) and the contact friction between the stator and rotors decreased, leading to the continuous rotation of rotors. Sequentially, the stator kept contracting (right figure), but the rotor still kept rotating in the same direction with the assistance of inertia.