# An Eight-Zonal Piezoelectric Tube-Type Thread Ultrasonic Motor Based on Second-Order Bending Mode
## Structure, Design Principle, and Working Mechanism
### Working Mechanism  
The eight-zonal stator was designed and then optimized with the finite element analysis (FEA) method by using ANSYS Mechanical APDL 15.0. To observe the movement, a time-dependent analysis was performed with an input voltage of 50 V at its resonant frequency as shown in Figure 4b. A random point at the helical portion of the nut, whose three-dimensional trajectory is shown in Figure 5, was selected. The displacement of the Z component (axial direction) was much larger than  
Figure 4: (a) Partition pattern of the piezoelectric ceramic tube for second-order bending vibration; (b) Electrical signal supply mode for the piezoelectric ceramic tube. Figure 3: Schematic diagram of vibration mode and strain curve for second-order bending mode. the other two components, as demonstrated by the projection on each plane. Thus, the simulation result shows that the designed stator can achieve fine elliptic curve motion to produce a \"hula hoop\" wobbling movement. In 1998, [PERSON] [18] theoretically analyzed the mechanical properties of the friction contact surfaces of the ultrasonic motor. According to his theory, we established a friction model for thread contact surface, and the threaded driving mechanism is discussed in Figure 6. Compared to the thread angle \(\alpha\) of a triangular thread in Figure 6a, the helix angle of the threads is very small and can be ignored. If there is no deformation at the contact surface, then the contact point is the peak of the elliptical motion curve. The considerable displacement of the Z component results in an included angle (\(\beta\) in the figure) between the plane where the elliptical motion lies and the \(XOY\) plane. When traveling waves are excited in the stator, the rotor would be driven at the peak of the elliptical motion curve by the frictional force. When the rotor is applied with an axial preload \(F_{a}\), the maximum load force output \(T\) can be calculated by Equation (1):  
\[T=F_{s}=\mu F_{a}cos\frac{\alpha}{2} \tag{1}\]  
where \(F_{s}\) is the maximum static friction, and \(\mu\) is the coefficient of static friction at the contact surface. Figure 5: Three-dimensional trajectory of a random point at the helical portion of the nut. Figure 6: Driving mechanism at the contact surface between the rotor and stator. (**a**) Front view. (**b**) Vertical section view. Moreover, as shown in Figure 6b, the ideal elliptic motion plane should be perpendicular to the contact surface. Thus, the contact area at the helical portion would produce a speed of \(v_{r}\) along the tangential direction of the contact surface. The relationship between the ideal included angle (\(\beta^{\prime}\) in the figure, between the ideal elliptical motion plane and the \(XOY\) plane) and the thread angle \(\alpha\) can be illustrated as Equation (2):  
\[\alpha=\pi-2\beta^{\prime} \tag{2}\]  
However, if the real included angle \(\beta\) is not equal to the ideal included angle \(\beta^{\prime}\) when the traveling wave is generated, there would be a radial component in the perpendicular direction of the moving speed \(v_{r}\) in the contact plane. This radial component would bring about a slip phenomenon in the contact plane, causing problems like movement instability and energy loss, and, thus, decreasing the output performance of the motor. In addition, from Equation (1), we can see that a smaller thread angle \(\alpha\) would contribute to a larger output load force; from Equation (2), a smaller thread angle \(\alpha\) demands a larger ideal included angle \(\beta^{\prime}\), which means larger displacement of the Z component in the elliptical trajectory. Finally, combining the theoretical and FEA analysis above and the machining difficulty, we chose \(30^{\circ}\) for the thread angle \(\alpha\) for relatively fine performance. By means of thread drive, the wobbling motion of the stator can be converted into linear motion along the axis without any additional mechanical conversion.