# Design and Impedance Control of a Traveling Wave Ultrasonic Motor with a Spherical Rotor
## 2 Operating Principle and Design of the TWUM  
As shown in Figure 1, the stator of the motor was pasted with a lead zirconate titanate (PZT) board, which was driven by two alternating voltages with a phase difference. The stator was excited to produce two standing waves, which superimposed into a traveling wave. As a consequence, the points on the stator surface underwent a slightly elliptical motion. Since there was a preload between the stator and rotor, the rotor was driven to rotate through the friction between the stator and rotor.  
Figure 1: Traveling wave ultrasonic motor (TWUM) with a spherical rotor. PZT: lead zirconate titanate.  
The driving torque \(T\) is expressed as:  
\[T=\int_{0}^{2\pi r_{s}}\frac{\mu r_{s}N\sqrt{r^{2}-r_{s}^{2}}}{2\pi r_{s}r}ds= \frac{\mu r_{s}N\sqrt{r^{2}-r_{s}^{2}}}{r}, \tag{1}\]  
where \(r\) is the radius of rotor, \(r_{s}\) is the radius of contact circle between stator and rotor, \(\mu\) is the friction coefficient of the contact layer between the stator and the rotor, \(N\) is the preload between the stator and the rotor. Let \(s\) be the slip rate of the angular velocity between the stator and the rotor, which is expressed as [18]:  
\[s=\frac{\omega_{s}-\omega_{r}}{\omega_{s}} \tag{2}\]  
where \(\omega_{s}\) is the ideal angular velocity corresponding to the average tangential velocity of the elliptical motion at the contact point between the stator surface and the rotor, and \(\omega_{r}\) is the actual angular velocity of spherical rotor. When the preload or the external torque is too high, the slip rate \(s\) should be 1, which means the rotor cannot rotate. When the preload is appropriate and the external torque is 0, the slip rate \(s\) should be close to 0, which corresponds to a no-load angular velocity. Once the preload \(N\) is set, the relationship between the output torque and the angular velocity should be determined via experiment.  
Considering the stick-slip phenomenon of the contact between the stator and the rotor, the friction coefficient \(\mu\) is a function of the slip rate, so the torque \(T\) is expressed as:  
\[T=\frac{\sqrt{r^{2}-r_{s}^{2}}}{r}c_{d}(\omega_{s}-\omega_{r})Nr_{s} \tag{3}\]  
where \(c_{d}\) reflects the influence of the angular velocity difference between the stator and the rotor on the friction coefficient and can only be determined via experiment.  
Figure 2 shows the ring-shaped stator designed in this paper, which had a 40 mm outer diameter and an \(r_{s}\) of about 15 mm. The stator consisted of a PZT board and an elastic body made of tin bronze. The PZT board was bonded on the bottom of the elastic body and was comprised of two polarized regions of phase A and phase B with a one-quarter wavelength difference in between them. The top of the stator was evenly distributed with 42 teeth to amplify the amplitude and improve the speed. The working mode of the stator was B\({}_{07}\), and the resonant frequency was tested to be 44 kHz using a frequency sweep of a laser vibrometer (PSV-400 Polytec GmbH., Karlsruhe, Germany). The spherical rotor had a diameter of 60 mm and was made of polyether ether ketone (PEEK450G, Jianbang Plastics, Changzhou, China), which has the advantages of high temperature resistance, abrasion resistance, corrosion resistance, low density, and easy processability.  
Figure 2: The ring-shaped stator: (**a**) the top of the stator and (**b**) the bottom of the stator bonded with a PZT board.