## 4 Optimization of Stator Characteristic Parameters  
### Realization of Low-Speed-High-Torque Characteristics  
For all types of ultrasonic motor, the stator amplitude is an important indicator of motor performance: the larger the amplitude, the greater the mechanical energy converted from electrical energy [20]. In practice, the speed is regulated mainly by driving signal frequency and voltage, which is essentially controlled by changing the amplitude [21]. If the motor structure is constant, then the driving frequency is adjusted from high to low in the design band; if the voltage is constant, then the motor speed is adjusted from low to high. The closer the driving frequency to the modal frequency, the larger the stator amplitude and the higher the motor speed. The stall torque measured at the maximum speed point is also the largest. Similarly, if the driving frequency is constant, then within a certain range, the larger the driving voltage, the larger the stator amplitude, the motor speed, and the stall torque.  
At present, low-speed-high-torque characteristics are reflected mainly in standing wave motors, especially the longitudinal-torsional hybrid motor, and are difficult to achieve in traveling wave motors because the normal and tangential vibrations of the stator elliptical motion of a hybrid motor are independent. However, the normal and tangential vibrations are coupled in a traveling wave motor; that is, the tangential displacement of a stator surface particle is induced by the normal bending vibration [22; 23]. Therefore, increasing (respectively, decreasing) the normal motion component, which means the normal amplitude and can represent the maximum stall torque, will inevitably increase (respectively, decrease) the tangential motion component, which means the tangential velocity can represent the limiting speed. From the perspective of the mechanism of elliptical motion, when the external drives are adjusted, an increase or decrease of the amplitude can be considered as enlargement or reduction of the elliptical motion trajectory, as shown in Figure 3a.  
However, for the longitudinal-torsional hybrid motor, the normal vibration and tangential vibration can be adjusted independently so that the size and shape of the elliptical motion trajectory can both change. Inspired by this, the size and shape of the elliptical motion in traveling wave motors should be changed to achieve low-speed-high-torque characteristics. Specifically, we should try to increase the normal amplitude as much as possible while reducing the tangential amplitude. That is to say, we must increase the ratio of the long axis (normal) to the short axis (tangent) of the elliptical motion trajectory. However, the stator characteristic parameters affect not only the elliptical motion directly, but also the amplitude. In other words, the amplitude is an intermediate dependent variable that affects the motor speed and output torque, whereas the characteristic parameters of the external drives and the stator are independent variables. According to the above analysis, adjusting the external drives can change only the size of the elliptical motion but not its shape. Therefore, to change the ratio of the long axis to the short axis, the stator characteristic parameters must be changed to make the elliptical motion trajectory tall and thin, as shown in Figure 3b.  
Accordingly, in the case of the same external drives and stator and rotor inner and outer diameters, to achieve low-speed-high-torque characteristics in a traveling wave motor, we should increase the stator amplitude \(w_{0}\) as much as possible for higher mechanical output power but reduce \(n\), \(h\), \(l\), and \(f_{m}\). In the final analysis, we adjust the four stator characteristic parameters of (i) modal order \(n\), (ii) stator base thickness \(H_{2}\), (iii) stator tooth height \(H_{1}\), and (iv) tooth spacing \(l\).