## 3 Motion and Power Transmission
### Limiting Speed and Stall Torque of the Motor  
According to Equation (1), the speed of point P in the \(x\) direction can be obtained:  
\[V_{\tau}=\frac{\mathrm{d}\xi_{p}}{\mathrm{d}t}=-w_{0}hk\omega_{n}\sin(kx- \omega_{n}t) \tag{3}\]  
When the stator and rotor are in contact with the traveling wave crest, the maximum tangential velocity of a stator particle can be regarded as being equal to the linear velocity of the rotor if there is no relative sliding between the stator and the rotor. Therefore, the theoretical limiting speed of the motor can be expressed as  
\[N_{\mathrm{max}}=V_{\tau\mathrm{max}}=-w_{0}hk\omega_{n}=-\frac{60 lnf_{m}w_{0 }}{{r_{c}}^{2}} \tag{4}\]  
where \(\omega_{n}\) is the vibration angular frequency and \(f_{m}\) is the modal frequency.  
Figure 2: Contact status between stator and rotor over one wavelength.  
When the motor is working, the rotor with friction material is pressed against the traveling wave stator under the preload. Because the friction layer is softer than the rotor material, it can be assumed that only the friction layer is subjected to contact deformation, with the rotor itself not undergoing elastic deformation. When the motor is not working, the compression of the friction layer is  
\[\Delta w=F_{n}/k_{n} \tag{5}\]  
where \(F_{n}\) is the preload acting on the rotor and \(k_{n}\) is the equivalent stiffness coefficient of the friction layer.  
According to contact force analysis, the force between the stator and rotor can be decomposed into a normal contact force and a tangential friction force. In the range of elastic contact, it can be assumed that (i) the normal contact force is proportional to the compression of the friction layer and (ii) the tangential friction force obeys Coulomb's law of friction. If the average dynamic friction coefficient is \(\mu_{d}\), then the theoretical limits of the normal contact force and the tangential friction force are  
\[f_{\text{max}}=k_{n}g_{m}=k_{n}(w_{0}+\Delta w) \tag{6}\]  
\[f_{\text{max}}=\mu_{d}f_{\text{max}} \tag{7}\]  
The limiting stall torque of the motor is  
\[M_{\text{Tmax}}=\sum_{\epsilon=1}^{Z}\iint\limits_{f_{\tau}(\epsilon)}r\text{d }f_{\tau}=\sum_{\epsilon=1}^{Z}\iint\limits_{S(\epsilon)}\frac{rf_{\text{max} }}{S(\epsilon)}\text{d}S=\sum_{\epsilon=1}^{Z}\iint\frac{f_{\text{max}}}{ \pi(r_{\epsilon}{}^{2}-r_{i}^{2})-Z\text{l}\Delta r}r^{2}\text{d}\theta\text{ d}r \tag{8}\]  
where \(r_{i}\) is the inner radius of the friction layer, \(Z\) is the number of contacted stator teeth, \(l\) is the tooth spacing, \(\Delta r\) is the radial width of the friction layer, and \(\theta\) is the circumferential angle of a single tooth.  
Equations (4) and (8) show the following. (i) Increasing the normal amplitude can increase both the stall torque and the limiting speed of the motor. (ii) Decreasing the axial distance \(h\) between point P and the neutral layer will reduce the tangential displacement \(\zeta_{p}\) so as to achieve a lower rotation speed of the motor. (iii) The tangential speed \(V_{\tau}\) is inversely proportional to the wavelength \(\lambda\); therefore, decreasing the modal frequency can enlarge the wavelength and reduce the speed.