# Modeling of Transmission Compliance and Hysteresis Considering Degradation in a Harmonic Drive
## 2 Mechanical Model
### Modeling of Compliance and Hysteresis  
Hysteresis behavior is usually caused by micro-sliding friction in mechanical systems [28]. An HD has three moving combinations: between the inner and outer races of the WG bearing, between the outer surface of the WG and the inner surface of the FS, and between the FS and CS teeth. The first two contacting combinations are mainly rolling contacts, and the frictional force is relatively small compared to that of the third sliding combination. Therefore, only the effect of friction in the tooth engagement area on hysteresis was considered. The gear-tooth interface operates in an almost mixed lubrication regime [30]. In this case, the external load was jointly carried by the asperity contact and the fluid film. \[W=W_{l}+W_{a}=(\lambda_{1}+\lambda_{2})W \tag{12}\]  
where \(W_{l}\) is the load carried by the fluid, \(Wa\) is the load carried through direct surface contact, and \(\lambda_{1}\) and \(\lambda_{2}\) are the fluid and asperity load ratio, respectively, which depend on the lubrication condition. The traction force in mixed lubrication can be written as the sum of two portions [31]:  
\[F_{f}=\left(F_{f}\right)_{l}+\left(F_{f}\right)_{a}=\mu_{l}W_{l}+\mu_{c}W_{a} \tag{13}\]  
where \((F_{f})_{l}\) and \((F_{f})_{a}\) are the hydrodynamic and asperity traction forces, respectively; \(\mu_{c}\) is the boundary friction coefficient; and \(\mu_{l}\) is the hydrodynamic friction coefficient. Substituting (12) into (13) yields:  
\[F_{f}=(\mu_{l}\lambda_{1}+\mu_{c}\lambda_{2})W=\mu W \tag{14}\]  
In previous works, a single stiffness element was usually included at the HD for simplicity, and the FS and the overall transmission compliance were not quantified into the individual locations of the HD. In practice, the deformation of the WG is also the main contributor to the total transmission compliance [10]. To adequately model HD behavior, in [10], a spring-mass system with two springs and a mass (assume that spring (a) represents the WG compliance, and spring (b) represents the FS compliance) was proposed to describe the stiffness and hysteresis behaviors of the HD transmission. It implied that the compliance of the WG needs to be considered when modeling the HD compliance. Previous researchers have observed that the number of teeth pairs in an HD that mate simultaneously increased with increasing resistance torque [21]. This is because  
Figure 4: A typical stiffness and hysteresis curve of a harmonic drive. the deflections of the FS and CS teeth gradually increase with the increasing load torque. This shows that the FS compliance is composed of the compliance of the FS cup and the FS teeth. The CS teeth are much more rigid than the FS teeth in conventional HDs. Therefore, the CS teeth contribution to transmission compliance was not considered in this paper. The stiffness of an FS tooth is expressed as follows:  
\[K_{Ti}=\frac{F_{i}}{\xi} \tag{15}\]  
where \(F_{i}\) is the gear loads, and \(\xi\) is the total deflection of an FS tooth. The total deflection is composed of deflections due to bending, deflections due to shearing, local deflections owing to Hertzian contact deformation, and deflections due to tooth foundation [20]. In particular, interferences occur at several tooth pairs even with no load, but the interference almost vanishes with an increase in the external load [20]. The no-load preload \(T_{0}\) due to inherent initial interference can be approximated by a piecewise linear function of the load:  
\[T_{0}=\left\{ \begin{array}{ll}c(T_{FS}-T_{i})&T_{FS}\leq T_{i}\\ 0&T_{FS}>T_{i}\end{array} \right. \tag{16}\]  
where \(c\) is constant to be determined, and \(T_{i}\) is the piecewise linear threshold. Based on the above discussions, we observed that the main sources of flexibility in HD transmissions included the FS cup, the FS teeth, and the WG. The hysteresis phenomenon resulted from the combined effect of the flexibility in the HD and micro-sliding friction in the tooth engagement area.