# Modeling of Transmission Compliance and Hysteresis Considering Degradation in a Harmonic Drive
## 2 Mechanical Model
### Modeling of Compliance and Hysteresis
In order to model the HD hysteresis, we considered its mechanical system analogy, and an HD model was proposed by using a translational motion model analogy (Figure 5). The behavior of the HD rotational compliance can be captured by the behavior of the spring and massless inclined plane system. The aggregate effect of multiple gear teeth meshing is instead achieved by a single continuous gear-tooth surface. As illustrated in Figure 5, \(K_{F}\), \(K_{T}\), and \(K_{W}\) represent the FS cup, FS teeth, and WG compliance, respectively; \(T_{f}\) is the HD friction torque; and \(T_{0}\) is no-load preload. When \(T_{FS}\) increased from zero, springs \(K_{F}\) and \(K_{T}\) were compressed, the number of working springs \(K_{T}\) increased, the no-load preload \(T_{0}\) decreased, and spring \(K_{W}\) stayed at its initial position as long as the driving force was less than friction torque \(T_{f}\). In order to model the FS teeth compliance (Figure 5), a series of springs with different lengths and stiffnesses were used to embody the FS teeth of the actual transmission. Varying spring lengths were intended to characterize the fact that the number of meshing tooth pairs varies with load. FS tooth stiffness was influenced by two main factors: the gear tooth geometry and parameters, and the point of application of load. [PERSON] and [PERSON] [32] showed that the stiffness had an almost parabolic dependence on the contact point position. Figure 5: A harmonic drive model with gear-tooth geometry. The combined stiffness of meshing tooth pairs can be simplified by a piecewise linear function of the load:  
\[K_{T}=\left\{ \begin{array}{cc}aT_{FS}+b&T_{FS}\leq T_{T}\\ aT_{T}+b&T_{FS}>T_{T}\end{array} \right. \tag{17}\]  
where \(a\) and \(b\) are constants to be determined, and \(T_{T}\) is the piecewise linear threshold. From the analysis of the HD model in Figure 5, the torque-displacement relationship of the HD gear can be modeled as  
\[T_{FS}=K_{F}\Delta\theta_{F} \tag{18}\]  
\[\Delta\theta_{T}\approx\left\{ \begin{array}{cc}\sum\limits_{i=1}^{\mathit{ ll}(T_{FS})}\frac{1}{\mathit{ai}+b}&T_{FS}\leq T_{T}\\ \sum\limits_{i=1}^{\mathit{ll}(T_{T})}\frac{1}{\mathit{ai}+b}+\frac{T_{FS}-T_ {T}}{\mathit{a}T_{T}+b}&T_{FS}>T_{T}\end{array} \right. \tag{19}\]  
\[T_{FS}\sin\alpha=K_{W}\Delta\theta_{W}\cos\alpha\pm(T_{FS}\cos\alpha+K_{W} \Delta\theta_{W}\sin\alpha+T_{0})\mu \tag{20}\]  
where \(\Delta\theta_{W}\), \(\Delta\theta_{T}\), and \(\Delta\theta_{F}\) denote the torsional angles of WG, FS teeth, and FS cup, respectively; \(\alpha\) is the gear-tooth angle, and \(\mathit{ll}(T_{FS})\) and \(\mathit{ll}(T_{T})\) are the largest integers less than \(T_{FS}\) and \(T_{T}\), respectively. Equation (20) incorporates information about the direction-dependent friction torque and no-load preload. When considering the compliance of the WG, the FS teeth, and the FS cup, according to the basic operating principle shown in Figure 5, the torsional angle of the HD can be obtained by using the following expression:  
\[\Delta\theta=\Delta\theta_{F}+\Delta\theta_{T}+\tan\alpha\cdot\Delta\theta_{W} \tag{21}\]