# Modeling of Transmission Compliance and Hysteresis Considering Degradation in a Harmonic Drive
## 5 Results and Discussion  
Considering the non-symmetric property of the HD's stiffness after testing, the transmission compliance and hysteresis model parameters were identified using the torque-displacement curves for \(T_{FS}\) > 0 (Figure 7). The identified parameters are listed in Table 2.  
The torsional angle was computed using the identified parameters and the transmission compliance and hysteresis model as given in Equations (25)-(28). The estimated torsional angle was next compared to the actual measured angle. Figure 10 shows the estimated and measured torque-displacement curves for different run times. The calculated curves closely coincided with those found experimentally. Therefore, the spring and massless inclined plane system can reasonably describe the transmission compliance and hysteresis behaviors of the HD. A dynamic model of hysteresis in HDs proposed by [PERSON] et al. (Appendix A) also was used to describe the transmission compliance and hysteresis phenomenon. As shown in Figure 10, when comparing the proposed model to the [PERSON]'s model, for the initial behavior, the estimation results formed two similar curves that are consistent with the experimental observation. However, for the HD after 100 h of operation, the model proposed in this paper can better characterize its hysteresis behavior. This is because the proposed model captures the behavior degradation mechanism of the reducer. The torque-displacement curve of the [PERSON]'s hysteresis model given in Appendix B, the [PERSON]'s hysteresis model given in Appendix A, and that of the proposed model are shown in Figure 11. In [PERSON]'s model, the behavior of the HD's rotational compliance was described by the behavior of a serial spring-mass system. They assumed that the linear spring (a) represents the WG compliance, the linear spring (b) represents the FS compliance, and hysteresis is caused by constant friction \(F_{f}\). It can be shown that [PERSON]'s model cannot capture the nonlinear flexibility of a harmonic drive, and the hysteresis curve is independent of the load torque. In [PERSON]'s model, they used the hereditary concept to \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **Initial Curve** & **Post-Test** & **Parameter** & **Initial Curve** & **Post-Test** \\ \hline \(T_{T}\) & 0 Nm & 30 Nm & \(K_{W}\) & 101,837.5 Nm/rad & 98,548.7 Nm/rad \\ \(A\) & - & 1833.5 & \(M\) & 0.066 & 0.076 \\ \(B\) & - & 5729.6 & \(C\) & \(-1.177\) & \(-1.152\) \\ \(K_{F}\) (\(K\)) & 50,923.2 & 78,197.8 & \(T_{i}\) & 115.5 Nm & 70.2 Nm \\  & Nm/rad & Nm/rad & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Experimentally identified parameter values used in the HD simulation. Figure 10: The measured and simulated hysteresis curves. represent the hysteresis behavior of the HD, and an analytical odd polynomial function was selected for the stiffness curve. It is important to observe that the hysteresis represented by this model decreased with an increase in load. Therefore, as shown in Figure 10, this model cannot accurately capture the hysteretic behavior of the harmonic reducer after degradation. Compared with the above models, the proposed model considers the influence of the multi-tooth engagement and interference characteristics of the HD on the transmission flexibility and hysteresis. The proposed model can capture the degradation of nonlinear stiffness and hysteresis behaviors of HD. Figure 12 shows an analysis of the effects of interference and multi-tooth meshing on the behavior of the HD. The hysteresis was clearly affected by the preload caused by the interference. At zero torque output, hysteresis decreased as preload decreased. This explains why the hysteresis decreased as the running time increased (Figure 7). Figure 12 also shows that the nonlinear stiffness of the reducer is attributed to the number of meshing teeth, which depends on the load. The number of meshing teeth reached the maximum (\(T_{T}=0\)) due to initial interference. Therefore, the initial stiffness was linear (Figure 7).