## 4 Numerical Examples and Discussion
### Result Presentation  
Figures 6-8 show the equivalent stress of the flexspline's tooth ring, which is at completed forced deformation (\(\beta=67.89^{\circ}\)) or at partial forced deformation in the working section in the double eccentric arc cam wave generator. The completed forced deformation is assumed under the next two conditions: (1) the neutral layer of the flexspline is the equidistant curve of the cam wave generator's profile after deformation; (2) the flexspline fits the wave generator completely after deformation. The partial forced deformation in the working section is assumed that under the next two conditions: (1) the flexspline in the working section fits the wave generator after deformation; (2) the flexspline in the non-working section is completely in free deformation. As shown in Figure 6, the whole equivalent stress of the flexspline would increase obviously as the wrap angle becomes larger. With a fixed wrap angle, the equivalent stress value changed regularly from low at major axis to high at connected point and to a little high at the minor axis. Moreover, stress concentration may occur with improper parameter selection. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Symbols** & **Values** \\ \hline Radius of the eccentric arc (mm) & \(r\) & 20.868 \\ Eccentric distance (mm) & \(e\) & 0.39 \\ The wrap angle (\({}^{\circ}\)) & \(\beta\) & 30\(\sim\)67.89 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The parameters of the wave generator. Figure 5: Structure parameters of the flexspline. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Material Parameters** & **Wave Generator** & **Flexspline** \\ \hline Steel type & 42 CrMo & SNCM439 \\ Density & 7850 kg/m\({}^{3}\) & 7870 kg/m\({}^{3}\) \\ Yield strength & 930 MPa & 835 MPa \\ Tensile strength & 1080 MPa & 980 MPa \\ Young’s modulus & 2.12 \(\times\) 10\({}^{11}\) Pa & 2.09 \(\times\) 10\({}^{11}\) Pa \\ Poisson’s ratio & 0.28 & 0.295 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The material parameters of the flexspline and wave generator in FEM. Figure 8: Equivalent stress with wrap angle from 30\({}^{\circ}\) to 40\({}^{\circ}\). Figure 6: Equivalent stress of flexspline in double eccentric arc cam with different wrap angle. Figure 7: Equivalent stress with wrap angle from 67.89\({}^{\circ}\) to 30\({}^{\circ}\). Figure 7 shows the equivalent stress on the outer end of the flexspline's tooth ring. We can observe that the stress at the connected point would be too large to exceed the allowable stress and an excessive stress concentration would appear when the flexspline is in completed forced deformation by using the double eccentric arc cam wave generator with the wrap angle at 67.89\({}^{\circ}\). Moreover, the equivalent stress at the minor axis is also too large for excessive deformation with excessive wrap angle. The main reason for this is the forced deformation of the flexspline in the non-working section by the cam wave generator. When we decrease the wrap angle gradually, the fitting area becomes less in the working area and a gap begins to appear in the non-working area between the flexspline and the wave generator; that is, the non-working section no longer fits together and begins to be in a state of free deformation, so the whole equivalent stress decreases rapidly. For the wrap angle from 67.89\({}^{\circ}\) to 50\({}^{\circ}\), the maximum equivalent stress of flexspline is reduced from 2300 MPa to 510 MPa. With the wrap angle decreasing to 50\({}^{\circ}\), the overall deformation stress of the flexspline is relatively moderate. This is mainly because decreasing the wrap angle results in the non-working area of the flexspline being in free deformation, which results in release of additional stress to further reduce the basic stress of the flexspline.