## 5 Results and Discussions
### Effect of Assembly Errors
According to Figure 16a,c, the misalignment deviations \(\varphi\) and \(\gamma\) will cause the mean number of contact pins to increase, and the number of new contact pins is different with the rotation angle of the cycloid gear. Theoretically, the load distributed on each pin should decrease when the number of contact pins increases, but in fact, the load on some contact pins increases instead, such as the 27 th and 28 th pins shown in Figure 16b when \(\theta_{\text{f}}=180^{\circ}\). In addition, even if the number of contact pins remains the same, the load distributed to each contact pin is different from that without the misalignment deviations, such as the 13 th to 18 th pins, shown in Figure 16d, when \(\theta_{\text{f}}=90^{\circ}\). To analyze the cause for the change of the load distribution on the contact pins, the instantaneous contact lines on the tooth surface of the contact pins are calculated for different \(\theta_{\text{f}}\), \(\varphi\), and \(\gamma\), as shown in Figure 17.  
Figure 16: Effect of misalignment deviations on (**a**) number of contact pins (\(\varphi=0.02^{\circ}\)), (**b**) load distribution on pins (\(\varphi=0.02^{\circ}\)), (**c**) number of contact pins (\(\gamma=0.02^{\circ}\)), and (**d**) load distribution on pins (\(\gamma=0.02^{\circ}\)). According to Figure 17a-c, the number of contact pins at the three different positions of the cycloid gear (\(\theta_{\rm f}=90^{\circ}\), \(\theta_{\rm f}=180^{\circ}\) and \(\theta_{\rm f}=270^{\circ}\)) is the same when the misalignment deviations are not considered, and the force condition on the tooth surface of the contact pins at the three different position is similar. As shown in Figure 17d-f, when \(\varphi=0.02^{\circ}\), there are two new pins involved in contact at the three positions, and a serious eccentric load occurs on the tooth surface of some pins, such as the 14 th to 16 th pins when \(\theta_{\rm f}=90^{\circ}\) (see Figure 17d). The contact line on the tooth surface of some pins is shortened, such as the 17 th to 20 th pins when \(\theta_{\rm f}=90^{\circ}\), which indicates that part of the tooth surface of these pins loses contact. A similar eccentric load phenomenon also occurs when \(\gamma=0.02^{\circ}\), as shown in Figure 17g-i. These phenomena explain why the number of contact pins is the same but the load distribution on each contact pin is different when the misalignment deviations are considered, and why the load on some pins increases instead of decreases when the number of contact pins increases. In addition, when the values of \(\varphi\) and \(\gamma\) are equal, the force condition on the surface of the contact pins shown in Figure 17d is similar to that shown in Figure 17h, and the force condition on the surface of contact pins shown in Figure 17e is also similar to that shown in Figure 17i. The difference between the \(\theta_{\rm f}\) in Figure 17d,h is \(90^{\circ}\), and the difference between the \(\theta_{\rm f}\) in Figure 17e,i is also \(90^{\circ}\). This is the  
Figure 17: Contact lines on the tooth surface of the contact pins: (**a–c**) without \(\varphi\) and \(\gamma\), (**d–f**) \(\varphi=0.02^{\circ}\), and (**g–i**) \(\gamma=0.02^{\circ}\). same as the above conclusion for the 90\({}^{\circ}\) phase difference of the TE curves (or the backlash curves) when \(\varphi\) and \(\gamma\) are considered (see Figure 15). To summarize, both the center distance deviation (\(\Delta y\)) and the misalignment deviations (\(\varphi\) and \(\gamma\)) will change the waveform of the curves of the TE and backlash, and the PPTE and PPB will increase with the increase in these deviations. The two misalignment deviations \(\varphi\) and \(\gamma\) have a similar influence on the TE and backlash, and there is a phase difference in the curves of the TE and backlash for the two misalignment deviations.