## 3 Model Verification and Discussion
### Effects of Assembly Error  
The effects of assembly error are investigated in this section. There are three aspects to illustrate illustrating the effects of assembly error. They are the TVMS, unloaded transmission error, and contact stress. There are four values for \(\Delta\)AP, \(\Delta\)AG and \(\Delta\)E, which are \(-0.5\) mm, \(-0.2\) mm, \(0.2\) mm, \(0.5\) mm, and four values for \(\Delta\Sigma\), which are \(-0.4^{\circ}\), \(-0.2^{\circ}\), \(0.2^{\circ}\), \(0.4^{\circ}\). Following this, the two chip areas are compared and the effects of tooth tip chipping discussed further. The TVMS of SBGs considering four types of assembly errors are shown in Figure 9. From this, the following four main points can be concluded: (1) It is evident that the TVMS generally increases with the increment of \(\Delta\)AP (see Figure 9a) with the exception of \(\Delta\)AP = 0.5 mm. In addition, the amplitude of variation of negative value has a greater increases than that of positive value. (2) The TVMS considering \(\Delta\)AG and \(\Delta\)E decrease with the increase of \(\Delta\)AG and \(\Delta\)E with the exception of \(\Delta\)AG = \(-0.5\) mm and \(\Delta\)E = \(-0.5\) mm. Moreover, the amplitude of variation for \(\Delta\)AG and \(\Delta\)E is larger compared to that of \(\Delta\)AP and \(\Delta\)E. (3) Regarding \(\Delta\Sigma\), there is a small reduction in TVMS with the increase of the absolute value of \(\Delta\)E. (4) Notably, when \(\Delta\)AP = 0.5 mm, \(\Delta\)AG = \(-0.5\) mm, \(\Delta\)E = \(-0.5\) mm, and \(\Delta\Sigma\) = \(-0.4^{\circ}\), the variations in TVMS deviate. Figure 8: Comparison of contact trajectories due to the assembly errors in [PERSON]’s paper [16]: (a) AAP; (b) \(\Delta\)AG; (c) \(\Delta\)E; (d) \(\Delta\Sigma\). Figure 7: Contact trajectories of different assembly errors on the rotating projection of the gear tooth surface: (a) \(\Delta\)AP; (b) \(\Delta\)AG; (c) \(\Delta\)E; (d) \(\Delta\Sigma\). From the variations and trends above, it can be seen that all exhibit reductions in TVMS and peak-to-peak values of TVMS. This discrepancy arises from the contact of the tooth root, which can lead to extreme vibrations, loud noises, significant contact forces, and even crack formation, severely impacting the lifespan of SBGs. In practical applications, many SBGs undergo modifications to the tooth root to prevent contact. Consequently, the presented method solely considers the contact of the tooth surface. If there is contact in the tooth root area, the method will not distribute force to potential points in the tooth root area, resulting in a reduction in TVMS due to the decrease in bearing area. The unloaded transmission errors considering different assembly errors are displayed in Figure 10. The range from the x-coordinate of the lowest point on the left side to zero is called the left meshing location and the range on the right side is called the right meshing location. There is a common phenomenon that the unloaded transmission errors will gradually equalize at the left meshing location and gradually reach their lowest point at the right meshing location. Due to the complexity of the SBG's tooth surface, assembly errors will lead to changes in the contact area, resulting in unpredictable variations in unloaded transmission error. As of now, no specific rules can be summarized regarding these variations. Assembly errors can induce changes in unloaded transmission errors. Favorable changes, such as \(\Delta\)AP = \(-0.5\) mm and so forth, can increase the lowest point of unloaded transmission error. The contact stresses of SBGs considering different assembly errors are shown in Figure 11. It is evident that the contact area, which consists of potential contact points wherein the contact stress exceeds zero, and the initial contact trajectory of \(\Delta\)AP and \(\Delta\Sigma\) move from heel and tip to toe and root. On the contrary, those of \(\Delta\)AG and \(\Delta\)E move from the toe and root to the heel and tip.