## 3 Model Verification and Discussion
### Effects of Toth Tip Chipping  
As Section 3.2 found, some assembly errors can result in edge contact and an increase in tip stress, ultimately leading to tooth tip chipping. Consequently, this section will discuss the effects of tooth tip chipping from three key aspects: TVMS, unloaded transmission error, and contact stress. Each assembly error selects a value characterized by edge contact and high tooth tip stress to introduce an identical area of chipping. They conclude \(\Delta\)AP = \(-0.5\) mm, \(\Delta\)AG = 0.5 mm, \(\Delta\)E = 0.5 mm, and \(\Delta\Sigma\) = \(-0.4^{\circ}\). In general, assembly errors will decrease TVMS in the influence range of tooth tip chipping (see Figure 12). A, B, D, E are the moments which have the maximal reduction of TVMS at \(\Delta\)AP, \(\Delta\)AG, \(\Delta\)E and \(\Delta\Sigma\), respectively. The reduction of A (see Figure 12a), B (see Figure 12b), D (see Figure 12c), and E (see Figure 12d), is 3.28%, 3.92%, 4.91, 3.88%, respectively. It is evident that the effect of \(\Delta\)E is the most significant when compared to \(\Delta\)AP and \(\Delta\)AG at the same error magnitude. \(\Delta\)AP exhibits the smallest effect, although its influence range on tooth tip chipping is the widest. An intriguing observation is that the TVMS of \(\Delta\)AP and \(\Delta\)E may exhibit increases at certain points in time. For example, the TVMS of moment C (see Figure 12c) increased 2.31%. That is because tooth tip chipping causes a change in the smallest clearance (see Figure 13a). Due to tooth tip chipping, the clearances at the chipping region increase, and the smallest clearance changes from M (0 mm) in tooth 1 to P (0.0113 mm) in tooth 1. It is worth noting that the clearance at P is nearly equal to N located in tooth 2. Tooth 1 initially bears a load of 12.45 kN (see Figure 13b), whereas tooth 2 carries 14.76 kN before tooth tip chipping occurs. However, following tooth tip chipping, there is a shift in load distribution, with tooth 2 now bearing a heavier load of 18.48 kN, surpassing by far tooth 1's load of 8.64 kN. As shown in Figure 14, The unloaded transmission error experiences varying degrees of reduction as a result of tooth tip chipping. The reduction of \(\Delta\)E = 0.5 mm (see in Figure 14c) is 131.3402 \(\upmu\)rad which is the most significant and the reduction of \(\Delta\)E = \(-\)0.4\({}^{\circ}\) (see in Figure 14d) is 62.1242 \(\upmu\)rad which is the smallest relative to the other three assembly errors. The effect of \(\Delta\)AP = \(-\)0.5 mm (see in Figure 14a) is similar to \(\Delta\)AG = 0.5 mm (see in Figure 14b). The comparison of contact stress considering the four assembly errors under tooth tip chipping is displayed in Figure 15. When tooth tip chipping occurs, the area of the chip does not bear the load force, thereby increasing the contact stress near the edge of the chipping area. For instance, with \(\Delta\)AG = 0.5 mm, the maximum contact stress reaches 1356.5 MPa. In contrast, the maximum contact stress for a healthy assembly is only 1186.9 MPa. The effects of different sizes of chips may lead to different changes in mesh characteristics. Figure 16 showcases the TVMS for different chip areas under healthy installation and E = 0.5 mm. Figure 17 illustrates the unloaded transmission error and contact stress for different chip areas. There are two sizes of tooth tip chipping to compare and analyze, Chip_1 and Chip_2 (see Figure 17). Firstly, when the installation is healthy, the maximum reduction of Chip_1 is 0.91% and this appeared at moment K (see in Figure 16a), and the maximum reduction of Chip_2 is 3.56% and this appeared at moment J (see in Figure 16a) under healthy installation. Following this, when the installation is \(\Delta\)E= 0.5 mm, the reductions at moments F, G, H, I are listed in Table 3 under \(\Delta\)E = 0.5 mm. The max reduction of Chip_1 is 0.8% and this appeared at moment H (see in Figure 16b) and the max reduction of Chip_2 is 4.92% and this appeared at moment G (see in Figure 16b).