## 4 Numerical Examples
The load distribution of the best ease-off tooth is shown in Figure 6c, and the value of loads is small in the addendum, root, heel, and toe, which could easily help to avoid stress concentration. The total loads on the contact line decreases for the best ease-off tooth and the conjugate tooth. The load-sharing coefficients are shown in Figure 6d. The peak value of the load-sharing coefficients reaches the maximum because of excessive mismatch for the theoretical tooth. The sum of the load-sharing coefficients is equal to 1 for the contact positions 1, 9, and 17. The maximum load-sharing coefficients come to the heel, and the curve is asymmetric for the conjugate tooth, because the length of the contact line from the heel to the toe tends to decline. Figure 5: TCA simulation and ease-off deviation. (**a**) TCA for the theoretical tooth. (**b**) TCA for the best ease-off tooth. (**c**) TCA for the conjugate tooth. (**d**) The ease-off deviation for the theoretical tooth. (**e0**) The best ease-off deviation for the optimization tooth. (**e1**) The ease-off deviation, dependent only on the length of the instantaneous contact line. (**e2**) The ease-off deviation, dependent only on the transmission error. Figure 6: Tooth load distribution based on LTCA. (**a**) The load distribution for theoretical tooth. (**b**) The load distribution for the conjugate tooth surface. (**c**) The load distribution for the ease-off modification tooth surface. (**d**) Loads sharing a coefficient of the contact lines. Conclusions drawn from Figure 6 show the smallest load value, which can be evenly distributed along the whole tooth for the best ease-off tooth. Besides, the best ease-off tooth avoids loads of the tooth edge. Loaded transmission error and its amplitude under loads are shown in Figure 7. The influences of the actual coincidence degree \(\epsilon_{e}\) on ALTE under loads is discussed as follows:  
(1) ALTE increases with gradually increasing loads for the conjugate tooth because \(\epsilon_{e}\) is a constant value with increasing loads and reaches the maximum value equal to \(\epsilon_{t}\) (Figure 7a). (2) As the load increases, \(\epsilon_{e}\) approaches the maximum value gradually from a small value due to excessive tooth mismatch for the theoretical tooth. So, the curve of the ALTE value fluctuates with increasing loads for the theoretical tooth. When the load form of 300 N-m increases to 600 N-m, the contact ellipse length increases (\(\epsilon_{0}\) decreases). Then, \(\epsilon_{e}\) increases, so ALTE decreases. When the load is \(\geq\)600 N-m, \(\epsilon_{e}\) reaches the maximum, so the ALTE gradually increases (Figure 7a). (3) The relationship between ALTE and loads for the best ease-off tooth shows that when the load increases to 300 N-m, \(\varphi_{r}/\varphi_{p}\) gradually increases, so \(\epsilon_{e}=\varphi_{r}/\varphi_{p}-\epsilon_{0}\) increases and then ALTE decreases. When the load increases from 300 to 600 N-m, the normal clearance decreases (\(\epsilon_{0}\) decreases). Therefore, \(\epsilon_{e}\) increases still and ALTE decreases. Similarly, when the load is \(\geq\)600 N-m, the actual coincidence \(\epsilon_{e}\) does not change (reaching the maximum), so the ALTE increases (Figure 7b). Conclusions can be drawn from Figure 7 that the actual coincidence, \(\epsilon_{e}\), refers to a variable value with increasing loads for the tooth with modification. ALTE decreases with increasing \(\epsilon_{e}\). After \(\epsilon_{e}\) reaches the maximum, ALTE increases with increasing loads.