## 3 The Model Validation
### Addendum Modification Coefficient
#### 4.2.1 The Influence on Stiffness  
The influences of the addendum modification coefficient on the tooth stiffness of the sliced gear are studied by using the parameters of Case 1, as shown in Figure 16. Attention should be paid to the position angle of the \(x\)-axis, which represents the whole mesh process. A negative value indicates that the meshing point has not reached the pitch point, and a positive value indicates that the meshing point has exceeded the pitch point. In the whole mesh process, the tooth stiffness curves of the sliced gear of a driving gear show a downward trend. The tooth stiffness of the sliced gear decreases gradually. Because the mesh point moves from the root to the top along the tooth profile, and the tooth thickness decreases, the tooth compliance of the sliced gear increases. The opposite is true for the driven gear. With an increasing addendum modification coefficient of the driving gear, that of the driven gear decreases, and the tooth stiffness curve moves to the positive \(x\)-direction, which indicates that the position of the meshing point is delayed because the tip diameter of the driving gear increases while the tip diameter of the driven gear decreases. With an increasing addendum modification coefficient, the tooth stiffness curves of the sliced gear of both the driving and driven gears move upward. Because the cutter is far from the gear center when positive addendum modification is adopted, the tooth thickness increases, which leads to an increase in stiffness. The change in tooth stiffness of the sliced gear increases with an increasing addendum modification coefficient when the tooth root bears the load, because the increasing addendum modification coefficient causes the thickness of the tooth root to increase. Another reason is that the tooth is considered as a cantilever beam whose root is fixed. The combination of these two reasons leads to high stiffness. In the whole mesh process, there are \"bulge\" phenomena in the middle part of the fillet-foundation stiffness curve of the sliced gear, because the two ends belong to the double-tooth engagement region, and the structure coupling leads to a large deformation of the fillet-foundation and low stiffness. With an increasing addendum modification coefficient, the fillet-foundation stiffness curves of the sliced gear of both the driving and driven gears move upward, because the root diameter becomes  
Figure 16: The tooth stiffness of the sliced gear with the parameters of Case 1. (**a**) Driving gear, (**b**) driven gear. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{2}{c}{**Case 1 (\(x_{1}+x_{2}=0\))**} & \multicolumn{2}{c}{**Case 2 (\(x_{1}+x_{2}>0\))**} & \multicolumn{2}{c}{**Case 3 (\(x_{1}+x_{2}<0\))**} \\ \hline \(x_{1}\) & \(x_{2}\) & \(x_{1}\) & \(x_{2}\) & \(x_{1}\) & \(x_{2}\) \\ \hline \(-0.8\)\(\sim\)0.8 & \(0.8\)\(\sim\)\(-0.8\) & \(0.1\) & \(0\)\(-0.8\) & \(-0.1\) & \(0\)\(\sim\)\(-0.8\) \\large and the normal component of the mesh force on the tooth surface changes, leading to an increase in the filled-foundation stiffness of the sliced gear, as shown in Figure 17.  
The tooth stiffness of the helical gear can be acquired by superposing the tooth stiffness of the sliced gear along the centerline of the gear. The sketch of superposition is shown in Figure 18.  
Due to the existence of a helical angle, the tooth stiffness curve of the sliced gear lagging in the engagement region is on the right side of the former one. By superposing the tooth stiffness of the sliced gear, the tooth stiffness of the helical gear appears to increase first and then decrease. As the addendum modification coefficient increases, the maximum value of the tooth stiffness of the helical gear increases.