## 6 Simulation of Meshing
### Simulation of Spur Gears Meshing with Out-of-Plane Misalignment  
The misalignment angle and plane in the case of out-of-plane misalignment are shown in Figure 4b. Table 3 shows the values of the maximum amplitude of TE functions (\(\Delta\varphi_{max}\)) and the modification parameters of the three crown types at an angular pitch misalignment of 0.01-0.4\({}^{\circ}\). Similar to Section 6.2, in the case of out-of-plane misalignment, the range of the modification required for all three crowning types to obtain a continuous TE function remains the same for any level of misalignment between 0.01\({}^{\circ}\) and 0.4\({}^{\circ}\). As for the results of the \(\Delta\varphi_{max}\), it can be observed that the logarithmic modification gives the smallest value of the \(\Delta\varphi_{max}\), followed by a circular and then a parabolic one. However, the difference in \(\Delta\varphi_{max}\) between all crowning is not significant. In order to compare the different crowning functions, the area of the TE function for each crowing case for 0.4\({}^{\circ}\) misalignment was evaluated. Figure 11 displays the TE function of each crowning type for 0.4\({}^{\circ}\) out-of-plane misalignment. By calculating \(S_{c,p,l}\) using Equation (44), the following results were obtained:  
\[S_{c}=-0.01180;\ S_{p}=-0.01192;\ S_{l}=-0.01161\]  
In this case, the area of the TE function with logarithmic modification is smaller than the parabolic by about 2.58% and than the circular by almost 1.59%, similar to in-plane misalignment. Figure 10: Path of contact of longitude circular, parabolic, and logarithmic crowned spur gear with 0.4\({}^{\circ}\) (7.0 mrad) of in-plan angular misalignment. The dashed line is the center of the tooth surface along gear width. In Figure 12, the same trend was observed as in the case of in-plane misalignment. The path of contact for the logarithmic function is slightly better than the others. Along the tooth width, the logarithmic function is closer to the center than the parabola by 0.0082 mm and the circular function by 0.048 mm. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Angular**} & \multicolumn{2}{c}{**Circular**} & \multicolumn{2}{c}{**Parabolic**} & \multicolumn{2}{c}{**Logarithmic**} \\ \cline{2-6}  & \(C_{\epsilon}\) **(mm)** & \(\Delta\varphi_{max}\) **(arcsec)** & \(c_{\mathbf{1}}\) **(mm\({}^{-1}\))** & \(\Delta\varphi_{max}\)**(arcsec)** & \(a_{\mathbf{1}}\) **(mm)** & \(\Delta\varphi_{max}\  
The three-dimensional simulations of circular and logarithmic modifications are illustrated in Figure 13. All the surfaces in the middle have the same surface curvature; however, compared to other modifications, the curvature of the logarithmic crowning has a rapid change at the edges, which helps to avoid edge contact even for extreme misalignment errors. Using the Hertz contact equations for the elliptical contact point described in Section 5, the MPR at various out-of-plane misalignment angles were calculated. Figure 14 presents the MPR for the three types of modification with varying misalignment of spur gears. At 0.01\({}^{\circ}\) (0.17 mrad) of out-of-plane misalignment, the parabolic crowning has the lowest MPR (\(MPR=11.7864\)), whereas the circular and logarithmic crowning has a slightly larger \(MPR\), 11.8041 and 11.8253, respectively. The maximum pressure ratio remains relatively constant for small misalignments up to 0.1\({}^{\circ}\) (1.7 mrad), but then, the logarithmic modification \(MPR\) start to increase, while the two traditional lead modifications decrease. At 0.4\({}^{\circ}\) (7.0 mrad) of misalignments, the \(MPR\) for the parabolic crowning is 11.7779, the circular one is 11.7959, and for the logarithmic one is 11.8292. These results indicate that with a larger angle of out-of-plane deviation, the contact area (or contact ellipse) becomes larger for the two conventional lead surfaces and smaller for the logarithmic modification.