## 6 Simulation of Meshing
### Helical Gear Pair with Out-of-Plane Misalignment  
In this section, an examination of the deviation of the helical gear system in the case of out-of-plane misalignment oriented along the \(x\)-axis is conducted (Figure 4b). Table 6 demonstrates the results required for each modification parameter (\(C_{c}\), \(c_{1}\), and \(a_{1}\)) of all three crown types for misalignment angles of 0.05-0.2\({}^{\circ}\) (0.87-3.5 mrad) to obtain continuous TE function. Moreover, \(\Delta\varphi_{max}\) was also obtained for every misalignment angle and crown type. It is observed that the amount of modification for all three crown types decreases with increasing misalignment. As for \(\Delta\varphi_{max}\), it is seen that the logarithmic modification gives the smallest value, followed by the circular and then the parabolic. To obtain a better understanding of the difference between modifications, the area \(S_{c,p,l}\) is obtained for all crown types for 0.2\({}^{\circ}\) (3.5 mrad) misalignment based on its TE function (Figure 20) and is calculated as:  
\[S_{c}=-0.645048;\;S_{p}=-0.642903;\;S_{l}=-0.509613\]  
It can be seen that the logarithmic function gives almost 21% less area compared to the circular and parabolic functions. This makes the logarithmic modification significantly better than the other two crowning types. \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{2-7}  & \(C_{c}\) **(mm)** & \(\Delta\varphi_{max}\) **(arcsec)** & \(c_{1}\) **(mm\({}^{-1}\))** & \(\Delta\varphi_{max}\) **(arcsec)** & \(a_{1}\) **(mm)** & \(\Delta\varphi_{max}\) **(arcsec)** \\ \hline \multirow{4}{*}{0.05\({}^{\circ}\)} & 0.804 & \(-\)0.4225422 & 0.00799 & \(-\)0.42257 & 0.873 & \(-\)0.42094 \\  & 0.830 & \(-\)0.424252 & 0.00820 & \(-\)0.424946 & 0.900 & \(-\)0.423014 \\  & 0.853 & \(-\)0.410921 & 0.00847 & \(-\)0.410987 & 0.927 & \(-\)0.409179 \\  & 0.360 & \(-\)0.845263 & 0.00359 & \(-\)0.845918 & 0.368 & \(-\)0.831488 \\
0.1\({}^{\circ}\) & 0.370 & \(-\)0.850059 & 0.00370 & \(-\)0.849776 & 0.380 & \(-\)0.836597 \\  & 0.381 & \(-\)0.823004 & 0.00380 & \(-\)0.825234 & 0.393 & \(-\)0.809249 \\  & 0.143 & \(-\)1.69059 & 0.00143 & \(-\)1.69062 & 0.046 & \(-\)1.4328 \\
0.2\({}^{\circ}\) & 0.146 & \(-\)1.70474 & 0.00146 & \(-\)1.7048 & 0.055 & \(-\)1.49707 \\  & 0.150 & \(-\)1.66515 & 0.00150 & \(-\)1.66508 & 0.081 & \(-\)1.48844 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Amount of modifications relative to out-of-plane angular misalignment for helical gears engagement. Figure 19: Path of contact of longitude circular, parabolic, and logarithmic crowned helical gear with in-plane misalignment: (**a**) 0.05\({}^{\circ}\) (0.87 mrad), (**b**) 0.1\({}^{\circ}\) (1.7 mrad), and (**c**) 0.2\({}^{\circ}\) (3.5 mrad). The dashed line indicates the center of the tooth surface along the width of the gear. The contact path for every crown type and misalignment (0.05-0.2\({}^{\circ}\)) is illustrated in Figure 21. It is seen that with increasing out-of-plane misalignment angles, the point of contact moves further away from the center. The contact path obtained is similar for 0.05\({}^{\circ}\) and 0.1\({}^{\circ}\) misalignment. However, at 0.2\({}^{\circ}\) misalignment, the difference is more apparent since the contact path for the logarithmic crowning is furthest from the center compared to the other two crownings. By comparing Figures 19 and 21 with Figures 10 and 12, it can be seen that the shift of the contact path from the center of the tooth surface is larger for the case of the helical gears. While for spur gears, there is a shift in the order of 0.3 mm and 1.0 mm for the in-plane and out-of-plane misalignment errors of 0.4\({}^{\circ}\), respectively, for helical gears, the respective values are 1.3 mm for in-plane misalignment and 5.0-6.5 mm (depending on the crown type) for out-of-plane misalignment even though the misalignment angle is only 0.2\({}^{\circ}\).