## 6 Simulation of Meshing
### Spur Gear Pair with In-Plane Misalignment  
In the case of in-plane misalignment, the deviation of the spur gear system is oriented along the \(y\)-axis (Figure 4a). As mentioned earlier, for each type of modification, the ranges of their main parameters, such as \(C_{c}\), \(c_{1}\), and \(a_{1}\), were defined, where \(C_{c}\) is the amount of the circular crown in the middle of the face width, \(c_{1}\) is the parabolic coefficient, and \(a_{1}\) is the logarithmic coefficient. As shown in Equation (20), the parabolic function has other \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Pinion** & **Gears** \\ \hline Number of teeth & 25 & 50 \\ Module, mm & 2.5 & \\ Pressure angle, degree & 20 & \\ Face width, mm & 25 & \\ Center distance, mm & 60 & \\ Poisson’s ratio & 0.3 & \\ Modulus of elasticity, GPa & 206 & \\ Total load, N & 150 & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters of spur gear pair. Figure 7: Path of contact of spur gear with 0.1\({}^{\circ}\) (1.7 mrad) of in-plane angular misalignment. two coefficients, namely \(c_{2}\) and \(c_{3}\), but in the simulation, they were assumed to be zero since only the longitudinal parabolic modification that produces a symmetric curve was considered. Similarly, the logarithmic function Equation (21) has a second parameter, \(a_{2}\), which is a surface factor that can have any value between zero and one (\(0<a_{2}<1\)), and it was chosen arbitrarily equal to 0.9. For the analysis, different levels of crowning modification were applied to the gear to obtain the continuous TE function presented in Figure 8. It can be seen from the figure that an insufficient amount of crowning will lead to a discontinuous TE, and therefore, the amount of crowning should be optimized to give a contact path close to the center of tooth surfaces or at a certain distance from the flank edges to obtain a continuous TE function. The results of the modification parameter values for all three types of crowing and different in-plane misalignment angles (0.01-0.4\({}^{\circ}\) or 0.17-7.0 mrad) that led to a continuous TE function is illustrated in Table 2. Moreover, the impact of the modification and misalignment on the maximum TE (\(\Delta\varphi_{max}\)) is given. Based on the findings, for the misalignment angle between 0.01\({}^{\circ}\) and 0.4\({}^{\circ}\), the modification parameter values that led to continuous TE function remained the same. In addition, the increase in the misalignment angle also increased the \(\Delta\varphi_{max}\), with the logarithmic modification having the least amount of TE. It is noted that although in practice, the magnitude of angular misalignment errors can rarely exceed a few hundreds of a degree (or equivalently a few tenths of a mrad), larger misalignment errors are also investigated in order to showcase the robustness and the effectiveness of the presented TCA model even in extreme cases of misalignment. Figure 8: TE functions with different amounts of crowning: (**a**) circular, (**b**) parabolic, and (**c**) logarithmic. Since the difference between \(\Delta\varphi_{max}\) is very small for all misalignment angles and modification values, the difference is assessed through the integration of \(\Delta\varphi\) to find the area of each TE function as follows:  
\[S_{c,p,l}=\int_{\varphi_{ii}}^{\varphi_{1}/}\Delta\varphi\partial\varphi_{1} \tag{44}\]  
where \(\varphi_{1i}\) and \(\varphi_{1f}\) are the angle of inclination of the pinion at the first and last point of engagement, and \(S_{c,p,l}\) is the area of circular, parabolic, and logarithmic functions.