## 6 Simulation of Meshing
### Helical Gear Pair with Out-of-Plane Misalignment
Consequently, a significant finding from the comparison of the simulations of spur and helical gear meshing is that in the presence of misalignments, the shift of the contact path from the center of the tooth is significantly larger for helical gears in comparison with spur gears. The three-dimensional mesh contact of helical gears in the case of circular and logarithmic modifications is presented in Figure 22. The parabolic crown is not illustrated as its surface is nearly identical to the circular crown. Looking at the figure, in the case of extreme misalignment between the two gear teeth, the surface with circular crowning will experience edge contact. In the logarithmic crowning case, for the same scenario, a much more extreme angle of misalignment is needed. The maximum contact pressure ratio of two helical gear pairs at different angles of out-of-plane misalignment was determined using the Hertz contact equations for an elliptical contact point. Figure 23 shows the \(MPR\) for all three types of modifications depending on the misalignment. Initially, at 0.01\({}^{\circ}\) (0.17 mrad) of out-of-plane misalignment, the logarithmic curve has the highest maximum contact pressure (\(MPR=15.1803\)), whereas the circular and parabolic curve has a lower pressure value (\(MPR\) of 15.0962 and 15.1661, respectively). Unlike the case of spur gears, where the lower MPR value was developed for the parabolic crowning at every misalignment angle, in the case of helical gears, the lower MPR value occurs for a different type of crowning for each case of misalignment. At 0.02\({}^{\circ}\) (0.35 mrad), the logarithmic crowning shows the lower MPR value (12.4127), while the circular crowning gives the highest MPR value (12.6628). Then, at 0.05\({}^{\circ}\) (0.87 mrad), the lowest value has the parabolic modification (MPR = 10.0767), while the highest value is  
Figure 20: TE of helical gear with three types of modification and 0.2\({}^{\circ}\) (3.5 mrad) of out-of-plane angular misalignment. found for the logarithmic modification (MPR = 10.1105). At 0.1\({}^{\circ}\) (1.7 mrad) of misalignment, the circular crown has the lowest value, and at 0.2\({}^{\circ}\) (3.5 mrad), the parabolic crown again has the lowest value. The values of MPR for different modifications at the misalignment angles among 0.01-0.1\({}^{\circ}\) were close to each other, but at 0.2\({}^{\circ}\) misalignment, the logarithmic crown was distinguished by a significantly higher MPR compared to the other two traditional crowns. All these differences are mainly related to the peculiarities of the curvature of the tooth, which is more noticeable in the helical gear. Figure 21: Path of contact of longitude circular, parabolic, and logarithmic crowned helical gear with out-of-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 scaled contact ellipses at the pitch point are shown at the 0.2\({}^{\circ}\) misalignment angle. Figure 22: Example of helical gear assembly models in space with: (**a**) unmodified tooth, (**b**) circular crowning, and (**c**) logarithmic crowning. Figure 24 shows an example of the contact path for the circular curve at an out-of-plane misalignment of 0.2\({}^{\circ}\) with contact ellipses. \(MPR\) in the upper part of the gear tooth surface is greater than in the lower part, gradually decreasing along the path of contact. For other crowning, the pressure contact behavior is the same. Figure 25 demonstrates the contact path in the case of out-of-plane misalignment with the minimum amount of longitudinal circular crowning and tip relief. It is clear that the overall path trace stays the same up until it reaches the tip relief, where contact switches with a significant rise in pressure and decrease in the contact ellipse. The relief length in this case is \(L=0.2\) mm, and the modification amount is \(c_{m}\)= 0.2 mm.