## 7 Conclusions  
In this study, a novel form of fundamental tooth contact equations for non-conjugate analysis is derived for a pair of gear teeth modelled as \(C^{1}\) continuous rotating parametric surfaces. The proposed model reduces the contact model to two equations with two unknowns and one independent parameter, thus giving more explicit, stable, and precise numerical solutions without convergence issues compared to conventional [PERSON]'s model [1], which consists of five typically nonlinear equations with five unknowns and one independent parameter. The developed model was applied to crowned spur and helical gears to assess parametrically their sensitivity to different in- and out-of-plane angular misalignment angles in terms of the path of contact, TE, and maximum contact pressure. For this purpose, three types of modification functions were applied, including the conventional parabolic crown, the conventional circular crown, and the less-common logarithmic crown. Modification functions were applied in the longitudinal direction and expressed mathematically in terms of the tooth surface planar coordinates (\(\overrightarrow{u_{i}}\), \(\overrightarrow{v_{i}}\)) of spur and helical gears. Metrics were defined in terms of the TE function and the contact pressure to facilitate a comparison of the performance of each crown design. In terms of these metrics, optimal designs are identified. Key conclusions are summarized as follows:  
1. The simulation showed the increase of TE and the deviation of the contact path from the tooth surface center is more pronounced in the out-of-plane misalignment compared to the in-plane. 2. In the case of spur gears, the logarithmic crowning modification produced a smaller negative effect on the TE function than the traditional types of crowning for both in-plane and out-of-plane misalignment although for the contact path, the improvement is not too substantial. In addition, the differences in results between the modifications are smaller in the in-plane misalignment, as demonstrated in Figures 9 and 10 and Table 2. For the out-of-plane misalignment, the differences are slightly larger (Figure 11, Table 3). 3. The simulations of the helical gear system show that the circular crowning modification function produces a smaller shift of the path of contact in the case of in-plane misalignment, keeping it closer to the center in comparison with the other modifications (Figure 19), but the differences are not too weighty. In the case of out-of-plane misalignment, parabolic crowning shows better results for the contact path (Figure 21). This is especially noticeable for the large out-of-plane misalignment of 0.2\({}^{\circ}\) (3.5 mrad), where the contact path of the logarithmic function is shifted closer to the edge than for the case of the parabolic or circular one. Still, for small misalignment angles, the  
Figure 25: The MPR of crowned helical gears with a minimum amount of circular and tip relief for an out-of-plane misalignment of 0.2\({}^{\circ}\) (3.5 mrad), with the location of contact ellipse at the contact point. differences in the path of contact are not so significant. However, for the TE, the logarithmic crowning in all cases gives smaller values (Tables 5 and 6). The integration of TE functions for the large misalignment angle of 0.2\({}^{\circ}\) (3.5 mrad) illustrates that the difference between the conventional modifications and the logarithmic ones is almost 21% (Figure 20). 4. The shift of the contact path from the center of the tooth during gear meshing in the presence of angular misalignments is almost an order of magnitude larger for the case of helical gears in comparison to spur gears. 5. The logarithmic modification is able to deal with extreme misalignments thanks to its increased curvature at the edges. In the case of circular and parabolic crowning, tooth surface contact under extreme misalignment will present a singularity at the tooth edge.