## 4 Tooth Surface Modification  
Modifications of a gear tooth, such as crowning, are used to reduce misalignment impacts during the assembly process. Crowning moves the peak load from the edges of the tooth flank and decreases the risk of edge contact pressures that could reduce the survivability of gears. Crowning is a longitudinally oriented modification that can localize the bearing contact. Such properly implemented modifications allow to transform discontinuous linear functions caused by misalignment to a continuous function of the TE of gear drives, decreasing vibration and noise [16; 17; 21; 43; 44]. The crowning of the tooth surface can typically be achieved by lightly altering the cutting tool path in relation to the gear frame of reference to remove more material at the edges of the tooth, according to a smooth, continuous function. Conventional choices for this function have been parabolic [30; 31] and circular [32] functions, while logarithmic ones were proposed later in [35]. Figure 5 presents the lead crown chart of gear teeth, where it can be seen that the curves of circular and parabolic modifications decrease more from the center of the tooth flank compared to the logarithmical curve. Moreover, the logarithmic function deviates more than the circular and parabolic modifications at the edges of the tooth face. The black line corresponds to an unmodified tooth with a simple edge fillet, shown here only for reference. The function of the circular, parabolic, and logarithmic curves of the modified tooth demonstrated in Figure 5 is presented in Equations (19)-(21), respectively, as follows:  
\[y=R-\sqrt{R^{2}-x^{2}},\text{ where }R=\frac{4C_{c}^{2}+b^{2}}{8C_{c}}, \tag{19}\]  
\[y=c_{1}x^{2}+c_{2}x+c_{3}, \tag{20}\]  
\[y=a_{1}Log\Bigg{(}\frac{1}{1-a_{2}\big{(}\frac{x}{b}\big{)}^{2}}\Bigg{)}, \tag{21}\]  
where, in this representation, the coordinate \(y\) is the extent of modification and \(x\)-axis is the position at the face width. In addition, at the gear tooth center, \(b\) is the half-face width length. To apply modifications to the meshing equation, Equations (19)-(21) must be rewritten in terms of the coordinate system of \(\overset{\rightarrow}{u}_{i}\) and \(\overset{\rightarrow}{v}_{i}\). The normal longitudinal relief of the parabolic modification \(\delta_{p}\) with \(\overset{\rightarrow}{u}_{i}\) and \(\overset{\rightarrow}{v}_{i}\) parameters are defined as:  
\[\delta_{p}=c_{1}\bigg{(}v-\frac{v_{max}-v_{min}}{2}\bigg{)}^{2}+c_{2}\bigg{(} v-\frac{v_{max}-v_{min}}{2}\bigg{)}+c_{3}, \tag{22}\]  
In the case when \(c_{2}=c_{3}=0\), the longitudinal parabola modification curve will be symmetrical. More details of the generation of parabola modification can be found in [PERSON]'s work [1]. The amount of parabolic crowning, as well as circular crowning, primarily depends on the length of the face width. The magnitude of circular modification  
Figure 5: Curves of different crowning modifications. \(\delta_{c}\) and radius of the curved circular crowning \(R\) (with parameters \(u_{i}\) and \(v_{i}\)) are determined as follows:  
\[\delta_{c}=R-\sqrt{R^{2}-\left(v-\frac{v_{max}-v_{min}}{2}\right)^{2}},\text{ where }R=\frac{4C_{c}^{2}+\left(v_{max}-v_{min}\right)^{2}}{8C_{c}^{2}} \tag{23}\]  
where \(C_{c}\) is the amount of crowning at the middle of the face width. The magnitude of longitudinal relief of the logarithmic modifications \(\delta_{l}\) is defined as:  
\[\delta_{l}=a_{1}Log\left(\frac{1}{1-a_{2}\left(\frac{2v-\left(v_{max}-v_{min} \right)}{v_{max}-v_{min}}\right)^{2}}\right), \tag{24}\]  
where coefficient \(a_{1}\) represents the degree of crowning curvature and \(a_{2}\) a surface factor that is a function of crown drop. To apply the modification to the surface, one of the crown functions \(\delta_{c,p,l}\) are added to the vector function \(\overset{\rightarrow}{f}_{i}(\overset{\rightarrow}{u}_{i},\overset{ \rightarrow}{v}_{i})\) of the parametric surface, and then, they are substituted into the rotation matrix.