# Influence of Metal Fear Tooth Geometry on Load and Wear within Metal-Polymer Fear Pairs
## 4 Evaluation of Local Load and Wear
### Local Load of the Tooth Flank  
The movement of the contact point along the tooth flank results in a varying sliding velocity and line load due to the changed leverage effect from tooth root to tooth tip. In addition, the overlap of one or two tooth pairs results in a doubling or halving of the line load, which leads to varying load on the tooth flank. The resulting load is described by the combination of the sliding velocity of the tooth flanks and the applied torque. The acting line load \(F_{T}\) is calculated from the torque \(T\), the diameter \(d\) and the tooth width \(b\)[19]. The line load does not take elastic behavior in the contact area into account. Figure 4: Schematic contact behavior and contact points of an involute gear pair. However, it is known from other investigations that the surface pressure resulting from tooth deformation is of secondary importance for tribological considerations of wear [20]. \[F_{T}=\frac{\left(\frac{2\times T}{d}\right)\times 1000}{b}\ [\text{N/mm}] \tag{1}\]  
Figure 5 shows the points of contact resulting from the contact behavior and the acting line load along the tooth flank (Figure 4). When two tooth pairs are in mesh (contact points A to B and D to E), the line load is halved compared to areas with overlap of only one tooth pair (contact points B to D). The line load is reduced from a maximum amount of 20.6 N/mm (one tooth pair in mesh) to 10.3 N/mm (two tooth pairs in mesh). The load decreases in the direction of the tooth tip due to the leverage effect of the increasing diameter. In the next step, the sliding stress is evaluated. The sliding stress results from a relative speed of the two tooth flanks due to different tangential speeds. Therefore, the local velocities of the tooth flanks of the pinion and wheel must be taken into account. The peripheral velocity of pinion (1) and wheel (2) are each derived from the rotational speed \(n\) and the gear diameter \(d\)[19]. \[v_{t1}=\frac{d_{1}*\pi*n_{1}}{60*1000}\ [\text{m/s}]\ \ \ \ v_{t2}=\frac{d_{2}*\pi*n_{2}}{60*1000}\ [ \text{m/s}] \tag{2}\]  
Using the peripheral velocity and the pressure angle \(\alpha=20^{\circ}\), the local tangential velocity \(w\) along the path of contact \(g_{y}\) is calculated [19]. \[w_{1}\big{(}g_{y}\big{)}=v_{t1}*(\frac{\sin\alpha\pm g_{y}}{\frac{d_{1}}{2}}) \ [\text{m/s}]\ \ \ \ w_{2}\big{(}g_{y}\big{)}=v_{t2}*(\frac{\sin\alpha\pm g_{y}}{\frac{d_{2}}{2}} )\ [\text{m/s}] \tag{3}\]  
The sliding velocities \(v_{g1}\) of the pinion and \(v_{g2}\) of the wheel along the position \(g_{y}\) on the path of contact are calculated from the difference of both tangential velocities [19]. \[v_{g1}=w_{1}\big{(}g_{y}\big{)}-w_{2}\big{(}g_{y}\big{)}\ \ \ \ v_{g2}=w_{2}\big{(}g_{y}\big{)}-w_{1} \big{(}g_{y}\big{)}\ \ [\text{m/s}] \tag{4}\]  
Figure 5: Resulting line load from the points of contact. The stress resulting from sliding is described by the specific sliding \(s\), which is calculated from the ratio of the sliding velocity to the radial velocity [19]. \[s_{1}=\frac{v_{g1}}{w_{1}\big{(}g_{y}\big{)}}\ [-]\quad s_{2}=\frac{v_{g2}}{w_{2} \big{(}g_{y}\big{)}}\ [-] \tag{5}\]  
Figure 6a shows the tangential velocities along the tooth flank and the resulting sliding stress by means of the specific sliding. The tangential velocity increases along the tooth flank toward the tooth tip due to the increasing diameter. The lowest speed is therefore at the tooth root (contact points A or E) and the highest at the tooth tip (E or A). In the pitch point (contact point C), the speed of both gears is identical (ideal rolling). In the tooth root and tooth tip area, however, the tangential velocities are different and sliding occurs. The sliding stress can be expressed by the specific sliding (Figure 6b). The specific sliding increases with distance from the pitch point (C) due to the increasing relative speed of both gears.