# On the Effect of DLC and WCC Coatings on the Efficiency of Manual Transmission Gear Pairs
## 2 Methodology
### Contact Mechanics of Coated Gear Teeth  
Although it may sound reasonable to assume that there is no difference (when calculating friction) between coated and uncoated gear teeth, this is not the case in reality. There is necessity for a different approach of the contact mechanics between a coated and a non-coated surface, which will consequently change the viscous friction formulae as well as boundary friction by changing the apparent contact area. The equations described in Section 2.2 originate from the assumption of Hertzian contact. However, when a coating layer is used the contact conditions may differ from Hertzian. It has been observed that the characteristics of the coating and base materials can change the type of contact. In the literature, there are several models proposed for modelling the contact of coated gear teeth. In the majority of cases, numerical models at conjunction level are proposed, which are computationally expensive and \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{**Height—Feature Parameters**} & \\ \hline Root-mean-square height & Sq & 0.408 & \(\mu\)m \\ Density of peaks & Spd & 0.00294 & 1/\(\mu\)m\({}^{2}\) \\ Arithmetic mean peak curvature & Spc & 0.669 & 1/\(\mu\)m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Surface parameters for the gear teeth. Figure 2: Surface asperities of the gear teeth. thus not possible to be coupled with gear dynamics [23; 24; 25]. An analytical method is developed in this work to model coated gear teeth. The contact between the coated and non-coated gear teeth during meshing will be modelled by considering the uncoated gear as cylindrical elastic solid and the coated one as layered flat elastic solid. The use of an equivalent cylinder on plane contact model for modelling gear teeth contacts is a common approach, as proposed by various researchers [23]. Figure 3 illustrates this approach. The layer of coating is considered to be perfectly adhered on the substrate. The main properties of the materials involved are the Poisson's ratio (\(\
u\)) and Young's modulus of elasticity (\(E\)). In the formulation, \"1\" refers to the layer of the coating, \"2\" to the substrate, and \"3\" to the indenter (counterpart in this case). The approach followed can be mainly summarized as a method that will track the non-Hertzian type of contact generated due to the presence of the coating and altering the methodology for the non-coated case to allow the calculation of frictional losses. Two additional ratios need to be determined, which will be used to track the shape of the contact and the pressure applied on it. The first one is the ratio of [PERSON]'s modulus of elasticity of the layer and the indenter [26]:  
\[\delta=\ \frac{E_{1}}{E_{3}} \tag{16}\]  
The second ratio is \(\gamma\), which is the ratio of the Young's modulus of elasticity of the layer and the substrate [26], described by:  
\[\gamma=\ \frac{E_{1}}{E_{2}} \tag{17}\]  
It should be d that for the case of \(\delta=0\), it is considered that the indenter is rigid whereas when \(\delta=\gamma\), the indenter is elastic. In addition, all materials involved share the same [PERSON]'s ratio, which helps us to avoid numerous calculations:  
\[\
u_{1}=\
u_{2}=\
u_{3} \tag{18}\]  
This assumption is generally applicable for many cases of coated gears, since the values of both steel and several types of coatings are comparable. Also, many methods to model the effect of coating (as found in the literature) are employing the same assumption [24; 25]. The next step requires the calculation of the maximum pressure and contact half width for pure Hertzian contact of a material with the properties of the layer. These values will be used as a benchmark for the actual values of the contact later. The maximum Hertzian pressure is given by:  
Figure 3: Description of approach of coated and non-coated gear. \[p_{0}=\left[\frac{E_{1}}{1-\
u_{1}^{2}}\frac{W}{(1+\delta)\pi R}\right]^{1/2} \tag{19}\]  
where \(W\) is the load and \(R\) is the equivalent radius of the two contacting bodies.