## 3 Results and Discussion
The first model (referred to as \"implicit EHL model\" or \"implicit\" in figures) implicitly solves the numerical EHL model within the multi-body formulation. The second model (referred to as \"no EHL solver\" or \"explicit\" in figures) represents the multi-body dynamic model with a simple analytical tribology model instead of full numerical EHL solution. The tapped velocity and contact force from implicit and explicit models are plotted in Figures 8 and 9, respectively. The results show the insignificant effect of using the implicit solution in comparison with the explicit one. The slight variation is observed in lightly loaded regions where the nature of the EHL regime is less pronounced, and the film thickness stiffness approached comparable values to those from solid contact. These are regions where the cam and tapped come into loaded contact or they are about to cease contact. They are shown schematically in Figure 8. This effect is more pronounced in acceleration presented in Figure 10. Overall, considering the insignificant effect of these regions as well as expected major effect of coatings in the highly loaded region, the explicit model is used for this investigation. \begin{table}
\begin{tabular}{c c} \hline \hline
**Steel** & \\ \hline Modulus of Elasticity & 210 MPa \\ Poisson’s Ratio & 0.300 \\ Pressure coefficient of boundary shear strength of asperities [31] & 0.185 \\ \hline DLC Coating & \\ \hline Modulus of Elasticity & 185 MPa \\ Poisson’s Ratio & 0.300 \\ Pressure coefficient of boundary shear strength of asperities [31] & 0.440 \\ Coating layer thickness & 4 \(\mu\)m \\ \hline \hline \end{tabular}
\begin{tabular}{c c} \hline \hline
**Problem** & 175 MPa \\ Poisson’s Ratio & 0.300 \\ Pressure coefficient of boundary shear strength of asperities [31] & 0.540 \\ Coating layer thickness & 4 \(\mu\)m \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Material properties. \begin{table}
\begin{tabular}{c c} \hline \hline
**Engine Speed** & **3000 RPM** \\ \hline Valve Mass & 0.07 kg \\ Spring Mass & 0.036 kg \\ Spring Constant & 33750 N\(\cdot\)m\({}^{-1}\) \\ Cam Base Circle Radius & 0.015 m \\ Cam Length & 0.012 m \\ Maximum Cam Lift & 0.0093 m \\ \hline \end{tabular}
\end{table} 
Table 7: Additional model parameters. Figure 8: Velocity of the tappet during engine cycle. Figure 10: Acceleration of the tappet during engine cycle. Figure 9: Contact force between the cam and the tappet during engine cycle. For the EHL solution, the contact load from Figure 9 and speed of entertaining motion in Figure 11 are used as inputs to the explicit coated EHL model. These results are used quasi-dynamically to simulate the behaviour of the lubricated contact during a complete cycle. For each case, the effects of coating are compared with uncoated steel. In addition, the effect of coating thickness is investigated. The main reason for the insignificant effect of implicit modelling can be explained by significantly smaller contribution of the lubricant film to contact stiffness under elasticity domination regimes on the Greenwood Chart, such as EHL [24]. For this purpose, the regimes of lubrication can be ascertained and verified using the Greenwood Chart [24]. This chart utilises working conditions (mainly speed-load combinations) to summarise the contribution of elasticity and piezo-viscosity effects in the form of two dimensionless numbers, \(G_{\mathrm{v}}=\frac{W^{*3/2}G^{*}}{U^{*1/2}}\) and \(G_{\mathrm{\varepsilon}}=\frac{W^{*}}{U^{*1/2}}\). The combination of these numbers specifies if the regime of lubrication is piezo viscous/elastic (PE), piezo viscous/rigid (PR), iso viscos-elastic (IE) or iso viscos-rigid (IR). PE corresponds to the classical EHL, and IR corresponds to the classical hydrodynamic regimes. IE is the regime for so-called \"soft-EHL\". Figure 12 shows the working conditions of this study on the Greenwood chart. It should be noted that the boundaries in this graph are indicative and based on simplified analytical formulas [24]. However, they roughly show the dominant effects.