## 1 Introduction
The model was extended by [PERSON] et al. [11], and the principles were applied to a four-cylinder engine, with results being validated experimentally. In these studies, the effects of tribo-dynamic couplings were neglected. Another model was developed and validated experimentally by [PERSON] et al. [12], which allowed the effects of surface roughness as well as thermal effects arising from asperity contacts to be considered. The results highlighted a direct correlation between frictional power losses and surface roughness. Further work was carried out by [PERSON] et al. [6], where a full EHL line contact was coupled with a multi-body dynamics-based full engine simulation and solved iteratively at every point in time. This method was developed for incorporation into simulation software. For this reason, the Reynolds equation and the film thickness equation were treated as modular decoupled systems so that the formulation could provide coverage of various load cases and versatility of hydrodynamic implementation whilst the results remain consistently accurate. Despite the significant development in this work, leading to a fully numerical implicit EHL solution in the tribo-dynamic formulation, the computational cost of the system was a limitation. One of the practical approaches to enhance the tribological performance of cam-tappet conjunctions is to apply appropriate coatings to control the surface behaviour. DLC (Diamond-like Carbon)-type coatings were first introduced on high-performance racing vehicles. However, in recent years, increasingly more manufacturers are incorporating DLC-coated components in commercial vehicles for their superior tribological behaviour compared to uncoated steel [13]. Their frictional benefits have been demonstrated experimentally in several studies [14; 15], whereby reductions in frictional losses of up to 40% were observed [16] depending on lubrication conditions. WC-C (Tungsten-Carbide Carbon) coatings are a type of metal-doped DLC coatings that have also shown further reductions in frictional losses, improvements in lubricity [17] as well as higher resistance to wear under heavily loaded conditions of sliding or rolling contact [18]. However, the application of WC-C coatings in automotive valvetrain applications is not reported in the literature. One of the first advancements was made in the field of the contact mechanics of coated surfaces in a study by [PERSON] and [PERSON] [19], whereby a numerical solution was obtained for the case of a cylindrical elastic body being in a loaded contact with a coated elastic substrate. Numerical methods were derived for the evaluation of the actual contact pressure and halfwidth, which originated from the Hertzian solution. This work was further developed by [PERSON] [20], with modifications being made to the formulation to incorporate an EHL film within the conjunction. Numerical results were tabulated for EHL film thickness depending on various contact parameters. Both studies, however, did not relate their findings to the dynamic behaviour of the cam-tappet conjunction. [PERSON] and [PERSON] [13] proposed a mathematical model that facilitated the evaluation of the behaviour of a cam-tappet conjunction when a thin-film DLC coating layer was introduced on the tappet. However, the model assumed dry contact conditions between the two components for a faster convergence to a solution. Hence, the tribo-dynamic coupling was missing. In this paper, considering the reported benefits of DLC and WC-C coatings in an EHL conjunction, their tribological behaviour in the cam-tappet application is evaluated numerically. The tribological benefits of a WC-C coating in cam-tappets have not been reported in the literature hitherto. The numerical solution comprises an explicit full numerical coated EHL of the contacts, coupled with an implicit analytical tribo-dynamic model. The implicit model which employs analytical EHL methods delivers realistic dynamically achieved load and speed conditions for the explicit method.