## 2 Methodology
### Flexible Multi-Body Dynamic Model  
For the purpose of this investigation, a single valvetrain model was used, thus facilitating efficient evaluation of the dynamic and tribological behaviour of the cam-tappet conjunction under coated and un-coated conditions. A 3D view of the system is shown in Figure 4, which was produced using EXCITE(tm) (Graz, Austria) Multi-Body Environment. All bodies comprising the system are flexible, thus allowing the incorporation of localised deformation in bodies. Contact deformations in the system are considered as restrains to account for deformations in the conjunction. Additional constraints and restraints are applied using joints, limiting the degrees of freedom of each body while also facilitating their motions and interactions. A list of bodies and their global degrees of freedom (DoF) are presented in Table 2. As mentioned above, these bodies are flexible. Hence, additional DoF are present due to the flexibility and localised deformations. The flexibility of the system is implemented using Component Synthesis Method [23]. Joints utilised to facilitate constraining the system are shown in Tables 2 and 3. The \"Axial Thrust Bearing\" (AXBE) joints enable the pairs of bodies to be constrained in three rotational and two translational DOFs. The remaining single translational DOF is restrained with discrete stiffness and damping parameters provided from experimental characterisation. The contact joint can be represented by a dry or lubricated model. In case of dry representation, the restrained DOF is governed by the contact stiffness. If the contact represents the presence of lubricant, the restrained DOF should consider the effective stiffness comprising contact and lubricant film compliances. Due to the load-speed combinations in the cam-tappet conjunction, the behaviour of the lubricated contact falls within the EHL regime. This will be shown in the results section using a \"Greenwood Chart\" [24]. Under this regime of lubrication, the stiffness of the lubricant film is several orders of magnitude higher than the one from contacting bodies. Since the lubricant film is in series with this contact, the effect of lubricant film stiffness can be neglected in the dynamic system [21; 22]. Consequently, the only coupling between dynamics and  
Figure 3: Example of post-processed surface topography. and modelled by Reynolds equation. The one-dimensional form of this equation, neglecting any side-leakage for the cam and follower lubrication, is presented below [25]:  
\[\frac{\partial}{\partial x}\left(\frac{\rho h^{3}}{\mu}\frac{\partial p}{ \partial x}\right)-12u\frac{\partial(\rho h)}{\partial x}-12\frac{\partial( \rho h)}{\partial t}=0 \tag{1}\]  
In above equation, both viscosity \((\mu)\) and density \((\rho)\) are functions of the hydrodynamic pressure. In this study, Dowson and Higginson equation is used to model the pressure-density relationship [10]. \[\frac{\rho}{\rho_{0}}=1+\frac{0.6\rho}{1+1.7\rho} \tag{2}\]  
For the pressure-viscosity relationship (piezo-viscosity), [PERSON]'s equation [26] is employed:  
\[\frac{\mu}{\mu_{0}}=\exp\biggl{(}(\ln\mu_{0}+9.67)\biggl{(}-1+\left(1+5.1\times 1 0^{-9}\times p\right)^{Z}\biggr{)}\biggr{)} \tag{3}\]  
\[Z=\frac{\alpha}{5.1\times 10^{-9}(\ln(\mu_{0})+9.67)}\]  
where \(\rho_{0}\) and \(\mu_{0}\) are the density and viscosity at atmosphere pressure, respectively. The Reynolds boundary condition is assumed for Equation (1) as [27]:  
\[p=0\Big{|}_{x_{in}},\,p=0\Big{|}_{x_{out}},\,\frac{dp}{dx}=0|_{x_{out}} \tag{4}\]  
In an explicit approach, the force equilibrium should be satisfied between the conjunction load resulted from the dynamic solution (which includes inertial forces) and the EHL pressure distribution obtained through Equations (1)-(4).