## 2 Methodology
### Flexible Multi-Body Dynamic Model
For this purpose, the hydrodynamic load-carrying capacity, provided by fluid film pressure is calculated as:  
\[W_{h}=l\int_{x_{in}}^{x_{out}}p(x)\;dx \tag{5}\]  
Under the EHL regime of lubrication in a cam-tappet conjunction, the presence of asperity interactions from both surfaces is likely. This interaction which contributes to the boundary friction and load-carrying capacity is usually characterised by the Stribeck parameter (\(\lambda\)), which is the ratio of lubricant film thickness over the composite roughness value of the surfaces. To consider this boundary interaction, the Greenwood and Tripp method is utilised [28]. In this method, the share of contact load carried by the asperities is calculated as (boundary load-carrying capacity):  
\[W_{a}=\frac{8\sqrt{2}}{15}\;\pi(n\beta\sigma)^{2}\;\sqrt{\frac{\sigma}{\beta} }\;\hat{E}AF_{\frac{5}{2}}(\lambda)\quad\lambda=\frac{h}{\sigma} \tag{6}\]  
The statistical function \(F_{\frac{5}{2}}(\lambda)\) can be expressed through the below polynomial function, which is based on the Gaussian distribution of surface asperity heights:  
\[F_{\frac{5}{2}}(\lambda)=\left\{ \begin{array}{cc}0.000421\lambda^{6}-0.00853 8\lambda^{5}+0.07162\lambda^{4}-0.3193\lambda^{3}+\\ 0.802\lambda^{2}-1.082\lambda+0.6166&\mbox{$for\;\lambda\leq 2.5$}\\ 0&\mbox{$for\;\lambda>2.5$}\end{array} \right. \tag{7}\]  
Hence, the abovementioned force equilibrium is obtained as:  
\[W=W_{h}+W_{a} \tag{8}\]where \(W\) is the total load on the contact (obtained from dynamics). The film thickness equation of lubricant utilised in Reynolds equation is presented as a function of geometry and the elastic deformation of surfaces [25]:  
\[h=h_{0}+\frac{x^{2}}{2R}+V(x) \tag{9}\]  
The main difference between the coated and uncoated model lies in deformation and contact mechanics. For an uncoated line contact (half plain), the normal displacement by a pressure distribution (elastic deformation) is controlled by the Flamant solution [29]:  
\[V(x)=-\frac{4}{\pi E^{\prime}}{\int_{-\infty}^{\infty}\ln\left|x-x^{\prime} \right|}p(x^{\prime})dx^{\prime} \tag{10}\]  
where \(1/E^{\prime}=0.5\left[\left(1-{v_{1}}^{2}\right)/E_{1}+\left(1-{v_{2}}^{2} \right)/E_{2}\right]\). The Green's function in the convolution of Equation (10) is [29]:  
\[g(x)=-\frac{4}{\pi E^{\prime}}\ln\left|x\right| \tag{11}\]  
Then, the corresponding frequency response function (FRF) in the frequency domain can be presented as [29]:  
\[\widetilde{g}(\omega)=\frac{4}{E^{\prime}\left|\omega\right|} \tag{12}\]  
For the coated line contact conjunctions, an explicit Green's function is not achievable for deformation in the space domain. However, its frequency response function is available as [29]:  
\[\widetilde{g}(m)=\frac{1-v_{c}}{\mu_{c}m}\frac{1+4 mh_{c}\kappa\theta-\lambda \kappa\theta^{2}}{1-\left(\lambda+\kappa+4\kappa m^{2}{h_{c}}^{2}\right) \theta+\lambda\kappa\theta^{2}}\hskip 36.135 ptm=\left|\omega\right| \tag{13}\]  
where \(\theta=exp(-2 mh_{c})\). The tilde represents Fourier transform with respect to \(x\). \(\mu_{c}\) is the shear modulus of the coating material (\(\mu_{c}=\frac{E_{c}}{2(1+v_{c})}\)), and \(h_{c}\) is the coating thickness. \(m\) is the frequency domain counterpart of radius in the space domain. \(v_{c}\) is the Poisson's ratio of the coating material. Other parameters in Equation (13) are expressed as [30]:  
\[\lambda=1-\frac{4(1-v_{c})}{1+\zeta(3-4v_{s})},\ \kappa=\frac{\zeta-1}{\zeta+(3-4v _{c})},\ \zeta=\frac{E_{c}(1+v_{s})}{E_{s}(1+v_{c})} \tag{14}\]  
There is a singularity problem of FRF at \(m=0\)[29]. This problem is resolved by the Gaussian quadrature. Deriving the FRF of Green function, the elastic deformation of coated contact can be obtained utilising the Discrete Convolution--Fast Fourier Transform (DC-FFT) fast algorithm and influence coefficients [31].