## 2 Thermohydrodynamic Lubrication Model  
The piston ring is subjected to various forces, including gas pressure, inertia force, frictional force, oil film pressure, and the ring's elastic force, which together result in relative motion between the piston ring and the ring groove. The basic modes of motion include circumferential motion, radial motion, and rotational motion. The force analysis of the piston ring is illustrated in Figure 1.  
The axial force equilibrium equation of the piston ring is shown in Equation (1):  
\[F_{contact\_ax}=F_{mas\_ax}+F_{gas\_ax}+F_{fric\_ax}+F_{bend} \tag{1}\]  
where \(F_{contact\_ax}\) is the axial component of contact force between the piston ring and cylinder liner, \(F_{mas\_ax}\) is the piston ring mass force, \(F_{gas\_ax}\) is the axial component of the gas force, \(F_{fric\_ax}\) is the axial friction force between the piston ring and the cylinder liner, \(F_{bend}\) is the bending force of the ring itself. The radial force equilibrium equation of the piston ring is shown in Equation (2):  
\[F_{contact\_rad}=F_{gas\_rad}+F_{fric\_rad}+F_{tension} \tag{2}\]  
where \(F_{contact\_rad}\) is the radial component of the contact force between the piston ring and the cylinder liner, \(F_{gas\_rad}\) is the radial component of the gas pressure force, \(F_{fric\_rad}\) is the radial friction force between the piston ring and the cylinder liner, \(F_{tension}\) is the elastic force of the piston ring. Under mixed or boundary lubrication conditions, when relative motion occurs between the two components, in addition to the oil film providing bearing capacity, the asperities on the surface also contribute to the load-bearing capacity due to micro-contact between rough surface peaks. Based on the micro-contact model proposed by [PERSON] and [PERSON] (2015), it is assumed that the surface roughness of the sliding surfaces follows a Gaussian distribution, and a contact model between the micro-asperities on the piston ring-cylinder liner surface is established. This equation describes the load-bearing capacity generated during the contact of asperities, incorporating factors such as surface roughness, material elastic properties, and film thickness. It is used to calculate the contribution of asperities under mixed or boundary lubrication conditions. In this equation, contact pressure, surface roughness, and film thickness are key parameters that influence the load-bearing capacity. The equation indicates that, under lubricated conditions, not only does the lubricant film contribute to load-bearing, but the contact between surface asperities also  
Figure 1: Piston ring force analysis diagram. provides additional support. The load-bearing equation of the micro-asperities is defined as shown in Equation (3):  
\[P_{asp}=\frac{16\sqrt{2}\pi}{15}(\eta_{s}\beta_{s}\sigma)^{2}E\sqrt{\frac{\sigma }{\beta_{s}}}F_{\frac{5}{2}}(H) \tag{3}\]  
where \(E=\frac{1}{\left(\frac{1-2^{2}}{12}+\frac{1-2^{2}}{12}\right)}\); \(\sigma=\sqrt{{\sigma_{1}}^{2}+{\sigma_{2}}^{2}}\); \(H=\frac{h}{\sigma}\); \(F_{\frac{5}{2}}(H)=\left\{ \begin{matrix}4.4086\times 10^{-5}(4-H)^{6804},H<4,\\ 0,H\geq 4\end{matrix} \right.\)  
\(P_{asp}\) is Contact pressure on the apparent contact surface, \(\eta_{s}\) is Density of asperity peaks, \(\beta_{s}\) is Radius of curvature of the asperity peaks, \(E\) is Equivalent elastic modulus, \(E_{1},E_{2}\) is Elastic moduli of the cylinder liner and piston materials, respectively, \(v_{1},v_{2}\) is [PERSON]'s ratios of the cylinder liner and piston materials, respectively, \(\sigma\) is Equivalent surface roughness, \(H\) is the Dimensionless film thickness ratio. The contact force of the micro-asperities on the apparent contact surface can be determined according to the pressure, as shown in Equation (4):  
\[W_{A}=\int_{A}P_{asp}dA \tag{4}\]  
The expression for the real contact area of the micro-asperities is shown in Equation (5):  
\[A_{c}=\int_{A}a_{c}dA \tag{5}\]  
\(\mathrm{a_{c}}\) is the distribution density function of the contact area during loading.