## 2 Methodology
### Theory  
In the model system, each direction of each particle in the elastic body obeys the laws of momentum and angular momentum, forming its dynamic equation. All the directions of all the particles in the elastic body constitute the dynamic equations [54; 55; 56]. The dynamic equation of the elastic body is as follows:  
\[[M]\big{\{}\ddot{x}\big{\}}+[D]\big{\{}\dot{x}\big{\}}+[K]\big{\{}x\big{\}}= \{f\} \tag{1}\]  
In Equation (1), \([M]\) is the mass matrix; \([K]\) is the structural dynamic stiffness matrix (varies with frequency); \([D]\) is the material dynamic damping (varies with frequency) and can be obtained by linear combination \([D]=a[M]+b[K];[x]\) is the displacement vector, reflecting global motion and local deformation; \([\ddot{x}]\) and \([\ddot{x}]\) are the velocity vector and the acceleration vector, respectively; and \([f]\) is the load vector, including external loads and interactions between bodies. \[ \begin{array}{c}\frac{\partial}{\partial x}\left(\frac{1}{ 12\eta}h^{3}\theta\frac{\partial p}{\partial x}\right)+\frac{\partial}{ \partial x}\left(\frac{1}{12\eta}h^{3}\theta\frac{\partial p}{\partial x} \right)=\\ \theta\frac{u_{1}+u_{2}}{2}\frac{\partial h}{\partial x}+h\frac{u_{1}+u_{2}}{ 2}\frac{\partial\theta}{\partial x}+\frac{\partial(h\theta)}{\partial t}\end{array}  \tag{2}\]  
Based on the extended Reynolds equation, the elastohydrodynamic lubrication theory is used to establish a lubrication model for the main bearings of the diesel engine, taking into account the influence of the elastic deformation of the journal, the bearing, and the oil-filling state. The extended Reynolds equation is shown in Equation (2). \(p\) is the oil film pressure; \(\theta\) is the oil filling rate; \(h\) is the oil film thickness; \(\eta\) is the oil viscosity; \(u_{1}\) and \(u_{2}\) represents the speed of the journal and the bearing bush in the circumferential direction, respectively; \(t\) is the time; \(x\) and \(z\) represent the coordinates along the circumferential direction and the coordinates along the axial direction, respectively (fixed bearing bush). From the calculation results of the elastohydrodynamic lubrication of the main bearings, the oil film state, the coupling effect between the elastic boundary and the oil film, and the oil inflow and outflow state can be found out in detail. When the minimum oil film thickness of the main bearing is less than the composite roughness between the main journal and the main bearing, frictional contact occurs. This study uses the Greenwood-Tripp [57; 58] model to describe the asperity contact. It is assumed that the random change of the surface profile height to the reference plane obeys the Gaussian distribution, and the nominal pressure on the contact surface can be expressed as follows:  
\[p_{a}=\frac{16\sqrt{2}\pi}{15}(\sigma_{s}\beta\eta_{s})^{2}E^{*}\sqrt{\frac{ \sigma_{s}}{\beta}}F_{\text{y}/2}\left(\frac{h}{\sigma_{s}}\right) \tag{3}\]  
In Equation (3), when \(\
icefrac{{h}}{{\sigma_{s}}}<4\), use the following formula to calculate \(F_{\text{y}/2}\):  
\[F_{\text{y}/2}=4.4086\times 10^{-5}\left(4-\frac{h}{\sigma_{s}}\right)^{6.804} \tag{4}\]  
When \(\
icefrac{{h}}{{\sigma_{s}}}\geq 4\), use the following formula to calculate \(F_{\text{y}/2}\):  
\[F_{\text{y}\over 2}=0 \tag{5}\]  
In Equation (3), use the following formula to calculate \(E^{*}\):  
\[E^{*}=\frac{1}{\left(\frac{1-\
u_{1}{}^{2}}{E_{1}}+\frac{1-\
u_{2}{}^{2}}{E_{ 2}}\right)} \tag{6}\]  
In Equation (3), \(\sigma_{s}\) is the variance of the compound gap between rough contact surfaces; \(\beta\) is the radius of contact point; \(\eta_{s}\) is the peak density of contact surface; \(h\) is the nominal gap between contact surfaces; \(E_{1}\) and \(E_{2}\) are the elastic modulus of upper and lower contact surface, respectively; and \(\
u_{1}\) and \(\
u_{2}\) is the Poisson's ratio of upper and lower contact surface, respectively. Equations (4) and (5) are used to explain how the coefficient \(F_{\text{y}/2}\) plays a role in Equation (3).