# Study on Temperature Characteristics of Lubrication Film of Valve Plate Pair in Axial Piston Pumps
## 2 Modeling  
### Modeling of Lubricating Film Shape  
Figure 1 shows the general configuration of a CY swash plate-type axial piston pump, comprising a main shaft, swash plate, cylinder block, pistons, and valve plate, among other components. In the pump, the cylinder block rotates relative to the valve plate fixed on the front housing. The Cartesian coordinate system is established in Figure 2. The origin \(O_{\text{C}}\) of the coordinate system is located at the intersection point of the spindle axis and the top surface of the valve plate. The \(Y_{\text{C}}\)-axis points towards the outlet port, and the \(Z_{\text{C}}\)-axis coincides with the center line of the spindle and points towards the right side of the spindle in Figure 1. The wedge angle \(\alpha\) is the angle between the valve plate and the cylinder block, while the azimuth angle \(\gamma\) is the angle between the \(Y_{\text{C}}\)-axis and the line between the minimum-film-thickness point and the origin \(O_{\text{C}}\). The positive direction is specified as clockwise in Figure 2.  
The cylinder block tilts relative to the valve plate under the influences of several factors, mainly including the forces acting on the cylinder block and torques due to the forces, eccentric wear of the valve plate, and the coaxiality error between the main shaft and the cylinder block [27]. Figure 1: Structure of the axial piston pump. Figure 2: The wedge lubrication film between the cylinder block and valve plate. (1) Modeling of the force balance of the cylinder block  
The force \(F_{\rm Pis}\) is  
\[F_{\rm Pis}=\frac{Z}{2}A_{\rm Pis}P_{\rm P} \tag{1}\]  
where \(Z\) is the number of pistons, \(A_{\rm pis}\) is the bottom area of the piston, and \(P_{\rm P}\) is the working pressure. The force \(F_{\rm fm}\) is  
\[F_{\rm fm}=\frac{1}{4}\Bigg{[}\frac{r_{4}^{2}-r_{3}^{2}}{\ln(r_{4}/r_{3})}- \frac{r_{2}^{2}-r_{1}^{2}}{\ln(r_{2}/r_{1})}\Bigg{]}\left(\pi-\frac{2\pi}{Z}+ \eta_{0}\right)P_{\rm P} \tag{2}\]  
where \(r_{1}\) is the inner diameter of the inner sealing belt of the valve plate, \(r_{2}\) is the outer diameter of the inner sealing belt of the valve plate, \(r_{3}\) is the inner diameter of the outer sealing belt of the valve plate, \(r_{4}\) is the outer diameter of the outer sealing belt of the valve plate, and \(\eta_{0}\) is the central angle occupied by the discharge waist groove. The cylinder block tilts slightly relative to the valve plate, and both of them make contact in the surrounding area of the minimum-film-thickness point. Mixed friction occurs in the area, and the micro-convex bodies in that area contact each other to generate the supporting force. The bearing capacity \(W\) and the supporting force \(F_{\rm mix}\) in the mixed-friction area between the cylinder block and valve plate are calculated as follows:  
\[\left\{ \begin{array}{l}W=\frac{4}{3}n_{\rm f}E_{\rm e}R_{\rm e}^{0.5}J_{h}^ {\rm\infty}\left(Z_{\rm f}-h\right)^{1.5}e^{-Z_{\rm f}/\sigma_{\rm e}}dz_{\rm f }\\ E_{\rm e}=1/\left[\left(1-\gamma_{1}^{2}\right)/E_{1}+\left(1-\gamma_{2}^{2} \right)/E_{2}\right]\\ \end{array} \right. \tag{3}\]  
\[F_{\rm mix}=\int_{\theta_{\rm i}}^{\theta_{\rm i}}\int_{r_{5}}^{\tau_{5}}Wdrd\theta \tag{4}\]  
where \(n_{\rm f}\) is the contact number of micro-convex bodies per unit area, \(z_{\rm f}\) is the maximum height of the micro-convex body, \(\sigma_{\rm e}\) is the surface roughness measured by the surface profiler, and \(R_{\rm e}\) is the equivalent radius of the micro-convex body at the contact interface. \(\theta_{\rm i}\) and \(\theta_{\rm j}\) are the angles of both ends of the mixed-friction area, \(E1\) and \(E2\) are the elastic modulus of the valve plate and cylinder block, respectively, and \(E_{\rm e}\) is the equivalent elastic modulus. \(\gamma_{1}\) and \(\gamma_{2}\) are the Poisson ratio of the valve plate and the cylinder block, respectively, and \(h\) is the oil film thickness.