## 2 Modeling
### Dynamic Model  
#### 2.5.1 The Force Exerted on the Piston  
It is desirable to analyze the motion of the piston to develop a dynamic model of the piston-cylinder interface. First, the piston revolves around the cylinder axis. At the same time, the piston makes a reciprocating motion in the cylinder along the axis. The velocity of reciprocating is related to the rotation velocity of the cylinder block. As shown in Figure 8, the \(z\)-axis coordinates (this coordinate system is different from that of the film) of one point on the piston can be calculated with:  
\[z=-R_{p}\cos\varphi\tan\beta+const \tag{18}\]  
The axial velocity and acceleration of the piston can be given as:  
\[\left\{ \begin{array}{l}\dot{z}=\omega R_{p}\sin\varphi\tan\beta\\ \ddot{z}=\omega^{2}R_{p}\cos\varphi\tan\beta\end{array} \right. \tag{19}\]  
where \(\omega\) is the angular velocity of the piston revolution; \(R_{p}\) denotes the distribution radius of the piston; \(\beta\) is the inclination angle of the swash plate; and \(const\) denotes a constant. It is also desirable to analyze the forces exerted on the piston. The piston is pressurized by the hydraulic fluid in the displacement chamber. The force can be calculated with:  
\[F_{d}=\pi R^{2}(p_{s}-p_{0}) \tag{20}\]  
At the same time, the piston reciprocates along the \(z\)-axis. The inertia force is derived as:  
\[F_{a}=\big{(}m_{p}+m_{s}\big{)}\tilde{z}=\big{(}m_{p}+m_{s}\big{)}\omega^{2}R_ {p}\cos\varphi\tan\beta \tag{21}\]  
where \(p_{s}\) denotes the pressure of the displacement chamber; \(p_{0}\) denotes the pressure of hydraulic fluid in the piston pump shell; \(m_{p}\) denotes the mass of the piston; and \(m_{s}\) denotes the mass of the slipper. As shown in Figure 8, a viscous friction force \(F_{pu}\) applied on the piston due to hydraulic fluid flow. Thus, the forces in the direction of the piston axis can be summarized as:  
\[F_{n}\cos\beta+F_{a}+F_{d}+F_{pa}=0 \tag{22}\]  
In Figure 8, the supporting forces of the film exerted on the piston are not shown. The supporting force of the film in the \(x\)-axis direction is defined as \(F_{px}\); the force in the \(y\)-axis direction is defined as \(F_{py}\). Relative to the coordinate system of the piston cross-section 1, the torque of the film exerted on the piston can be defined as \(M_{px}\) and \(M_{py}\). \(F_{st}\) denotes the  
Figure 8: The motion of the piston and forces exerted on the piston. friction force exerted on the slipper. \(F_{\omega}\) denotes the centrifugal force exerted on the piston, which can be calculated with:  
\[F_{\omega}=\big{(}m_{p}+m_{s}\big{)}\omega^{2}R_{p} \tag{23}\]  
As shown in Figure 8, in the \(x\)-\(y\) plane, two force balance equations can be written as:  
\[\left\{ \begin{array}{c}-F_{st}\cos\varphi+F_{\omega}\sin\varphi+F_{px}=0\\ F_{st}\sin\varphi+F_{\omega}\cos\varphi+F_{n}\sin\beta+F_{py}=0\end{array} \right. \tag{24}\]  
In addition, two torque balance equations can be written as:  
\[\left\{ \begin{array}{c}-F_{st}\sin\varphi L_{z}-F_{\omega}\cos\varphi(L_{z} -L_{c})-F_{n}\sin\beta L_{z}+M_{px}=0\\ -F_{st}\cos\varphi L_{z}+F_{\omega}\sin\varphi(L_{z}-L_{c})+M_{py}=0\end{array} \right. \tag{25}\]  
where \(L_{z}\) denotes the distance from the center of the piston ball head to the piston section 1 and \(L_{c}\) denotes the distance from the center of the piston ball head to the piston center of mass. The forces of the film can be obtained by integrating the pressure field. According to the Reynolds equation, the film pressure field is determined by the film height, squeeze rate, and piston position. The film height and squeeze rate can be determined by the eccentricity and the shifting velocity of the piston radial movement. Therefore, the forces of the film exerted on the piston can be assumed a function of the eccentricity \(e\), the shifting velocity \(\dot{e}\), and the piston position \(\varphi\). Except for the forces of the film, other forces are only related to the piston position \(\varphi\). Thus, Equations (24) and (25) can be written as:  
\[\left\{