# Simulation and Experimental Analysis of Pressure Pulsation Characteristics of Pump Source Fluid
## 2 Theoretical Analysis of Fluid Pressure Pulsation of the Plunger Pump  
### Analysis of Plunger Kinematics  
The relative position of the plunger swash plate is shown in Figure 1. The \(\text{OX}_{1}Y_{1}Z_{1}\) coordinate system is a fixed coordinate system located at the center of the plunger pump. At the intersection of the axis of the plunger pump shaft, the \(\text{X}_{1}\) axis is perpendicular to the axis of the shaft and points to the top dead center of the plunger movement, and the \(Z_{1}\) axis coincides with the axis of the plunger pump shaft. When the inclination angle of the swash plate rotates along the Y\({}_{1}\) axis by angle \(\alpha\), the OX\({}_{1}\)Y\({}_{1}\)Z\({}_{1}\) coordinate system rotates along the Y\({}_{1}\) axis by angle \(\alpha\) to obtain the OX\({}_{2}\)Y\({}_{2}\)Z\({}_{2}\) coordinate system. Since the plunger rotates with the swash plate, the trajectory of the center of the ball head will change from a circle to an ellipse, that is, it will stretch in the X\({}_{2}\) axis direction and remain unchanged in the Y\({}_{2}\) axis direction. Therefore, the OX\({}_{3}\)Y\({}_{3}\)Z\({}_{3}\) coordinate system is obtained by stretching the OX\({}_{2}\)Y\({}_{2}\)Z\({}_{2}\) coordinate system along the X\({}_{2}\) axis. The OX\({}_{4}\)Y\({}_{4}\)Z\({}_{4}\) coordinate system rotates with the cylinder, which is obtained by rotating OX\({}_{3}\)Y\({}_{3}\)Z\({}_{3}\) by an angle \(\phi\) equal to the cylinder rotation angle. When the plunger pump is at the original position, that is, the swash plate inclination angle is zero and the pump is not rotating, the OX\({}_{4}\)Y\({}_{4}\)Z\({}_{4}\) coordinate system completely coincides with the OX\({}_{1}\)Y\({}_{1}\)Z\({}_{1}\) coordinate system.  
From the basic theoretical knowledge of coordinate transformation, the transformation relationship can be obtained as follows:  
\[\left(\begin{array}{c}x_{1}\\ y_{1}\\ z_{1}\end{array}\right)\ =\ \left(\begin{array}{ccc}\cos(-\alpha)&0&-\sin(- \alpha)\\ 0&1&0\\ \sin(-\alpha)&0&\cos(-\alpha)\end{array}\right)\left(\begin{array}{c}x_{2}\\ y_{2}\\ z_{2}\end{array}\right) \tag{1}\]  
\[\left(\begin{array}{c}x_{2}\\ y_{2}\\ z_{2}\end{array}\right)\ =\ \left(\begin{array}{ccc}1/\cos\alpha&0&0\\ 0&1&0\\ 0&0&1\end{array}\right)\left(\begin{array}{c}x_{3}\\ y_{3}\\ z_{3}\end{array}\right) \tag{2}\]  
\[\left(\begin{array}{c}x_{3}\\ y_{3}\\ z_{3}\end{array}\right)\ =\ \left(\begin{array}{ccc}\cos\phi&-\sin\phi&0\\ \sin\phi&\cos\phi&0\\ 0&0&1\end{array}\right)\left(\begin{array}{c}x_{4}\\ y_{4}\\ z_{4}\end{array}\right) \tag{3}\]  
By organizing Equations (1)-(3), the coordinate transformation matrix from coordinate system OX\({}_{4}\)Y\({}_{4}\)Z\({}_{4}\) to coordinate system OX\({}_{1}\)Y\({}_{1}\)Z\({}_{1}\) can be obtained as:  
\[\left(\begin{array}{c}x_{1}\\ y_{1}\\ z_{1}\end{array}\right)\ =\ \left(\begin{array}{ccc}\cos\phi&-\sin\phi&\sin \alpha\\ \sin\phi&\cos\phi&0\\ -\cos\phi\tan\alpha&\sin\phi\tan\alpha&\cos\alpha\end{array}\right)\left( \begin{array}{c}x_{4}\\ y_{4}\\ z_{4}\end{array}\right) \tag{4}\]  
When analyzing the movement of the plunger in the OX\({}_{4}\)Y\({}_{4}\)Z\({}_{4}\) coordinate system, the coordinates of the center of the plunger ball head are:  
\[\left(\begin{array}{c}x_{4}\\ y_{4}\\ z_{4}\end{array}\right)\ =\ \left(\begin{array}{c}R\\ 0\\ 0\end{array}\right) \tag{5}\]  
Figure 1: Relation coordinate diagram of the plunger swash plate.  
From Equations (4) and (5), the displacement of the center of the plunger ball head along the \(Z_{1}\) axis at any rotation angle can be obtained as:  
\[Z_{1}\ =\ -R\tan\alpha\cos\phi \tag{6}\]  
A derivative of the above equations, the speed of the center of the plunger ball head in the direction of the \(Z_{1}\) axis is:  
\[Z_{1}\ =\ R\big{(}\ -\sec\alpha\cos\phi\ \ \ \tan\alpha\sin\phi\ \big{)}\left( \begin{array}{c}\dot{\alpha}\\ \dot{\phi}\end{array}\right) \tag{7}\]