# Dynamic Analysis for a Reciprocating Compressor System with Clearance Fault
## 2 Model of the reciprocating Compressor
### Load Model  
During the working process of the reciprocating compressor, the loads acting on the mechanism mainly include gas force acting on the piston, the inertia force, and the frictional force, which make up the multiple piston loads. The multiple piston forces equal the vector sum of them and can be calculated with Equation (5):  
\[F=F_{g}+F_{i}+F_{r} \tag{5}\]  
where \(F_{g}\) denotes the gas force, \(F_{i}\) is the inertia force, and \(F_{r}\) represents the frictional force. The gas force comes from the cylinder cavity, for which the direction is perpendicular to the face of the piston, and the value is proportional to the cylinder pressure. Note that the working process of a compressor is a complex thermodynamic process, which consists of a suction process, compression process, discharge process, and expansion process. Therefore, the cylinder pressure shows four different variations in one cycle. In order to simplify the computation, this paper assumes that there is no power loss in the suction and discharge stages, the compression and expansion stages are isentropic, there is no leakage, and the medium gas is an ideal gas. Figure 3 displays the ideal diagrammatic drawing of the cylinder pressure. After simplification, the relationship between the gas force and the equivalent displacement of the translational mechanism can be obtained with Equation (6):  
\[F_{g}=p\cdot\pi\cdot r^{2}=\left\{ \begin{array}{ll}p_{s}\cdot\pi\cdot r^{2} &\text{\emph{Suction phase}}\\ p_{s}\cdot\left(\frac{s+s_{0}}{x}\right)^{m}\cdot\pi\cdot r^{2}&\text{\emph {Compression phase}}\\ p_{d}\cdot\pi\cdot r^{2}&\text{\emph{Exhaust phase}}\\ p_{d}\cdot\left(\frac{80}{x}\right)^{m}\cdot\pi\cdot r^{2}&\text{\emph{Expantion phase}} \end{array} \right. \tag{6}\]  
where \(p\) represents the cylinder pressure, \(p_{s}\) and \(p_{d}\) are the cylinder pressure in the suction stage and the discharge stage, respectively, \(r\) denotes the radius of the piston, and \(s\) represents the piston stroke. The inertial force of the mechanism consists of the reciprocating inertia force and the rotational inertia force. The reciprocating inertia force is produced by the crosshead and piston assembly that move in a reciprocating motion, for which the value is proportional to the mass and acceleration of the moving parts, and the direction is the opposite of the acceleration of the piston. Figure 3: PV indicator diagram of a reciprocating compressor. The rotational inertia force is mainly caused by the dynamic imbalance of the crankshaft, which can be offset by adding a balance block. Thus, the inertia can be calculated with Equation (7):  
\[\left\{ \begin{array}{l}F_{i}=m\cdot\tilde{x}=m\cdot\omega^{2}\cdot(\cos\theta+ \lambda\cdot\cos 2\theta)\\ m=m_{p}+0.5m_{cr}\end{array} \right. \tag{7}\]  
where \(m_{p}\) represents the mass of the crosshead and piston and \(m_{cr}\) denotes the mass of the linkage. The frictional force is mainly roused by the relative motion between the piston and the cylinder, crosshead, and sliding way. Compared with the air force, the friction force is so small that it can be ignored. According to the above analysis, the piston force mainly depends on the gas force, which is difficult to simulate. To achieve simulation of the piston force, this paper designed a load model composed of a plate cam, a translating roller follower, and a cylindroid helical coil compression spring as shown in Figure 1. The cam angle corresponded to the crankshaft angle; in other words, the cam angle in the rise phase, the high-dwell phase, the fail phase, and the low-dwell phase corresponded to the crankshaft angle in the suction process, compression process, discharge process, and expansion process, respectively. The spring was installed between the follower and piston. The minimum load of the spring equaled the piston force during the suction process, and the maximum load of the spring equaled the piston force during the discharge process.