# Dynamic Analysis for a Reciprocating Compressor System with Clearance Fault
## 3 Computational Dynamic Analysis
### Computational Simulation of the Design Model under a Fault State  
Due to error of assembly, wear, and other factors, excessive clearance is the most common failure of the reciprocating compressor movement mechanism. The typical representatives are the clearance failures on the big end and little end of the linkage. These clearance failures are the major reasons behind abnormal vibration of the reciprocating compressor body. Therefore, this paper establishes a clearance fault model based on the design model to investigate the dynamic behavior of the moving parts under fault conditions.  
The clearance faults in the joint between the crankshaft and linkage or between the linkage and crosshead can be classified as faults of the revolute joint with clearance. These faults can be simulated by setting different clearance sizes at the corresponding revolute joints (the revolute joint model can be referred to in [20]). For the contact between the journal and bearing, we applied the contact method based on the impact function available in the ADAMS function library [21]. The contact model treats the contact as a nonlinear spring damper, including a contact force model based on the Hertz contact law and a friction model based on the Coulomb friction model. The contact force is calculated by the penetration and the speed of penetration as in the following equation [22; 23; 24]:  
\[F_{N}=\left\{\begin{array}{ll}K\delta^{n}+STEP(\delta,\;0,\;0,\;d_{\max},\;C _{max})\dot{\delta}&\delta>0\\ 0&\delta<0\end{array}\right. \tag{14}\]  
where \(K\) is the stiffness coefficient that depends on the material property and the radius, \(\delta\) is the penetration, \(n\) is the deformation force exponent that equals 1.5 for the metallic contact, \(d_{max}\) is a positive real value of the boundary penetration, and \(C_{max}\) is the maximum damping coefficient, and the following is also true:  
\[\left\{\begin{array}{ll}K=\frac{4}{3\pi(h_{1}+h_{2})}\sqrt{\frac{R_{1}R_{2} }{R_{1}+R_{2}}}\\ \\ h_{i}=\frac{1-p_{i}^{2}}{\pi E_{i}}(i=1,\;2)\end{array}\right. \tag{15}\]  
where \(R_{i}\) is the radius, \(\mu_{i}\) is the Poisson's ratio, and \(E_{i}\) is the modulus of the elasticity.  
The Coulomb friction force can be given by Equation (16) [22; 23; 24]:  
\[F_{t}=-c_{f}F_{N}\frac{v_{t}}{|v_{t}|} \tag{16}\]  
where \(c_{f}\) is the friction coefficient, \(F_{N}\) is the normal force, and \(v_{t}\) is the relative tangential velocity.  
We established the fault model based on the design model in ADAMS, as Figure 5a shows. In the case of faults in the joint between the linkage and crankshaft, we set the revolute joint between the crankshaft and linkage to clearances of 0.1 mm, 0.2 mm, and 0.3 mm and set the other joints as the ideal joints. In the case of a clearance fault in the joint between the linkage and crosshead, we set the revolute joint between the linkage and crosshead to clearances of 0.1 mm, 0.2 mm, and 0.3 mm and set the other joints as the ideal joints. In processing the simulations of these cases, the contact model based on the IMPACT function was used to simulate the revolute joint with clearance. Table 3 depicts the characteristic parameters of the impact function.