## 2 Hydraulic Simulation and Theoretical Analysis
### Theoretical Analysis of the Low-Frequency Characteristic
#### 2.2.2 Analysis of Pressure in Different Stages with Internal Leakage  
Considering the conditions of dual-cylinder parallel and large stiffness of the mechanism, the speed and displacement of the dual-cylinder parallel system are still the same in the case of the internal leakage. Therefore, for intuitive analysis, the cylinder with the internal leakage is used to analyze the fault feature. The leakage rate equation is as follows:  
\[q_{leak}=l_{r}\Delta P \tag{16}\]  
where \(l_{r}\) is the leakage coefficient, \(\Delta P\) is the leakage pressure difference.  
In Phase 1, \(P_{1}\) and \(P_{2}\) gradually increase to \(P_{1}^{max}\) and \(P_{2}^{max}\), and \(P_{1}^{max}>P_{2}^{max}\). Then, in Phase 2, The leakage pressure difference is \(P_{1}^{max}-P_{2}^{max}\), which can cause a leakage \(q_{leak}\). According to the Equation (11), \(q_{leak}\) can lead to a leakage speed \(v_{leak}\) in the retraction direction of the piston rod to maintain the stable pressure in both rod chamber and rodless chamber. Then, we have:  
\[\dot{P_{1}}=\frac{K_{B}\Big{[}q_{1}-q_{leak}-(v_{s}-v_{leak})A_{1}\frac{\rho(P _{1})}{\rho(0)}\Big{]}}{A_{1}x+V_{1}}=0 \tag{17}\]  
\[\dot{P_{2}}=\frac{K_{B}\Big{[}-q_{2}+q_{leak}+(v_{s}-v_{leak})A_{2}\frac{\rho( P_{2})}{\rho(0)}\Big{]}}{V_{2}-A_{2}x}=0 \tag{18}\]  
where \(q_{leak}\) is the leakage, \(v_{s}\) is the lifting speed of piston rod.  
In Phase 3, the rapid closure of the valves can lead to a sudden drop in pressure. At the end of Phase 3, \(P_{1}^{max}\) and \(P_{2}^{max}\) gradually decrease to \(P_{1}^{s}\) and \(P_{2}^{s}\), and \(P_{1}^{s}\geq P_{2}^{s}\) holds. Meanwhile, according to Equations (17) and (18), the net flow into the rodless chamber and rod chamber of the internal leakage cylinder are, respectively, as follows:  
\[-q_{leak}+A_{1}v_{leak}\frac{\rho\big{(}P_{1}^{s}\big{)}}{\rho(0)} \tag{19}\]  
\[q_{leak}-A_{2}v_{leak}\frac{\rho(P_{2}^{s})}{\rho(0)} \tag{20}\]  
The velocity \(v_{leak}\) of the piston rod is in the direction of retraction. Then, in Phase 4, the initial pressure difference \(P_{1}^{s}-P_{2}^{s}\) causes the internal leakage, and the pressure difference between the two chambers gradually decreases [31]. As a result, according to the Equations (16) and (19), \(q_{leak}\) continues to decrease, which can lead to the increase in the net inflow into the rodless chamber and the rise of \(P_{1}^{s}\). A similar situation also occurs in the rod chamber of the internal leakage cylinder, which can lead to the rise of \(P_{2}^{s}\). This process continues until \(P_{1}^{s}=P_{2}^{s}\), \(v_{leak}=0\) and \(q_{leak}=0\), that is, the internal leakage disappears, and the piston rod of the cylinder is stable.  
In addition, according to Equations (19) and (20), the increase in the internal leakage can intensify the growth trend of the pressure in Phase 4. Therefore, the severity of the internal leakage can be evaluated by the pressure growth trend of the two chambers in Phase 4. Moreover, according to simulation verification, the pressure growth trend in the rodless chamber is more significant than that in the rod chamber. Therefore, the pressure signal of the rodless chamber was used in the subsequent analysis. Furthermore, the slope and pressure increase rate of the pressure signal were selected as the features of the internal leakage.