## 2 Hydraulic Simulation and Theoretical Analysis
### Simulation Analysis  
The simulation analysis was implemented in the Simcenter Amesim 2021.1 environment, and the batch processing function in it was used to improve simulation efficiency. Specifically, to meet the structural characteristics and load conditions of the dual-cylinder parallel balance oil circuit, the hydraulic simulation model mainly refers to the relevant manuals of the container front crane [32]. The hydraulic simulation model consists of hydraulic structure and mechanical structure. In the hydraulic structure, the HJ020 model and the HL012 model is used for the hydraulic cylinder and the pipeline. In the HL012 model, the compressibility of the fluid and the expansion of the pipe or hose wall under pressure are considered by using the effective bulk modulus. The friction factor based on Reynolds number and relative roughness is considered for pipeline friction. The pipeline material is galvanized iron. The hydraulic structure is consistent with that in Figure 1, and the corresponding parameters are shown in Table 1. The mechanical structure simulates the cantilever beam support structure as shown in Figure 2, which is common in mechanical equipment [1; 2; 3; 4]. All simulations start when the lifting displacement of the hydraulic cylinder is zero. The lifting stage lasts for 5 s, consisting of the acceleration phase (Phase 1) and the stable phase (Phase 2). In Phase 1, high-pressure oil enters the rodless chambers A and B, forcing the piston rod to accelerate and rise. In Phase 2, the acceleration of the piston rod has ended, and the lifting speed has stabilized. The pressure maintenance stage lasts for 10 s, consisting of the pressure fluctuation phase (Phase 3) and the pressure maintenance phase (Phase 4). In Phase 3, the balance valves enter the closing process, which cause the sharp drop in pressure in the rod chamber of the hydraulic cylinder A and B [33]. In Phase 4, the balance valves are completely closed, and the pressure in the rod chamber of the hydraulic cylinder A and B is only related to the leak. Moreover, the simulation process from retraction stage to the pressure maintenance stage is also verified, and the results are similar. To simulate internal leakage, one orifice is connected between the rod chamber and the rodless chamber of hydraulic cylinder B, and four orifices with different hole diameters (0 mm, 0.4 mm, 0.6 mm, 1 mm) are used to separately simulate healthy (no internal leakage), minor leakage, medium leakage and serious leakage. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Value** & **Unit** \\ \hline Piston diameter & 200 & mm \\ Rod diameter & 50 & mm \\ Length of stroke & 400 & mm \\ Counterbalance valve setting pressure & 50 & bar \\ Check valve cracking pressure & 10 & bar \\ Relief valve cracking pressure & 150 & bar \\ Pipe diameter & 25 & mm \\ Pipe wall thickness & 4 & mm \\ Young’s modulus for Pipe material & \(2.06\times 10^{6}\) & bar \\ Pipe absolute roughness & 150 & \(\mu\)m \\ Load mass & 1000 & kg \\ Beam length & 3 & m \\ Motor shaft speed & 5000 & rec/min \\ Pump displacement & 30 & cc/rec \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters. Figure 2: The cantilever beam support structure. Figure 3 shows the rodless chamber pressure of hydraulic cylinder A and B with different internal leakage. In Phase 1, the directional valve opened quickly, and the pressure increased sharply from 0 s. The pressure gradually stabilized at 1 s, and the simulation entered Phase 2. In Phase 2, the pressure remained stable, and the directional valve and the balance valves began to close at 5 s, and the simulation entered Phase 3. After the sharp drop of pressure, the balance valves were completely closed, and then the simulation entered Phase 4, In Phase 4, when no leakage occurs, the pressure fluctuation was the most obvious and persistent.