## 3 Results and Discussion
### Static Linear Analysis
#### 3.2.1 Critical Position 1  
In the first place, an analysis was carried out by applying a pressure of a value of 0.65 MPa in both pistons in order to observe the distribution of von Mises stresses and the safety factor and so determine how the machine behaves for a high-pressure value and determine, in a first estimate, the most critical area, thus confirming or negating the hypothesis. Figures 13 and 14 show the von Mises stress distribution and the safety factor distribution, respectively. After this first analysis (pressure of 0.65 MPa) it can be observed that the maximum values of the von Mises stress are located on the surface of the rod cap of both pistons; therefore, compliance with the aforementioned hypothesis is confirmed, also noting that the safety factor is greater than unity. However, paying attention to this safety factor and reaching the conclusion that the pressure value applied would be optimal would be a mistake, since these results were obtained for a first mesh, without carrying out a convergence analysis, that is, without performing a mesh refinement in the most critical areas. Similarly, it was verified that if a refinement of the mesh were carried out, the value of the von Mises stress would increase, and a safety factor of less than unity would be obtained. Thus, since the working hypothesis was confirmed, independent pressure values can be applied to the plunger of each one of the pistons. In this way, after carrying out several simulations, the maximum admissible pressure values were obtained: 0.3 MPa in the high-pressure piston plunger and 0.15 MPa in the low-pressure piston plunger. Additionally, the pressure applied to the piston plunger is the difference between the maximum admissible pressure difference and that determined in the low-pressure cylinder, that is, the maximum admissible in the low-pressure cylinder intake, this being manometric. \begin{table}
\begin{tabular}{c c} \hline \hline
**Mode of Vibration** & **Natural Frequency (Hz)** \\ \hline
1 & 256.206 \\
2 & 691.678 \\
3 & 1386.978 \\
4 & 1614.984 \\
5 & 1725.334 \\
6 & 1793.178 \\
7 & 2064.468 \\
8 & 2105.807 \\
9 & 2172.460 \\
10 & 2256.000 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Values of the natural frequency for each mode of vibration. Finally, Figure 15 shows the convergence analysis of the mesh, and Table 4 shows the values of the mesh size of the refined elements, the iteration number, the von Mises stress and the relative error obtained. As can be seen, the relative error obtained is 0.64% in the fourth iteration, that is, a negligible error and less than 10%; therefore, it can be concluded that the mesh converges in this iteration. Figure 16 shows the distribution of the von Mises stress in the assembly for the maximum admissible pressure values indicated previously, where it can be seen that the maximum von Mises stress is located in the high-pressure piston rod cap, with a value of 96.57 MPa. Figure 14: Safety factor distribution: isometric view (**left**) and location of the lowest value (**right**). Figure 13: Von Mises stress distribution: isometric view (**left**) and location of the highest value (**right**). Similarly, Figure 17 shows the distribution of the displacements, where the displacements are displayed with a magnification factor of 10 in order to observe the deformed position of the machine more clearly. As can be seen, the maximum displacement has a value of 4.106 \(\times\) 10\({}^{-3}\) mm in the high-pressure piston plunger, this being a negligible value. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**High-Pressure Piston Rod** & **High-Pressure Piston Rod** & **Iteration** & **Von Mises Stress** & **Relative Error (\%)** \\
**Cap Element Size (mm)** & **Cap Element Size (mm)** & **Iteration** & **(MPa)** & **Relative Error (\%)** \\ \hline
0.75 & 2.75 & 0 & 47.85 & - \\
0.45 & 1.25 & 1 & 75.12 & 36.30 \\
0.425 & 1.125 & 2 & 95.95 & 21.71 \\
0.38 & 0.9 & 3 & 96.57 & 0.64 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Mesh convergence analysis data.