## 2 Material and Methods
### Analysis from a Mechanical Engineering Standpoint
#### 2.2.8 Linear Static Analysis  
Once it has been verified that none of the natural frequencies of the vibration modes correspond to 0 Hz, a linear static analysis is performed. To achieve results that closely approximate reality, it is essential to ensure mesh convergence until the maximum von Mises stress value shows no significant variations. Additionally, accurately determining the stress envelope is crucial for conducting a precise analysis and avoiding incorrect results. The steam conditions at the cylinder inlet depend on the machine's geometry, the dimensions of the piston and chamber, and its intended application. The desired operating pressure should be as high as possible, theoretically achievable by increasing the number of molecules, raising the steam temperature, and reducing the space, i.e., decreasing the cylinder diameter. This study is based on several fundamental assumptions and mathematical formulations similar to those in other studies [38]. Despite the absence of specific operational data for the machine, an initial simulation was conducted assuming a working gauge pressure of 2 MPa, which was arbitrarily chosen to begin obtaining results. This pressure was applied to both cylinders but on the opposite faces of the pistons. Additionally, the selected gauge pressure considers the atmospheric pressure in the opposite chamber. Thus, as the steam expands, the exhaust valve opens, releasing steam at atmospheric pressure, while steam is admitted at working pressure in the other chamber. After applying a working pressure of 2 MPa, the machine's performance was evaluated. It was confirmed that the minimum safety factor was not less than one, indicating that  
Figure 17: Critical position 2: axonometric view (**left**) and front view (**right**). the yield strength was not exceeded, which would otherwise cause the steam engine to fail under static load. Consequently, iterative analyses are necessary to identify a pressure value that results in a maximum von Mises stress below the yield strength and a safety factor greater than one to ensure safe operation. The goal was to define the range of working gauge pressures that provide safety factors for both critical positions as close as possible to the values of 2 or 4, which is the optimal range in modern machine design. After simulations, it was found that the most restrictive position for obtaining the maximum operating pressure (3.550 MPa) is the lower dead center of the left cylinder, yielding a safety factor close to 2. In contrast, to achieve the minimum operating pressure such that the safety factor does not exceed 4, the most restrictive position was determined to be the upper dead center of the left cylinder, with a pressure of 1.885 MPa. However, this study focuses on the results corresponding to the maximum allowable pressure (3.550 MPa), as this value yields the highest possible engine efficiency. The results for the minimum allowable pressure (1.885 MPa) were also checked to determine the acceptable pressure range since it may not always be possible to operate at maximum or minimum admission pressure, depending on the required needs. Figure 18 shows the application of pressure loads when the engine is in both critical positions. After applying the working pressure loads, a linear static analysis was conducted. This required a convergence analysis along with the steam engine's response, characterized by the von Mises stress, displacement, and safety factor values. Additionally, a manual mesh convergence analysis was carried out to ensure that the results were reliable, given that the numerous components in this engine led to a complex mesh. This manual analysis involves an initial mesh with a predetermined global size followed by simulation. Finally, the location where external deformations generate the  
Figure 18: Front view of pressure loads at the critical positions: critical position 1 (**left**) and critical position 2 (**right**). maximum von Mises stress is identified, recording its value, the iteration number, and the element size.