## 2 Material and Methods
### Analysis from the Mechanical Engineering Standpoint
#### 2.2.8 Linear Static Analysis  
Once it has been confirmed that none of the natural frequencies of the vibration modes are equal to 0 Hz, a linear static analysis can be executed. For a better understanding of this stage, a brief introduction is set out below. This analysis is based on several fundamental assumptions: linear elasticity, small deformations, and static equilibrium conditions. [PERSON]'s law states:  
\[\sigma=E\cdot\epsilon, \tag{3}\]  
where \(\sigma\) is the stress, which is linearly proportional to the strain \(\epsilon\), with \(E\) being the material's modulus of elasticity. The assumption of small deformations implies that changes in the system's geometry are negligible, thereby preserving the original configuration. Additionally, static equilibrium requires that the sum of all forces and moments acting on the system equals zero. The equations governing linear static analysis involve equilibrium equations, stress-strain relationships, and strain compatibility. For a system in static equilibrium, internal forces must balance external forces. The stress-strain relationship is derived from the linear elastic behavior of materials, while compatibility conditions ensure that deformations conform to boundary conditions and interactions between different structural components [33]. The mathematical formulation for this scenario involves differential equilibrium equations for an axial load \(P\) and an internal pressure \(p\) acting on a differential element and other regions where the load is applied. Axial equilibrium is represented by : \[\frac{dN}{dx}+q=0,\] (4)  
Figure 8: Critical positions: lower dead center (**left**) and upper dead center (**right**). transverse equilibrium is represented by : \(\dfrac{dV}{dx}+pA=0\), (5) and the equilibrium of moments is represented by : \(\dfrac{dM}{dx}-V=0\), (6) where \(N\) is the normal force, \(V\) is the shear force, \(M\) is the bending moment, and \(q\) is the distributed load. Likewise, the strain-displacement relationships for axial deformation are given by:  
\[\epsilon_{x}=\dfrac{du}{dx} \tag{7}\]  
where \(u\) is the axial displacement. For bending deformation, the strain is:  
\[\epsilon_{xx}=\dfrac{d^{2}v}{dx^{2}} \tag{8}\]  
where \(v\) is the transverse displacement. On the other hand, stress-strain relationships adhere to [PERSON]'s law for normal and shear stresses, which are defined by the following equations:  
\[\sigma_{x}=E\cdot\epsilon_{x}, \tag{9}\]  
\[\tau_{xy}=G\cdot\gamma_{xy} \tag{10}\]  
where \(G\) is the shear modulus and \(\gamma_{xy}\) is the shear strain [34]. Finally, by solving these differential equations, taking into account the boundary conditions and applied loads, the displacements and stresses at each point are obtained. To achieve the most accurate results, mesh convergence must be performed until the maximum von Mises stress value stabilizes and shows minimal variation. Additionally, accurately determining the stress envelope is crucial to ensure a correct analysis and to avoid erroneous results. Steam conditions at the cylinder inlet vary depending on the geometry of the machine, as well as the dimensions of the piston and chamber, and its intended application. The working pressure to be obtained is the highest possible; this can be achieved by increasing the number of molecules, by increasing the steam temperature, and, finally, by reducing the space, that is, by reducing the diameter of the cylinder. Despite the absolute absence of precise machine operating data, an initial simulation was carried out assuming a gauge working pressure of 0.62 MPa (data received from real examples of steam engines in locomotives) [35], which is equivalent to a force of 800 kg for a piston of 0.040 m diameter. This pressure was applied to various components, including the slide valve block and piston face, depending on their positions. On the other hand, the gauge pressure chosen took into account the atmospheric pressure in the opposite chamber.