## 2 Materials and Methods
### Analysis from the Mechanical Engineering Standpoint
#### 2.2.6 Critical Positions  
There are infinite positions that can be simulated, depending on the angle of rotation of the crankshaft. However, the analysis must be carried out in the critical operating positions, these being when the stresses to which the machine is subjected are at their maximum. For that reason, it is necessary to determine the critical positions, reducing the number of simulations and simulating only those that give rise to the highest value of von Mises stress. In this way, it is possible to determine the maximum values of pressure in the invention, ensuring the operation of the engine under safe conditions. For the present invention, two critical positions were determined:  
* Critical position 1: This corresponds to the position of the high-pressure piston plunger in its movement from top dead center to bottom dead center after the intake closes, just when the expansion of the water vapor begins. * Critical position 2: This corresponds to the position of the high-pressure piston plunger in its displacement from bottom dead center to top dead center after closing the intake, just at the beginning of the expansion of the water vapor. Figures 7 and 8 show an isometric view and a front view (both sectioned) in each of the critical positions in order to better visualize the position of each of the piston plungers. The difficulty in determining the optimum pressure values is that this machine has two cylinders, a high-pressure and a low-pressure cylinder, so that the pressure at the exhaust of the high-pressure cylinder is the pressure at the intake of the low-pressure cylinder (disregarding possible pressure losses in the intake and exhaust valves, as well as those due to friction). This fact relates the pressure in the cylinders, making it difficult to calculate the pressure applied to the plunger of each piston. Figure 8: Critical position 2: isometric view (**left**) and front view (**right**). Figure 7: Critical position 1: isometric view (**left**) and front view (**right**). In addition, there is one more unknown factor, which is the angle formed by the eccentric pulley that transmits the rotary movement of the crankshaft to the valves and which has an alternative movement and regulates the intake and exhaust (as explained in Section 2.1). Depending on this angle, the valve closes the intake earlier or later, and, therefore, the expansion of the steam may be one or the other. This is because, if the intake valve closes earlier, the steam expands to reach lower pressure values, and if it closes later, it does not reach such low values after the expansion. Therefore, studying the thermodynamic cycle that occurs inside the cylinder would be the subject of another investigation, since it depends on the aforementioned angle, and determining its optimal value in order to obtain the highest possible thermodynamic performance would be a complex process. Thus, to focus on the objective of this investigation, which was to determine the optimal pressure values so that failure does not occur, some working hypotheses were generated to determine the maximum admissible pressures applied on the faces of the plungers. The first hypothesis is that, once the pressures on the faces of the plungers are applied with a certain value in order to carry out a first iteration and perform the analysis, the critical area of the greatest von Mises stress cannot be located in an area that both cylinders influence. For example, if, after applying a certain pressure to each of the plungers, it turns out that the von Mises stress is located on the high-pressure piston rod, the value that was applied to the low-pressure cylinder does not matter, since the value of the von Mises stress on the high-pressure piston rod depends only on the pressure applied on the high-pressure cylinder and not on the low-pressure one.