**3. Results and Discussion**

Before showing the results of the static analysis of the double-acting steam engine (stress distribution, safety coefficients, deformations and displacements), it is convenient to perform a modal analysis of the engine to determine if there exist any rigid body modes.

Autodesk Inventor Professional performs this simulation by subjecting the structure to vibrations at different frequencies. If the modal frequencies obtained in the analysis are close to 0 Hz then the element to be studied behaves as a mechanism and therefore it would not make sense to perform a static analysis on it.

The eight modal frequencies obtained for the steam engine are: F1: 0.60 Hz, F2: 0.63 Hz, F3: 0.74 Hz, F4: 0.76 Hz, F5: 2.21 Hz, F6: 2.31 Hz, F7: 3.86 Hz and F8: 4.32 Hz. The simulation shows that the first four (slightly lower) frequencies cause displacements in two free counterweights that the invention has whose function is to transmit certain inertia that facilitates the opening and closing of the valves. These counterweights could be excluded in the simulation for a static analysis but this exclusion would affect the solicitation that affects the opening valves of the steam boxes. The dynamism of these elements will therefore be considered so that they do not contaminate interpretation of the results.

The static analysis of the invention has contemplated the study of the two cases indicated above: when the piston of the steam cylinder follows a downward movement and when it moves following an upward movement.

A mesh convergence study has also been performed in order to establish the credibility of the results, since the high stresses are concentrated in very narrow regions of the mechanism. As explained in Section 2.2.5, the discretization of each of the pieces directly affects the results of von Mises stress analysis. The software used allows a refinement of the mesh according to the places where the stress is greater. This process is cyclical since once the regions where the stress is greater are determined, the mesh is refined and the von Mises stress is recalculated. In addition, there are some convergence criteria and in the present study it has been defined that the maximum number of cycles is 10, specifying that when the difference between results is less than 5% the refining process of the mesh is stopped.

Figure 13 shows the convergence curve for the two cases under study. When the piston moves downward, the convergence rate is 4.373% in the fifth iteration (Figure 13a). On the other hand, when the piston moves upwards the convergence rate is lower and therefore more reliable, with a value of 0.013%, although this result is obtained in the seventh iteration (Figure 13b).

**Figure 13.** Convergence curve: (**a**) downward movement and (**b**) upward movement.

The analysis shows that von Mises stresses are generally low, not reaching 5 MPa (Figure 14), although there are a series of singular points where the stress is higher. This is the case of the opening axle of valve D, one of the valves that diverts water vapor into the piston or into the condensation pipe and occurs when the piston descends, reaching a value of 188.4 MPa (Figure 14a). Also, when the piston rises the maximum stress is located at the same point with a somewhat lower value of 129.6 MPa (Figure 14b). Although these values are high they are not too high, considering the elasticity limit of the material with which the piece is made (cast iron). Figure 15 shows in more detail the point at which the maximum load in the upward direction of the piston is recorded.

**Figure 14.** Von Mises stress distribution: (**a**) downward movement and (**b**) upward movement.

**Figure 15.** Location of the point where the von Mises stress is maximum.

If the parts that regulate the opening of the valves are excluded, the next set of parts subjected to higher stresses are the rods that join the parallelogram to the piston of the steam cylinder. In Figure 16 this greater stress is located specifically in the second rod, just at the point of insertion of the rod with the frame that serves as support. The von Mises stress value for that point is 47.45 MPa (Figure 16a) when the piston of the steam cylinder moves in a downward direction and somewhat higher with a value of 47.57 MPa (Figure 16b), when moving in an ascending direction, which on the other hand makes sense. The values of this second main stress are already relatively low for the metallic materials with which they are made.

**Figure 16.** Location of the second highest von Mises stress: (**a**) downward movement and (**b**) upward movement.

Another aspect to be studied is the safety coefficient, which is defined as the relationship between the stress to which a part is subjected and the elasticity limit of the material with which it is manufactured. This parameter shows which elements of a structure work with stresses close to the elastic limit of the material and therefore run the risk of breaking and which elements work below it within a particular safety threshold.

Currently, the parts function with a safety coefficient with values between 2 and 4. Parts that work below 2 are too close to the limit of elasticity and suffer significant fatigue, while if the value is above 4, the pieces work far from that limit and therefore are oversized.

In the time of Agustín de Betancourt, the knowledge that existed on the resistance of materials was not very broad and in addition tests were not realized to determine the limits of elasticity of the materials, the reason why mechanisms were generally quite over-dimensioned. The double-acting steam engine is no exception to this rule.

Figure 17 shows the distribution of safety coefficients, it being possible to observe that almost all the elements of the invention have a safety coefficient above 12 and that only a few elements work within a smaller range of values but in any case well over 4.

**Figure 17.** Distribution of safety coefficients.

Furthermore, the point that gives a lower value for the safety coefficient is the opening axle of valve D. The detailed study of the safety coefficient in that axle shows that valve D works with greater stress when the piston of the cylinder is descending and therefore closed and preventing the passage of steam at high pressure to the lower chamber. In this situation the valve axle has a minimum safety coefficient of 4.02 (Figure 18a), above the optimum working values. Similarly, when the valve is open allowing the passage of steam at high pressure the safety coefficient of the valve axle is greater with a value of 5.85 (Figure 18b).

**Figure 18.** Location of the lowest safety coefficient: (**a**) downward movement and (**b**) upward movement.

As indicated previously, if a study is performed excluding the valves the element with the lowest safety factor is the engine speed regulator, with a value of 8.67 when the piston of the steam cylinder falls (Figure 19a) and another of 8.62 when it ascends (Figure 19b). Thus, since the rest of the elements have higher coefficients it is completely clear that the engine is largely oversized.

**Figure 19.** Second lowest safety coefficient: (**a**) downward movement and (**b**) upward movement.

On the other hand, the study of the deformation of the elements that make up a mechanism is important, since even if an element does not work in a range of stresses close to the elastic limit of the material, due to its slenderness it can deform its geometry excessively, compromising the correct contact between these elements. Autodesk Inventor Professional shows the equivalent deformation of each element as a relationship between the deformation of the element and its length. In the present study, the maximum deformation is located in the element that suffers the highest stress, that is the opening axle of valve D. However, its deformation is 0.14% with respect to the size of the element when the piston descends (Figure 20a) and 0.10% when it rises (Figure 20b), so it can be considered negligible.

**Figure 20.** Equivalent deformations: (**a**) downward movement and (**b**) upward movement.

Finally, we should analyse the displacements of some singular elements such as mobile counterweights, which have the highest values. This aspect was indicated previously when carrying out the modal analysis of the invention, since when the counterweights had an inertial function they suffered the greatest displacements.

Thus, when the piston of the steam cylinder moves in a downward direction the counterweight will undergo a displacement of 18.73 mm (Figure 21a) and when it moves in an upward direction of 18.74 mm (Figure 21b). Not in vain, both the material with which the counterweights were made and the elliptical design of the same show that they were designed to bear the wear caused by the continuous movement to which they were subjected.

**Figure 21.** Displacements: (**a**) downward movement and (**b**) upward movement.
