2.2.5. Meshing

The last step before beginning the static analysis consists of the geometric discretization or meshing of the engine. Most software that performs an FEA uses tetrahedral elements, type tetra 10 (4 physical points and 10 nodes). If the simulation is more complex, for example, in dynamic analysis or in static analysis of complex geometries, some software uses hexahedral volumes, such as hexa 8, hexa 20 or hexa 27 for meshing. The use of the hexahedron allows the mesh to be formed by elements of greater volume, and therefore it needs a smaller number of elements to generate all of the geometry of the model. The choice of hexahedral meshes decreases the computational requirements and saves time, but results in less accurate results, since the total number of nodes is reduced and the mesh is less suited to the real geometry. In an analysis such as the present study, there is no problem in using tetra 10, since the total number of elements to analyze is not excessive.

Therefore, in FEA analysis, the elements used were tetrahedrons with 4 nodes and first order integration (constant interpolation of the stress and strain). The elements are formulated in a 3D scheme with three degrees of freedom per node (translational in X, Y and Z directions). However, a sensitivity analysis of the size of the element has not been carried out, since the structure is oversized.

On the other hand, the specific software used in the analysis of thermal parameters or lighting uses bar elements or shell elements. These elements of lower order than those used in the present study could also be used for the static or modal analysis of a mechanism, but it would require a much larger number of nodes to simulate its geometry, and therefore for a computer with a standard processor, these meshes would need an excessive time in their calculations. Regarding the shell elements, although they are capable of capturing the stress gradients in a more efficient way, they have not been used in

this study, since the mechanism is oversized, and because it would provide an accuracy that would not change the conclusions in a significant way.

Autodesk Inventor Professional can automatically generate a mesh based on the size of the element to be discretized. By default, it generates tetrahedra whose average size is 10% of the length of the element, a minimum size of the tetrahedron of 20% of the average size, a maximum variation between tetrahedra of 1.5 and a maximum angle of rotation of 60◦.

However, it is advisable to pay more attention to the places where the geometry is complex, since in these parts, the mesh usually gives results that differ greatly from the actual geometry of the element. For these singular places, the software offers the possibility of carrying out a manual meshing of the element in which the mesh size is determined. In addition, in general lines, it should be mentioned that in places where the concentration of stress is greater, it is also convenient to increase the density of the mesh. Figure 14 shows the meshing result that is generated automatically by Autodesk Inventor Professional.

**Figure 14.** Automatic mesh obtained from the Hay inclined plane.

In the present research, it was also necessary to modify the discretization in some places, performing a refinement of the mesh. Specifically, this was done in the contacts between the teeth of adjacent gears (Figure 15) and on the surfaces of the ends of the axles in contact with their supports (Figure 16). In the first case, the reason for this intervention was the complicated geometry of the tooth of the gear, while in the second case, the reason lay in the concentration of the stresses.

**Figure 15.** Refinement of the mesh in the teeth of the gears.

**Figure 16.** Refinement of the mesh at the ends of the shafts in contact with the supports.

To conclude this section, it must be emphasized that for static analysis, it is necessary to establish a convergence criterion since iterative processes are used in this analysis. Thus, the result obtained is compared with the previous analysis and when it differs by less than 5% the process stops. In this research, and taking into account the computational resources available, the analysis used a mesh of 1,028,444 elements and 1,660,194 nodes in the blocking situation, and in the situation of emergency braking a mesh of 1,417,521 elements and 2,237,388 nodes.

## **3. Results and Discussion**

On the interpretation of the results, it is convenient to point out that von Mises stresses were used instead of principal stresses. Although these show somewhat lower values than the principal stresses for the elements made of wood, it does not mean that these values are not valid, but they provide important indicative values, which, moreover, will not be too far from the value maximum stress in the most unfavorable direction.

Furthermore, the main direction of the wooden element (its greater length) is the direction parallel to the grain (the most resistant), and the engineer thought of wooden elements to work in compression. This indicative data, which shows the calculation of the von Mises stress on the wooden elements, is far from the elastic limit of the structure, whereas it shows large concentrations of stress on metal (isotropic) elements, and in this case, the von Mises stress is a precise datum of the load which the mechanism supports.

#### *3.1. Modal Analysis*

Before carrying out the static analysis, it is necessary to check previously, by means of a modal analysis, whether the engine behaves like a rigid solid. If it had such behavior, the software would provide erroneous results.

To do this, the model is subjected to a series of vibrations at different frequencies, which produces some deformations so that at high frequencies, elements appear that have a deformation greater than the reference deformation adopted by the software. Autodesk Inventor Professional provides the 8 lowest frequencies at which any part of the model is deformed, and if those frequencies are close to 0 Hz, it means that the element behaves like a rigid solid, and therefore, it would not make sense to perform a static analysis.

For the blockage situation, the following frequencies were obtained: F1: 2.00 Hz, F2: 2.81 Hz, F3: 2.83 Hz, F4: 5.78 Hz, F5: 6.25 Hz, F6: 8.11 Hz, F7: 9.00 Hz and F8: 9.04 Hz. This means that, in this situation and as expected, the model behaves statically and, therefore, the study can proceed.

For the emergency braking situation, the frequencies were different since some elements do not come into action. The results obtained were: F1: 3.39 Hz, F2: 4.25 Hz, F3: 6.37 Hz, F4: 6.42 Hz, F5: 6.71 Hz, F6: 9.08 Hz, F7: 9.26 Hz, F8: 9.46 Hz. Thus, the modal analyses of both situations were very similar without loads.

On the other hand, as explained previously, the mesh size affects the exact places where stresses are concentrated, so the software allows the possibility of establishing a study on the convergence of the results by iterating the results in the regions that are considered more convenient. Therefore, for the static analysis a maximum of 10 cycles of iteration has been established, as long as the results vary more than 5% in each iteration. If the results in a region are less than that 5%, it stops iterating. In the same way, the greater the maximum number of iterations and the smaller the variation between results, evidently, the greater the computational requirements and the more prolonged the simulation time. In any case, this study is necessary in order to obtain a reliable result.

The results of the convergence curves for the two situations studied are shown in Figure 17. In the blockage situation the convergence rate (0.035%) is reached at the fifth iteration (Figure 17a), and in the emergency braking situation, the convergence rate is lower (0.335%), reached at the third iteration (Figure 17b). Therefore, the evolution of both graphs allows us to determine that the results obtained is very reliable.

**Figure 17.** Convergence curve: (**a**) blockage situation and (**b**) braking situation.

#### *3.2. Static Analysis*

As can be supposed, the results of the static analysis also vary depending on the situation in which the mechanism is studied.

In the blockage situation, the maximum stresses are located on the supports of the horizontal axles. At these points, the von Mises stress reaches a maximum at the insertion of the DD axle with the inertia flywheel with a value of 186.9 MPa (Figure 18a). Similarly, for the rest of the contacts between the axles and the supports, there also appear high stress values: the support of the AA axle is subjected to a stress of 169.5 MPa (Figure 19a), and the support of the intermediate BB axle to a stress of 77.8 MPa (Figure 20a). Outside of these points, stresses are generally low.

In the braking situation, the maximum stress is located at the point of attachment of the support bar of the first brake, with a value of 1025 MPa (Figure 18b). This stress, as was observed in the previous sections, could not be achieved, since the operator would need to use a force equivalent to 2 tons to be able to stop the loaded tugboat. Therefore, the support bar of the first brake would break. However, it should be emphasized that the rest of the structural elements would be subject to reasonable stress. Thus, the second highest stress value would be reached in the support of the bar with a value of 390.7 MPa (Figure 19b), and the third highest value (287.1 MPa), would be reached at one part of the drive shaft, and would be located at the point of contact between the shaft of the largest gear of the DD shaft and that of the BB shaft gear (Figure 20b).

**Figure 18.** Location of the highest von Mises stress: (**a**) blockage situation and (**b**) braking situation.

**Figure 19.** Location of the second highest vonMises stress: (**a**) blocking situation and (**b**) braking situation.

**Figure 20.** Location of the third highest von Mises stress: (**a**) blocking situation and (**b**) braking situation.

Another result of the static analysis is that relative to the safety coefficient. The safety coefficient at a given point in the model is calculated as the ratio between the von Mises stress and the elastic limit of a material for that point. In other words, the safety coefficient shows graphically whether the material from which a piece is made is capable of supporting a certain stress or not. In principle, a safety factor close to one unit would indicate that the material is working very close to its elastic limit; if it is less than one, it would indicate that the material exceeds this limit and would break, and if it is greater than one, it would indicate that it works without problems when subjected to said stress.

Nowadays, the materials and their elastic limits are much better known than at the end of the 18th century, and for this reason, engineers aim for the elements to work within a range of safety coefficient of 2 to 4 units. At the time of the *Hay inclined plane*, on the contrary, machines tended to be oversized, since those limits were unknown. For this reason, it is not surprising that, in general, almost the entire model works well above the elastic limit of the materials.

In the blockage situation, it can be observed, however, that the axle supports work with a value of 3.51 (Figure 21a). These elements have in common that they are in contact with the most stressed elements, and therefore, William Reynolds used wooden supports, because they are elements that can be easily replaced, since there was much wear due to the excessive friction of the axles (the bearing had not yet been invented).

However, in the braking situation, the point of maximum stress (point of union of the support bar of the first brake) coincides with the one with the lowest safety factor, with a value of 0.74 (Figure 21b). Outside this bar, the points that have the lowest safety coefficient are located at the joints between the links that form the brake, with a value of 1.7 in the tenth link (Figure 22b). The rest of the links have somewhat higher values, also around 2.

**Figure 21.** Location of the lowest safety coefficient: (**a**) blockage situation and (**b**) braking situation.

**Figure 22.** Location of the second lowest safety coefficient: (**a**) blockage situation and (**b**) braking situation.

Also, in the blockage situation, it must be mentioned that the union between the inertia flywheel and the axle is another delicate place. The flywheel, which is the element for which movement had been restricted, suffers an important torque in its axle. However, the safety factor of the highest stress point is 4.05, and therefore it is well dimensioned, but it is by far the point of the transmission shaft that works with the lowest safety coefficients (Figure 22a).

Finally, and also for the blockage situation, the distal support of the AA axle is shown. In this place, there is the element with the third lowest safety coefficient with a value of 4.13 (Figure 23), but already forming part of the set of oversized elements, as happens with the rest of the pieces of the mechanism.

**Figure 23.** Third lowest safety coefficient in the blockage situation.

Another aspect that static analysis studies is that of the deformations suffered by the various elements, as these play an important role in the proper functioning of the mechanism. Thus, an excessively deformed element can generate problems for the correct performance of its function, affecting the proximate elements. For example, those elements inserted in guides, crossed by bolts or those that need to keep their contacts well defined in order to gear without gaps are very sensitive to deformations.

In the blockage situation, the maximum equivalent deformation, located at the junction of the inertia flywheel with the DD axle has a value of 0.13% with respect to the size of the element, and, therefore, is negligible (Figure 24a).

**Figure 24.** Location of the highest deformation: (**a**) blockage situation and (**b**) braking situation.

However, in the braking situation (Figure 24b), the maximum equivalent deformation, located at the insertion of the braking bar with the support of the first link of the brake and according to the point of maximum stress, is somewhat greater, with a value 0.74%, although very far from representing a danger for the operation of the mechanism.

Finally, the displacements are analyzed. For reasons similar to those mentioned in the previous section, the elements subjected to a stress point can present important displacements with respect to their usual working position. However, and in general, it can be seen that most of the elements hardly suffer any displacement when subjected to normal loads.

In the blockage situation, the piece subjected to the greatest displacement, with a value of 22.98 mm, is the inertia flywheel (Figure 25a). Although this is an acceptable displacement, this data indicates again that this piece is affected by the stopping of the transmission shaft and suffers like no other the consequences of this effort. The second point where the displacement is greatest is located at a point in the structure (Figure 26a) with a value of 6.27 mm, which indicates that the displacements in the rest of the elements of the transmission shaft are negligible, and that the structure works correctly.

In the braking situation, the end of the actuated lever is the point registering the greatest displacement with a value of 247.1 mm (Figure 25b). It is a solid bar of cast iron 2 m in length, so although such a displacement is not acceptable, it is understandable given the characteristics of the force that causes it (close to 22 kN). The second maximum displacement is located at the inertia flywheel, as also happens in the blockage situation, with a value of 10.28 mm (Figure 26b). The rest of the elements present negligible displacements.

**Figure 25.** Location of the highest displacement: (**a**) blockage situation and (**b**) braking situation.

**Figure 26.** Location of the second highest displacement: (**a**) blockage situation and (**b**) braking situation.
