## 3 Materials and Method
### Finite Element Analysis  
The 27 gear geometries presented in the previous section have been simulated via FEM in MG and STBF condition with the aim of obtaining the stress histories (\(\overline{\sigma}(t)\)) in the \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{3}{*}{**Gear Geometry**} & \multicolumn{8}{c}{**Design Parameters**} \\ \cline{2-10}  & \(m_{n}\) & \(\alpha_{n}\) & \(x^{*}\) & \(z\) & \(b\) & \(h_{JP}^{*}\) & \(h_{aP}^{*}\) & \(\rho_{JP}^{*}\) \\ \cline{2-10}  & **mm** & \(\circ\) & **-** & **-** & **mm** & **-** & **-** & **-** \\ \hline
19 & 8 & 15 & 0 & 24 & 10 & 1.30 & 1 & 0.380 \\
20 & 8 & 15 & 0.2 & 24 & 10 & 1.25 & 1 & 0.380 \\
21 & 8 & 15 & 0.4 & 24 & 10 & 1.25 & 1 & 0.380 \\
22 & 8 & 20 & 0 & 24 & 10 & 1.25 & 1 & 0.380 \\
23 & 8 & 20 & 0.2 & 24 & 10 & 1.25 & 1 & 0.380 \\
24 & 8 & 20 & 0.4 & 24 & 10 & 1.25 & 1 & 0.380 \\
25 & 8 & 25 & 0 & 24 & 10 & 1.25 & 1 & 0.317 \\
26 & 8 & 25 & 0.2 & 24 & 10 & 1.25 & 1 & 0.317 \\
27 & 8 & 25 & 0.4 & 24 & 10 & 1.25 & 1 & 0.317 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._  
Figure 4: 27 different tooth profiles by varying \(m_{n}\) (rows), \(\alpha_{n}\) (columns), and \(x^{*}\) according to Table 1. It is represented also the base circle (in a continuous line) and the pitch circle (in a dash-dotted line); their distance is related to \(\alpha_{n}\). The dimensions have been scaled to facilitate visibility. tooth root radius. In this way, as it is explained in more detail in the next section, it has been possible to study stress histories by applying [PERSON]'s fatigue criterion. Each gear geometry has been modelled through the open-source software Salome-Meca/Code_Aster. In the present study, 2D simulations have been performed to speed-up the calculation time of the 54 simulations (27 modeling the MG conditions and 27 modeling the STBF ones). In Figure 5, it is possible to see an explanatory model of STBF condition and, in Figure 6, an explanatory model of MG condition is represented. Two-dimensional simulations are also supported by the fact that in previous studies (in which fatigue criteria based on critical planes were applied to gears, e.g., [62; 63]) it has always been observed that the critical plane is perpendicular to the frontal view of the gear. Moreover, in this study, ideal geometries have been exploited and defects are not taken into account. In both the configurations, the loaded teeth have been modeled through quadrangular elements. The tooth root radii have been discretized by 31 nodes as well as the teeth flanks. The remaining part of the gears have been modeled through triangular elements having larger dimensions. This allowed reducing the number of total mesh elements taking into account the overall stiffness of the simulated gears. In MG simulations (Figure 6), the gears have been simulated in their whole geometry. On the teeth involved in the gearing, the mesh density has been increased as mentioned above. In the STBF loading condition (Figure 5), it was possible to exploit the symmetries. Indeed, in STBF, the gear is stationary and there is a symmetry in the plane passing through the center of the gear and parallel to the anvil's contact faces (yellow line in Figure 5). Since the teeth are loaded cyclically due to the meshing with other teeth (MG) or with anvils (STBF), non-linear simulations have been set up discretizing the loading cycle in 40 time steps. The modeled material is linear isotropic having a Young modulus \(E=205\) GPa and a Poisson's ratio \(v=0.3\). In all the simulations, the state of stress never exceeded the yielding. Therefore, the non-linearity of the simulation is related to the contact that, as visible from Figures 5 and 6, involves the points (highlighted in green) of the two components according to the simulation. In other words, the nodes on which contact can occur are the tooth flanks in MG and between the  
Figure 5: Finite Element Modelling of STBF tests. In yellow, it is shown the symmetry line of the system. The force is applied to the anvil.