## 2 Material and Methods: Fine-Module Gears  
In order to evaluate the influence of the gear size on the load-carrying capacity for tooth bending, Single Tooth Bending Fatigue (STBF) [27] tests were performed on a gear having normal module \(m_{n}=2\) mm made of 39 NiCrMo3. The tests were performed on a STEPlab UD04 electromechanical monoaxial pulsator capable to apply up to 5 kN. To perform STBF tests, a special apparatus consisting of 2 anvils, a supporting fork, and a centering pin was developed (Figure 2). The STBF tests take advantage of the Wildhaber distance W5 [28] to simultaneously load two teeth, ensuring that the contact forces are on one hand parallel and coaxial among them, on the other hand perpendicular to the teeth flanks. In this manner, during the test the friction between anvils and gear flanks is enough to keep the gear sample in position without the need of external supports. The fork and the centering pin are required during the mounting stage only to ensure that the contact points between gear flanks and anvils are in the correct positions. On the other side of the coin, with this configuration it is not possible to perform tests keeping the stress ratio \(R=0\) as in engaging gears. A minimum compressive load should be always present to ensure that the gear remains in the correct position [29]. Based on literature [30; 31; 32; 33; 34], a value of \(R=0.1\) is sufficient to ensure that no undesired movement will take place.  
Table 2 summarizes the geometrical properties of the gear samples. The quality of the gear is 5 according to DIN 3962 [22] (EN).  
The tests were performed according to the stair-case approach [35]. If a test at a certain force level \(F_{i}\) ended with a failure, the force for the successive test was decreased by \(\Delta F\). If the test ended with a Run-Out (the sample withstand 5 M cycles) the force was increased by \(\Delta F\). The force interval \(\Delta F\) was set to 100 N.  
The Up-and-Down Method for Small Samples with Two Specimens \"in Series\" according to [PERSON] [36] was used to post-process the results. It is based on the original work of [PERSON] [37] which allows the estimation of the fatigue limit starting from a small number of samples thanks to the adoption of statistics. This work was successively developed by  
\begin{table}
\begin{tabular}{c c c c c c} \hline \(m_{n}\) [\(mm\)] & 2 & \(d_{en}\) [\(mm\)] & 52.000 & \(x\) [\(-\)] & 0.300 \\ \(\mathbf{a}_{n}\) [\({}^{\circ}\)] & 20 & \(\mathbf{d}_{b}\) [\(mm\)] & 48.846 & \(\mathbf{h}_{I\!\!P^{\ast}}\) [\(mm\)] & 1.250 \\ \(z_{n}\) [\(-\)] & 26 & \(\mathbf{d}_{a}\) [\(mm\)] & 57.200 & \(\rho_{I\!\!P^{\ast}}\) [\(mm\)] & 0.380 \\ \(\mathbf{b}\) [\(mm\)] & 10 & \(\mathbf{d}_{f}\) [\(mm\)] & 47.960 & \(\mathbf{h}_{a\!\!P^{\ast}}\) [\(mm\)] & 1.000 \\ \hline \end{tabular}
\end{table}
Table 2: Gear data—2 mm-module gear.  
Figure 2: Scheme of the testing apparatus.  
Little in order to take into account the effect of samples mounted in series. An STBF test is a typical example of this configuration.  
Fatigue failure occurs generally before about 1 M cycles having all the same amplitude. However, it is common testing practice to select an arbitrary test duration (Run-Out). In this manner, a substantial portion of the testing time is spent where failure is unlikely, but the run-out condition has not yet been reached. Interrupting individual tests when the first specimen fails (one of the 2 tested gear teeth) means off course saving time, but also losing information regarding the subsequent response (if any) of the second tooth. The method proposed by [PERSON] takes into account this effect statistically.  
The fatigue limit \(F_{FP_{STBF50\%}}\) is estimated as the sum of the first meaningful load level \(F_{1}\) (Equation (8)).  
\[F_{FP_{STBF50\%}}=F_{1}+k\cdot\Delta F \tag{8}\]  
where \(k\) a statistical parameter depending on the Failure (F)--Run-Out (RO) sequence.