## 4 Method 1: FVA Approach Based on FZG Research  
The first estimation method discussed here had been developed at FZG and is defined as a recommended procedure within FVA (Research Association for Drive Technology, in German: _Forschungsvereinigung Antriebstechnik e.V._) research according to FVA guideline 563 I [25]. In [25], a practical approach for the evaluation of the tooth root load-carrying capacity, including the gear specimen, pulsator equipment, and test planning, was described. It is based mainly on the work of [PERSON] et al. (FVA research project 304) [23], where data elaboration is discussed, laying the fundamentals on the analysis of a large number of experimental data performed by FZG. The methodology according to [25] has been developed by examining a symmetric pulsator test configuration for case-hardened gears. Further investigations and detailed statistical analysis of an extended experimental FZG database can be found in [PERSON] et al. [24; 26] and [PERSON] et al. (FVA research project 610 IV) [56]. Refs. [24; 56] also reaffirmed the validity of this methodology. The FVA approach based on FZG research aims to estimate a gear SN curve such as the one described in Figure 4. Here, different considerations are made regarding the limited-life and long-life regions. On the one hand, for each load level, the first region presents a different lognormal deviation of the failure load cycle number. In other words, the dependency of the variance with respect to the load level is considered (see also Figure 6). On the other hand, the long-life region is described by an endurance limit, whose dispersion is in the load/stress direction. For both the experimental and gear SN curves, these two regions are estimated separately using two different calculation procedures. According to this method, the experimental endurance limit is estimated according to [PERSON] [57]. It is assumed that the experimental endurance limit follows a normal distribution. In the case of reduced sample size (\(n<10\)), this method suggests estimating the experimental endurance limit by adopting the modified probit method as defined by [PERSON] and [PERSON] [58]. Figure 5 shows an example of the application of the [PERSON] [57] approach, displaying a staircase test sequence together with an additional theoretical test run. The experimental endurance limit \(\sigma_{F0\infty,50\%}\) is defined starting from the determination of the number of tests \(f_{i}\) on load level \(i\), leading to the sums \(F\) and \(A\). According to [PERSON] [57], \(\sigma_{F0\infty,50\%}\) is defined as:  
\[\sigma_{F0\infty,50\%}=\sigma_{F0,i=0}+d\cdot\frac{\sum{i\cdot f_{i}}}{\sum{f _{i}}}=\sigma_{F0,i=0}+d\cdot\frac{A}{F} \tag{1}\]  
Once the \(\sigma_{F0\infty,50\%}\) has been estimated, the gear endurance limit \(\sigma_{F0\infty,1\%,MC}\) (under the MG loading condition) is calculated by means of corrective coefficients [23]:  
\[\sigma_{F0\infty,1\%,MC}=f_{1\%,F0}\cdot f_{P\to MG}\cdot\sigma_{F0 \infty,50\%,P} \tag{2}\]  
Figure 4: SN curves for different failure probabilities by assumption of separated consideration of long-life (endurance) and limited-life ranges [24]. Extracted from AGMA 18 FTM26, Reliability of Gears—Determination of Statistically Validated Material Strength Numbers, with the permission of the publisher, the AmericanGear Manufacturers Association, 1001 North Fairfax Street, Suite 500, Alexandria, Virginia 22314. Figure 5: Exemplary evaluation according to Hück staircase method; example extracted from [25]. where \(f_{1\%FD}\) is equal to 0.92 for peened gears and 0.86 for unpeened gears according to [PERSON] [23]. An additional coefficient \(f_{P\to MG}\) equal to 0.9, defined by [PERSON] [22], is adopted to take into account the load differences. \(f_{P\to MG}\) was not applied in this paper; that is, \(\sigma_{F0\infty,50\%}\) must be reduced by 0.77 or 0.83 to estimate \(\sigma_{F0\infty,1\%,MG}\).