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Effect of Gear Design Parameters on Stress Histories Induced by Different Tooth Bending Fatigue Tests: A Numerical-Statistical Investigation  
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permissible bending stress in the fillet region as \(\sigma_{FP}\) and the maximum stress due to pure bending in the fillet region as \(\sigma_{F}\), the relation \(\sigma_{F}\leq\sigma_{FP}\) has to be satisfied (considering the life factor \(Y_{NT}=1\)) to ensure an infinite life to the designed gear. On the one hand, different analytical methods to calculate \(\sigma_{F}\) can be found, e.g., ISO 6336-3 [16], ANSI/AGMA [17]. On the other hand, \(\sigma_{FP}\) is proportional to the material strength \(\sigma_{Film}\) (method B of ISO 6336-3 [16]). According to ISO 6336-5 [18], for specific materials and/or treatments \(\sigma_{Film}\) has to be estimated experimentally. In the literature, it is possible to distinguish three different types of tests to estimate \(\sigma_{Film}\): (1) Meshing Gears (MG) tests, e.g., [15; 19], (2) Single Tooth Bending Fatigue (STBF) tests, e.g., [20; 21; 22], and (3) notched specimens tests, e.g., [23; 24; 25]. With respect to MG tests, the specimen is itself a gear made with the material and treatment to be characterized. It meshes with (at least) one more gear, see [15; 19]. Through MG tests, it is possible to reproduce the actual loading conditions during operation and, thus, the exact stress history in the fillet region [26; 27]. Accordingly, it is possible to characterize the fatigue behavior of the tested material (and treatment) very reliably using this method [28]. Indeed, the TBS obtained via MG tests (\(\sigma_{FilmMG}\)) is the value of \(\sigma_{Film}\) as defined in the standards. However, running tests on MG is relatively expensive, time consuming, and requires specific test rigs, e.g., [15; 29]. The above-mentioned limitations are mainly due to the fact that, for each gear tested, a single value of \(\sigma_{FilmMG}\) can be estimated since the breakage of a tooth makes the specimen unusable for the acquisition of other data [26; 27]. Specimen costs and the duration of the experimental campaign (with respect to MG tests) can be drastically reduced exploiting notched specimens. This kind of test can be performed on any universal testing machines. However, since there is no meshing, these tests do not reproduce a stress history that is faithful to the gears operating conditions. In addition, the different geometry (with respect to the gears teeth) further compromises the accuracy of the results that, to be used in the standards, require the implementation of a set of corrective coefficients [30]. An excellent compromise between the time required for the experimental campaign, the costs related to the specimens (and the test equipment) and the accuracy/reliability of the results can be found in STBF tests [20]. STBF tests consist of the application of two coaxial forces (fixed directions and sinusoidally variable amplitudes) to two teeth of the sample (a gear made of the material and treatment to characterize). More specifically, in STBF tests, two anvils (having parallel faces and mounted on a universal testing machine) load two specific teeth of the above-mentioned gear, see [31; 32; 33]. The loading condition leads to the forces to be tangent to the base circle and, at the same time, normal to the tested teeth flanks (Wildhaber property [1]). Since in STBF loading condition, the gear has not to mesh and/or to rotate, the damage/absence of specific teeth (not the tested one) does not compromise the application of the load. Therefore, through STBF tests, it is possible to perform multiple measurement on the same gear. This is a clear economic advantage towards the experimentation on MG. In addition, in STBF tests the lubrication is not required since there is no meshing [34]. However, in STBF tests, the stress history induced in the fillet region presents some differences with respect to those induced in MG and to those modeled in the standard to estimate the TBS.