Influence of Geometry Deviations on the Tooth Bending Strength

In the following section, the influence of the geometry deviations on the tooth bending strength is examined. Using the two calculation methods presented, the effect of the measured geometry deviations on tooth root stress is first determined, and the difference between the two calculation methods is examined. Furthermore, the influence of considering the measured tooth root contours in the determination of the tooth bending strength is shown by means of experimental investigations on the measured test gears.

The results of the investigation into the influence of production-related geometry deviations on tooth root stress are shown in Figure 6 as an example for variant II (20MnCr5 – bainitization). The tooth root stresses were calculated for all measured tooth gap contours of variant II at a uniform pulsator force of FPuls = 44 kN with four clamped teeth. A median normal distance of nDev,median ≤ 1 μm was chosen as the stopping criterion for the iteration of the hypothetical manufacturing tool for the calculation method according to ISO 6336 3:2019 Method B (Ref. 14). For the tooth root stress calculation using the tooth contact analysis Stirak (version 4.3.2.6), a resolution of 30 rolling positions per pitch was used. This is a compromise between the required calculation time and the resolution accuracy. Any increase in the number of rolling positions reduces the distance between the selected contact line and the desired pulsator diameter dPuls, thus improving the calculation accuracy. On the other hand, the required calculation time is increased as a higher number of rolling positions is calculated in the tooth contact analysis. To compare the results between the two calculation methods, the tooth contact analysis evaluates the maximum tangential tooth root stress under tensile load.


Figure 6—Influence of manufacturing scatter on the tooth root stress.
The bar chart in Figure 6 shows the calculated tooth root stresses of five different tooth gap contours with both calculation methods. The five selected tooth gap contours are the gap contours with the minimum and maximum tooth root stress of both calculation methods, as well as the tooth root stress of the median tooth gap contour from Figure 4. As can be seen from the bar chart, the two calculation methods provide different tooth root stresses for any of the five selected tooth gap contours. Furthermore, the minimum and maximum tooth root stresses of the two calculation methods occur on different tooth gap contours. These differences can be attributed to the fundamental differences between the two calculation methods as well as to the different fitting quality of the tooth gap contour (see Figure 6—right). Due to the iteration of the hypothetical manufacturing tool for the standard calculation, a highly accurate fit of the actual tooth root contour is not possible for every tooth gap contour, as can be seen, for example, in the contour comparison of gap contour G1 in Figure 6. The fitting quality is defined by the stop criterion for the iteration nDev,median ≤ 1 μm, and by the rolling simulation of the hypothetical manufacturing tool for generating the tooth root contour. Deviations between the measured and the used tooth root contour for calculation cannot be excluded. For the FE-based simulation, the measured tooth root contours are only smoothed to achieve a higher fitting quality (see Figure 6). Due to these limitations, the FE-based calculation tends to calculate higher tooth root stresses than the standard calculation (see Figure 6—boxplot diagram).

In the following, the influence of the tooth root contour on the determination of tooth bending strength in the pulsator is investigated for the presented variants I to IV (see Figure 7). For this purpose, pulsator tests were carried out on the measured and presented variants at an ambient temperature of TTest = 180°C. The limiting number of load cycles in the pulsator was NL = 3·106 load cycles and the test frequency was approximately fTest ≈ 35 Hz. The tests were evaluated using the IAGB/Hück staircase method for a failure probability of PA = 50% (Ref. 37). The pulsator forces were converted for the corresponding tooth gap contour using the calculation method presented according to method B of ISO 6336 3:2019 (nominal contour vs. median tooth root contour of the respective variant) (Ref. 14). The fracture surfaces of all variants showed no conspicuity. Final material tests on the influence of the operating temperature T = 180°C on the material structure are still pending.


Figure 7—Influence of the tooth root contour on the determination of tooth bending strength in the pulsator.
Assuming that all variants have the same tooth gap contour, the test results of all variants can be plotted on the same graph with the ordinates of the pulsator force FPuls (load) and the ordinate of the equivalent tooth root stress σF0-B (stress) (see Figure 7—top center). With the same load for all variants and assuming the same tooth gap contour, the tooth root stress is the same for all variants. However, this is not the case when considering the measured mean tooth contour of the variants. In this case, the variants can only be plotted on a graph with only one ordinate, the load or stress (see Figure 7—top right). For example, the consideration of variant IV shows that by considering the median tooth root contour, the top load level is reduced by ΔσF0-B,IV = -44.5 N/mm2 compared to the assumption of the nominal contour. On the other hand, considering the median tooth root contour makes only a small difference in variant III. This is due to the smaller deviation between the nominal contour and the median tooth root contour (see Figure 4). This difference becomes particularly clear when comparing the variants (see Figure 7—bottom center). Considering the median tooth root contour, the mean bending strength of variant IV is reduced by ΔIV = 3.3%, whereas the tooth bending strength of variant III remains almost unchanged. The difference in tooth root stress between variant III and IV is ΔσF0-B,IV III,nominal contour = 54.8 N/mm2, using the nominal contour, and is reduced to ΔσF0 B,IV III,mean tooth root contour = 10.6 N/mm2 when the respective mean tooth root contour is considered.

This means that the two variants are much closer to each other due to the consideration of the mean tooth root contours. This change shows that the geometrical influence should be included in the evaluation of durability tests focusing on material and heat treatment, as it is not a material influence but an influence of the manufacturing process chain. A further improvement in the evaluation of material tests for tooth bending strength could be achieved by considering the actual tooth gap contours. However, this would require an adaptation of the test procedure to obtain a constant step distance, as a constant step distance is a requirement for the evaluation according to IAGB/Hück (Ref. 37). An alternative is to use another evaluation method, such as maximum likelihood, which does not require a constant step distance for the evaluation (Ref. 38). However, statistical research has already shown that a maximum likelihood estimation does not provide estimates of mean and variance that are true to expectation, making this method of limited use for determining fatigue life values (Ref. 39).