Gear root bending strength: statistical treatment of Single Tooth Bending Fatigue tests results
Source: Politecnico di Milano, Italy
Gear tooth failure due to tooth root bending fatigue is one of the most dangerous failure modes in gears. Therefore, the precise definition of gear bending fatigue strength is a key aspect. Indeed, STBF tests have been developed in order to estimate the gear load carrying capacity by testing directly the spur gear teeth. Here, with a specific focus on symmetric STBF test, a methodology based on two different statistical tools (i.e. Maximum Likelihood and statistic of extreme) is proposed with a view to estimate the gear tooth root bending load carrying capacity.
Tooth root bending fatigue is considered as one of the most dangerous gear failure mode because, typically, it results in the interruption of the power flow within the gearbox. Standards, such as ISO 6336-3  and ANSI/AGMA 2001 , provide designers with an analytical framework, to assess a gear train, in respect to this failure mode, by comparing the maximum tooth root stress with the limit value of the gear itself. Data of several typical materials are present within the standards (e.g. , ), defined at 1% gear failure probability (or 99% reliability) for tooth root bending fatigue. Different reliability level can be reached using literature coefficients (e.g. , ).
However, when specific material data are not available, extensive gear testing is needed, especially if a certain reliability level, after a given service life, must be assured.
Within this context, a gear SN estimation approach, based two different statistical tools (i.e. maximum likelihood method and statistic of extreme) will be shortly presented here. More details on the proposed statistical methodology will be presented at the International Conference on Gears 2022.
Short remarks about STBF gear testing
Tooth root bending fatigue strength can be evaluated by means of running gears tests (e.g. , ), Single Tooth Bending Fatigue (STBF) tests (e.g. –), and tests on notched specimens (e.g. ISO 6336-3 annex A , –). Amongst them, due to its higher efficiency and capability to directly work with gears, STBF tests are the most adopted .
STBF test are performed in two configurations: asymmetric and symmetric. The first, mainly adopted in the U.S. , , is described in the SAE recommended practice J1619 . In a asymmetric configuration, one tooth is directly loaded by the machine, while a second tooth and a pin work as supports. Here, when the loaded tooth breaks, the test is stopped.
Figure 1: Example of symmetric STBF test rig of an aerospace grade gear.
The symmetric STBF test configuration is the one mainly adopted in Europe , , . Here, both tooth roots are subject to the same nominal toot root stress. In a symmetric STBF test configuration, each test is stopped when one of the two teeth breaks, as it is physically impossible to load again the same tooth roots. An example of symmetric STBF test rig is shown in Figure 1.
In the last two decades, the authors have estimated the tooth root bending fatigue load carrying capacity via a symmetric STBF approach for different kind of gear specimen. Some examples: gears for aeronautical applications ,  , for small planetary gearboxes , for wind turbine gearboxes ,  also additively manufactured gears , . All of them have been performed on a Schenck pulsator, adopting R=0.1. For more details concerning the adopted testing procedure, the interested reader is referred to the aforementioned references
The classical approach to elaborate STBF test data
Unfortunately, STBF tests results cannot be directly applied to assess a gear pair. Indeed, the loading condition of a STBF test rig is not representative of the real case. The literature provides two different approaches for the treatment of STBF test data. Therefore, they can be used within the existing calculation methods. They are proposed within FVA report no. 304  and within the works of Rao and Mc Pherson , .
Both authors highlight two main issues that must be considered in order to treat STBF test data:
- The testing and service failures are different. In STBF test, a failed test corresponds to the breakage of some tested teeth, while in the meshing case, the failure of the specimen (i.e. the gear itself) corresponds to the failure of the weakest tooth.
- The testing and service loading conditions are different . In the real case, the force varies both, the application point, and its magnitude, while in STBF the force is a sinusoidal load whit a fixed amplitude and a given application point.
Stahl and Rao faced the problem with two different approaches. On the one hand, Stahl, K.  proposed several coefficients, based on experimental evidence: three different coefficients deal with the statistical difference: f_(p→Z) and f_(Z→ZR) translates STBF results to the case of a meshing gear while f_(1%ZR) is adopted to reach the 99% reliability level. A further coefficient f_korr based on Rettig experimental evidence  is used to adress the different loading conditions (the same is adopted here).
On the other hand, Rao and Mc Pherson ,  define a simple statistical relation between the teeth and the gear, and by means of a ‘fit by eye’, they translate STBF test data, corrected to the real case load, to the desired reliability level. Concerning the different loading conditions, Allowable Stress Range (ASR) diagrams are used.
The proposed approach: teeth SN curve estimation by means of maximum likelihood
Figure 2: Typical gear SN curve
In the gear field, both ISO 6336-3 and ANSI-AGMA 2001-D04 reports classical gear SN curve, suggesting that gears do not present a fatigue limit (see Figure 2). However, the standards do not suggest a typical slope for the long-life region. Anyhow, in the gear field, the estimation of such secondary slope is becoming relevant if the ass
essment is based on a load spectra calculation, thus requiring a proper estimation of both slopes.
Therefore, the developed methodology proposes a model based on a curve shape compatible with the ones proposed by the standard, i.e. a curve presenting two slope, one for the limited life region, one for the long life one. This kind of curve has been firstly proposed by Spindel et. al. :
where N is the number of cycles, S is the applied load, N_e and S_e represent the knee position, k is the slope of the limited life region and k_1 is the slope of the long-life region.
A first attempt to estimate this curve can be done by simply considering all the data as a failure and then fitting this curve. However, through a deep analysis of symmetric STBF test data, it is possible to notice that each experimental point “hides” the presence of another loaded tooth, which is survived. In other words, in a symmetric STBF test rig, each test interrupted by the failure of one of the two teeth is actually a failure and a survival. On the other hand, the case of a runout represents two runouts. This important presence of survived teeth can be properly handled only adopting MLE as a tool for the estimation of the teeth SN curve.
Indeed, MLE is an estimation technique typically applied in the determination of the S-N curve, as it has great advantages to consider, at the same time, different kind of data. Within MLE, exact data (e.g. failures) and censored points (e.g. runouts and survivals) are considered, with a different statistical meaning, within the same calculation procedure , , . MLE has been applied for the estimation of the S-N curve of specimens (e.g. , –), as well as in other components, e.g. welded structures (e.g. , ).
Figure 3: Different estimated teeth SN curves.
The proposed methodology applies MLE in order to estimate the experimental S-N curve, that is the S-N curve of the teeth, and its associated dispersion.
The proposed approach: statistic of extreme to estimate the gear SN curve
Once the S-N curve of the teeth has been estimated, STBF result elaboration is necessary to make it representative of the gear. In the real case scenario, where gears are meshing and rotating, gear failure due to tooth root bending fatigue is ruled by failure of its weakest gear tooth , , .Therefore, the gear S-N curve can be defined as the one describing the weakest tooth amongst the z gear teeth. In other words, the CDF (Cumulative Density Function) of the teeth has to be elaborated in order to define the gear CDF, that is the CDF of the weakest tooth among the z gear teeth.
By means of a mathematical passage , statistic of extremes allows to define the CDF of the minimum value over n extractions of X. Focusing on the case of gear tooth root bending fatigue, it can be used to define the distribution of the tooth with the smallest resistance over z gear teeth.
The article header image explains graphically the procedure: the STBF CDF is elaborated, by means of statistics of extremes, to define the gear CDF.
Figure 4: Estimated gear SN curves
For gear failure due to tooth root bending fatigue, ISO 6336-5  proposes a typical failure probability of 1% (i.e. a reliability of 99%), hence it is possible to define the curve at the required reliability level. In other words, estimating the load carrying capacity with the 99% percent probability of being exceeded for the weakest tooth over z teeth. However, different SN curve, at different reliability levels can be obtained by calculating different percentiles.
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Luca Bonaiti, M. Sc., Department of Mechanical Engineering, Politecnico di Milano, Italy