#### Abstract

Being able to properly predict gear failure is a key aspect to achieve a reliable light-weight gearbox. Among the several gear failures, tooth root bending fatigue is considered as the most dangerous one because it implies the stoppage of the whole gearbox. In order to characterize a gear for this phenomena, Single Tooth Bending Fatigue (STBF) tests are the most performed ones. However, as in STBF test THERE IS no sliding/rolling contact and as the specimens are teeth rather than gears, some differences occur between the test conditions and those of the real case. This paper deals with the statistical ones that is the estimation of the gear SN curve starting from the teeth one. The teeth SN curve has been estimated by means of a statistical model developed considering Murakami’s idea of nonpropagating crack. Then, a methodology based on statistic of extreme is adopted for the purpose of estimating the gear SN curve.

#### 1. Introduction

Gears are mechanical components that, through the meshing of profiles, transmit power between two axes with an almost constant angular speed ratio [1, 2]. Due to their peculiar working principles, several failure mechanisms may occur in gear. Both ISO/FDIS 10825-1 [3] and ANSI/AGMA 1010-F14 [4] enlist several failure mechanisms that can affect gears. ISO/FDIS 10825-1 [3] classifies the most classical failure mode within two main macrocategories: tribological damages (i.e., scuffing and wear) and fatigue damages (i.e., contact fatigue and bending fatigue). In the design phase, several analytical calculation methods are adopted in order to avoid those damage phenomena (e.g., ISO 6336 series). Furthermore, during the exercise, it is possible to rely on monitoring technique as a means to control the advancement of the aforementioned phenomena (e.g. [5–9]).

On the one hand, scuffing, wear, and contact fatigue are generated by the rolling-sliding loaded contact between tooth flanks. On the other hand, the tooth root bending fatigue phenomenon is generated by the normal force that, during gear meshing, varies both in terms of load entity (due to the load sharing between teeth) and position (as the contact point moves along tooth active profile). As a result, the stress occurring within the tooth root is not constant, but it varies between a very small negative value (due to the extension of the stress field of the tooth root adjacent field), to a maximum (at the outer point of single tooth in contact). Furthermore, the tooth root itself implies the presence of a notch, which implies very high stress values [10, 11].

Among all gear failure modes, designers consider the tooth root bending fatigue failure mode as the most dangerous one. Indeed, the failure of a single tooth root implies a sudden stoppage of the power flow within the gearbox; thus, it is no more able to operate.

In order to avoid the tooth root bending fatigue failure, designers rely on several standardized calculation methods (e.g. ISO 6336-3 [12] and ANSI/AGMA 2001 [13]). According to them, a gear pair is assessed by comparing the tooth root stress with the limit value for the desired life. Moreover, adopting the classical damage accumulation framework provided by the Palmgren-Miner rule, both ISO 6336-3 [12] (together with ISO 6336-6 [14]) and ANSI/AGMA 2001 [13] allow designers to perform a load spectra analysis within the classical gear assessment framework.

Apart from the knowledge of the load spectra, any damage accumulation framework requires also the knowledge of the component SN curve. Both ISO 6336-5 [15] and ANSI/AGMA 2001 [13] propose typical endurance limit that, together with corrective coefficients (i.e., for ISO 6336-3 [12] and for ANSI/AGMA 2001 [13]), defines the gear SN curve. It is worth mentioning that the same concept is applied for the macropitting case too.

Figure 1 shows the typical dimensionless SN curve for several typical gear material proposed by ISO 6336-3 [12]. On the one hand, the static and the limited life region are well defined. On the other hand, the long life region is represented by a shaded area whose limits are an inclined line (that implies the inexistence of the fatigue limit) and a horizontal one (representing the gear fatigue limit). Nevertheless, ANSI/AGMA 2001 [13] proposes a slightly different curve where there is not a fatigue limit, but two inclined lines. All those curves are defined at 1% gear failure probability. Different reliability levels can be adopted by embracing further coefficients that shift the SN curve. ANSI/AGMA 2001 [13] includes this kind of coefficient while designers adopting ISO 6336 series can adopt those suggested by Hein et al. [16]. Nevertheless, both ANSI/AGMA 2001 [13] and ISO 6336 series strongly encourage to perform specific experimental campaign in order to estimate the gear SN curve.

This paper is aimed at discussing how to elaborate tooth root bending fatigue data in order to being able to estimate the component SN curve. Here, a statistical framework through which analyse gear fatigue data is discussed and adopted in order to estimate the gear SN curve. Finally, results are compared with the ones obtained with other literature data.

This article is aimed at continuing some previous works of the authors [17, 18]. There, the statistical framework is based on [19, 20], where a curve without fatigue limit is proposed. Nevertheless, the existence or not of the fatigue limit is sill under discussion (e.g. [21, 22]). Therefore, the proposed statistical framework features the presence of the fatigue limit.

The rationale behind this choice is that recent development in tooth root bending fatigue characterization [23–27] suggests that Murakami’s idea of nonpropagating cracks (e.g., [28–30]) can be apply also in the case of tooth root bending fatigue (thus, the gear tooth root bending phenomena can be modelled considering the existence of the fatigue limit). Among several models to describe the fatigue behaviour of a component (e.g., [19, 31, 32]), an SN parameter estimation technique, developed by Loren, S. [33, 34], also based on Murakami’s idea of nonpropagating cracks, is adopted as a means to estimate the specimen SN curve.

#### 2. Gear Testing for Tooth Root Bending Fatigue

ISO 6336-5: 2016 [15] presents four different methodologies for tooth root bending fatigue testing. Method A refers to test performed on gears, whose dimension and loading conditions are as close as possible to the actual case. Method B describes those tests performed on reference gears, whose dimensions and loading conditions differ from the actual one; Single Tooth Bending Fatigue Test (STBF) (also called pulsator test), which will be described later, belongs to this category. Methods and denote test conducted on notched and unnotched uniaxial specimen, respectively. Gear testing literature respects those categories. Indeed, it is possible to find test performed on rotating gears (e.g., [25, 35–37]), by means of the STBF procedure (e.g. [25, 26, 36–42]) as well as test performed on notched specimen (e.g. [43–45]).

The STBF procedure seems to be the most adopted tooth root bending fatigue testing methodologies. Indeed, as mentioned by McPherson and Rao [46], the STBF testing procedure has several advantages. Firstly, STBF tests can be performed on any uniaxial testing machine. Secondly, as there is no rolling/sliding contact, only the tooth root bending fatigue failure mode is present. Therefore, it is possible to perform tooth root bending fatigue test without the risk of falling into other gear failure modes (e.g., scuffing). Finally, by testing teeth rather than directly the gear, it is possible to obtain more specimen from the same component.

STBF tests replicate the tooth root bending phenomena by applying, by means of two anvils, a force far away from the tooth root. Two are the configuration typically adopted in order to perform STBF test. The first one is the symmetric configuration, in which two anvils apply the load on the teeth flanks. Typically, the anvil spans over 3/5 teeth, and thanks to the same principle of the Wildhaber measurements [47], the contact between anvils and teeth flanks occurs at the same nominal diameter. Hence, both teeth result to be subject to the same nominal tooth root stress. The latter configuration is the asymmetric one, deeply described in [48, 49]. Here, the machine load is directly only one tooth, while a second tooth, together with a centring pin, works as supports, reacting to the machine load. Therefore, only one tooth is tested at the time. Both test configurations must keep a minimum compressive force with a view to maintain the specimen in position.

As test data are evaluated according to the force applied by the test rig (i.e., a mechanical pulsator in this case) on the gear teeth, in order to move STBF force data into meshing gear tooth root stress, any approaches that to translate STBF experimental points to running gears have to deal with different passages: (1)Calculate the STBF tooth root stress. This is typically realized by means of FEM (e.g., [50, 51]) or using analytical methods (e.g., ISO 6336-3 method B [12])(2)Calculate the teeth SN curve in meshing gear condition starting from the teeth STBF data points(3)Define the gear SN curve, in meshing gear condition, at the proper failure probability level (e.g., 1%)

However, the advantages of the STBF testing procedure imply that two differences arise between the STBF loading case and the actual case [23, 36–39, 52, 53]. On the one hand, the lack of the rolling/sliding condition implies that the loading condition occurring within the toot root is different [11, 54, 55]. On the other hand, the fact that the tests are performed on the teeth rather that gear implies a statistical difference [17, 18]. Therefore, the results of STBF test have to be corrected in order to be used for the assessment of a gear pair under the real case loading scenario. Four different approaches can be found in literature, one developed by Seabrook and Dudley [53], the FVA report no. 304 [39], the work of Rao and McPherson [36, 37], and, recently, the one proposed by Hong et. al. [52]. Interested readers are referred to [17, 52] for a comprehensive review about how the literature deals with the aforementioned passages.

The authors [17, 18] adopt statistic of extreme in order to address the statistical difference while high-cycle multiaxial fatigue criteria (coupled with numerical simulations) are adopted to address the problem of the different loading conditions by defining a coefficient [11]. The same is applied here.

In the last twenty years, the authors have adopted a symmetric STBF procedure for the estimation of the tooth root bending fatigue resistance on several types of gears material/geometry. Some examples: austenite ductile iron gears [56], aeronautical-grade materials [50, 51], gears with very large modulus [57, 58] and 17-4 PH additively manufactured gears [59, 60]. All of them have been performed on a Schenck pulsator. For each experimental campaign, a specific equipment, whose features depend on the examined gear specimen, has been designed and adopted. Figure 2 shows the equipment adopted for the present experimental campaign. In order to maintain the gear specimen in position, a load ratio R=0.1 has been adopted. For additional details regarding the adopted testing procedure, the interested reader can refer to the aforementioned references. Result data discussed in [11, 17, 18, 61] are adopted here as reference case.

#### 3. Estimation of the Specimen SN Curve

Under the assumption of fatigue limit existence, when estimating the fatigue behaviour of a component/specimen, both the finite life and the infinite life are considered as two random variables, typically assumed to follow a log-normal distribution: where the terms and refer to the logarithm of the applied stress and the number of cycle , respectively. is a normally distributed random variable, whose mean depends on the applied stress. Also, the variance can be described as dependent on the applied stress too [32, 62].

Finite life implies that the higher the stress, the lower will be the number of cycles to failure. Therefore, can be any decreasing continuous function. Following Basquin’s approximation of the SN curve, can be a linear one. The same is adopted here. The variance is considered as constant. On the other hand, infinite life (i.e., the long life region) implies that there is a stress below which the component has an infinite life (i.e., the fatigue limit). and are mean and variance.

Therefore, Equation (1) can be rewritten as follows:

where and are the parameter describing the linear relationship between (i.e., the log of the applied stress) and (i.e., the log of the number of cycles).

A first attempt to estimate a component/specimen SN curve can be accomplished by investigating the two regions separately; defining two different sets of experimental data, each one adopted to estimate a single region. The finite life region behaviour is estimated performing test at high load levels. On the other hand, the fatigue limit can be estimated adopting a staircase procedure.

However, despite being relatively easy, this procedure has two main disadvantages: (1)The whole SN curve is biased by how the points have been collected. Point defined while investigating the infinite life can be adopted to describe the finite life behaviour. On the other hand, points obtained while looking into the finite life provide also information about the infinite life region(2)The staircase procedure is a sensitivity analysis aimed to estimate an endurance limit that is the load required to have 50% of failure at a certain lifetime (i.e., the runout level). On the other hand, the fatigue limit is the load level under which 50% of the tested component will show infinite life. The endurance limit can describe the fatigue one only if the runout threshold is sufficiently high

Surely, it would be better to adopt an SN curve estimation procedure that is able to estimate both finite and infinite life within the same calculation procedure. Therefore, a framework is able to overcome the below disadvantages while still working with a runout level . Fatigue literature presents several models able that satisfy the aforementioned requirement (e.g., [31]). Among them, the model proposed by Loren [33, 34] is adopted here. This model has been preferred in respect to the random fatigue limit (RFL) as it lays its basis on both statistical and experimental evidences, while the RFL [31] as the Loren’s model lays its basis on both statistical and experimental evidence, while the RFL is based only on statistical consideration.

The model proposed by Loren, S. [33, 34] is based on Murakami’s idea of nonpropagating cracks, which suggests that the limited life and the fatigue limit are related to two different phenomena (e.g., [28–30]). In the region ahead the fatigue knee (i.e., the limited life region), the cracks nucleate, and then, they propagate until the component/specimen breakage. On the other hand, in the region after the fatigue knee (i.e., the long life region), cracks can nucleate but their propagation stop (if the stress level is below the fatigue limit). Experimental evidence confirms that also tooth root bending fatigue follows Murakami’s idea of nonpropagating cracks [23–27].

Loren, S. model [33] works within a Maximum Likelihood (ML) framework, which is a technique to estimate distribution parameters based on the idea of finding, mathematically, those parameters that are more likely to represent the data. More precisely, the estimated distribution parameters are those that maximize the likelihood , which is based on the probability of observing the data. For further details concerning the ML estimation as well as likelihood properties, the interested reader is referred to [62–64].

According to Loren, S. [33], the component/specimen may have failed because the stress level is above the stress limit and/or it belongs to the finite live region. , i.e., the likelihood function for an observed value (i.e., a failure), is defined as follows:

where the symbol refers to a probability density function (PDF) while refers to a cumulative density function (CDF).

On the other hand, the component/specimen may present a life above the runout level for two reasons: at the stress level, it has a finite life above or the stress level is below . , i.e., likelihood for a right-hand censored (i.e., a survival) data, is as follows:

It is worth mentioning that, due to the compresence of the two phenomena, both Equation (3) and Equation (4) present terms related to the two different PDFs/CDFs.

The two likelihood terms are combined in the likelihood :

where is equal to 1 for observed data and equal to zero for censored data.

However, in order to perform the calculation procedure, it is more common to work on the log likelihood , that is the natural logarithm of the likelihood :

Equation (3) and Equation (4) are then included in Equation (6), thus obtaining the final log-likelihood formulation:

The parameters that are estimate by ML are those that maximize and, subsequently, also . However, as the classical calculation software includes only minimization algorithm, those parameters are obtained by minimizing . The term is a constant term that is removed from the calculation procedure.

Figure 3 reports the estimated model parameters and how they interact with the experimental data. By expressing each parameter within its confidence interval, likelihood ratio (LR) has been used in order to verify the model assumption [62, 63, 65].

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As two separate failure mechanisms are present, the PDF/CDF representing the failure probability of the specimen/component is described by the combination of both finite life and infinite life PDFs/CDFs [34]:

Different curve, at different failure rate, can be obtained by calculating, for different level, the number of cycles of the corresponding percentile. Furthermore, Equation (8) allows to calculate the probability of observing a failure after number of cycle when the specimen/component is subject to a certain stress level. This 3D surface is the so-called PSN curve. Figure 4 report the estimated teeth PSN curve. Similarly, Figure 5 shows different SN, at different reliability level.

The experimental data discussed here have been obtained adopting a symmetric STBF. Therefore, each experimental point represents two teeth, which have been tested at the same time at the same nominal load. This implies that each experimental points contains information about both the tested teeth. On the one hand, if one teeth fails after cycles, the other one shows a life that is greater than , and if the test stops because has been reached, both teeth shows a life greater than . If one focuses on the information provided by each single tooth, data are subject to randomly right censoring, i.e., there is no a fixed value after which teeth testing is stopped. Indeed, the test of a single teeth results to be stopped both due to tooth failures after cycles (which is random) or for having reached the runout level. On the other hand, if the focus is on the STBF test point, the test is stopped only because the runout level has been reached.

Within [17, 18], the adopted model allows the usage randomly right censored data. Therefore, it has been possible to estimate the SN curve referred to one single tooth. However, within [33, 34], there is no mention whereas it is possible to use randomly right censored data. Therefore, the estimation has been stopped to the STBF one. However, as will be shown in the next section, it is possible to estimate the gear SN without losing any information.

#### 4. Estimation of the Gear SN Curve

In order to estimate the gear SN curve, it is necessary to find a statistical relation between gear and teeth failure. This relation is based on the idea that, for the tooth root bending failure, a gear is considered failed when a single tooth breaks. The broken tooth will be the weakest one [36–38, 52, 53, 64].

By means of a mathematical passage [65], it is possible to use SoE in order to define the CDF of the minimum value observed over extractions of (the distribution of which is known) as follows:

where is the smallest value distribution and is the parent one. Figure 6 graphically explains Equation (9) effect: the smallest value distribution is narrower and with a lower scale parameter.

Accordingly, the gear CDF can be defined as the CDF of the weakest tooth among the gear teeth, the CDF of which is :

However, the CDF defined in the previous section (i.e., Equation (8)) does not describe the teeth themself but the STBF experimental data, in which the failed teeth is weakest one among the teeth pair. Therefore, Equation (10) can be rewritten as follows:

That is, is considered as a system of teeth pair, within which the smallest value over the teeth pair defines the gear load carrying capacity. In other words, the gear is considered as a system teeth pair among which the weakest pair rules the gear failure. Figure 7 reports the estimated gear PSN curve. Similarly, Figure 8 shows different SN, at different reliability level.

#### 5. Results and Conclusion

Figure 9 reports different SN curves, at different failure probabilities, for both the STBF and the gear case. It is possible to notice that, as already anticipated, the gear PDF results to be narrower that the teeth one; indeed, gear percentile is much closed than the teeth one (see Figure 6).

Looking at 50% percentile of the teeth SN curve (i.e., Figure 10), it is possible to observe, graphically, the effect of the combination of the two phenomena. At high stress level, the probability of having infinite life is close to zero; therefore, the predominant phenomena is the crack propagation. Here, the curve is a line that almost fits the data, as imposed by the model adopted in order to estimate the finite life region. On the other hand, at stress level almost close to the STF fatigue limit (e.g., around 1000 MPa), there is the compresence of both phenomena. Indeed, the curve at 50% failure probability passes in between those points failed by crack propagation (on the left) and those points failed because of load higher than the fatigue limit. Finally, at stress level close to the fatigue limit, the number of cycle to failure tends to the infinite life.

Figure 10 shows a comparison between curves; one has been estimated with the adopted model, and the other has been obtained from literature approaches. All the curves are coherent with each other.

The curves named “double slope” have been estimated according to [17, 18]. The one named “FVA, STBF” has been obtained by estimating the finite life behaviour by fit of experimental data; on the other hand, its long life one has been estimated by means of a staircase procedure. “FVA, Gear” has been obtained applying Sthal’s corrective coefficient [39] on “FVA, STBF”. All the curves adopt in order to deal with the load difference effect.

Focusing on the limited life region of the STBF curves, results are similar. Furthermore, due to Equation (8) implication, in the region close to the STBF fatigue limit, the adopted model estimates a higher life because of the compresence of two failure mechanisms, one occurring at high number of cycles, one at lower number cycle. Indeed, both failure modes have a similar “weight.” On the other hand, in the STBF long-life region, both endurance limit and the fatigue limit have the same value. This implies that the selected runout level is higher enough to represent the infinite life phenomena.

However, in the gear finite life region, it is possible to notice a difference between the estimated gear life. This different prediction is due to the different estimated variance, which is estimated with a higher value by the adopted Loren model. On the other hand, in the gear long-life region, the predicted gear life is in between the two curves. The estimated gear fatigue limit at 1% probability of failure is slightly higher than the one predicted adopting the FVA coefficients. The reason behind is that the estimated variance is slightly lower than the ones adopted to calculate those coefficients.

#### 6. Conclusions

A statistical framework, through which estimating the gear tooth root bending fatigue SN curve starting from STFB test data, is proposed here. The estimation framework has been developed considering a symmetric Single Tooth bending Fatigue (STBF) test configuration, and it is based on the hypothesis of fatigue limit existence.

STBF test results have been elaborated by means of a Maximum Likelihood (ML) model based on Murakami’s idea of nonpropagating crack. Statistical consideration, based on statistic of extreme (SoE), has been adopted in order to estimate the gear SN curve.

The estimated curves have been compared with those obtained adopting other models. Results seem to be coherent. However, each estimation techniques have its own biases that result in a different estimation of the gear life.

It is worth underlying that a similar statistical framework can be developed in the case of an asymmetric STBF test configuration, where only one tooth is tested, by estimating the tooth SN curve with Equation (7) and using Equation (10).

#### Acronyms

CDF: | Cumulative density function |

ML: | Maximum Likelihood |

MG: | Meshing gear |

PDF: | Probability density function |

STBF: | Single Tooth Bending Fatigue |

SoE: | Statistic of extreme |

PSN: | Probability-Stress-Number (of cycle). |

#### Data Availability

The STBF test data supporting this article are from previously reported studies, which have been cited.

#### Conflicts of Interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

#### Acknowledgments

The authors express their gratitude to Professor Stefano Beretta for his invaluable support during the validation of this statistical model.