iKonPro®: A software for the probabilistic prediction of rolling contact fatigue

This work develops a software for the probabilistic prediction of rolling contact fatigue in multiple-row ball bearings subject to any loads and rotations. iKonPro®, as it has been named, calculates the failure probability of the raceways and the bearing, taking as input the material fatigue properties, contact-load and -angle distributions for discrete load cases, and the subsurface stress responses to different ball loads. The software is based on the determination of ball kinematics and equilibrium, raceways stress states, and local and global failure probabilities. Here, iKonPro® is applied to a wind-turbine pitch bearing and the results demonstrate that it correctly captures the size effect.


Introduction
Mechanical components, such as bearings, commonly suffer from Rolling Contact Fatigue (RCF).RCF produces a progressive material deterioration on both the raceways and the rolling elements, due to the variable stresses exerted through the contact [1].Eventually, RCF may lead to spalling, that is, the removal of material fragments by cracks that initiate sub-superficially and grow to the surface [1].In turn, spalling causes failures in the form of vibrations and noise [2], so reliable RCF prediction methodologies and tools are needed to guarantee a correct operation and to prevent failure.
There exist standards and guidelines [3,4] that are useful for approximately estimating the expected bearing performance, but rely on many assumptions and simplifications.Alternatively, recent works (e.g.Portugal et al. [5]) have proposed more accurate methodologies that rely on the determination of: 1) the temporal location of rolling elements, 2) the magnitude and orientation of the contact loads exerted by each rolling element at each time, 3) the stress signal at all the analyzed raceway points, and 4) the overall damage [5].These methodologies consider both the particular raceway material (through experimentally characterized SN curves) and the influence of surrounding flexible structures (if contact data is extracted by finite element -FEanalyses).However, these methodologies still present at least two shortcomings: 1) they do not include the effect of the lubrication and 2) they are in essence deterministic, because failure is quantified by Miner's rule.This work addresses the second shortcoming.
On the other hand, through the probabilistic approach, alternative methodologies have been presented.For example, Menck has proposed the Finite Segment Method [6], where the bearing is divided in angular segments, the survival (or failure) probability of each segment is obtained based on an equivalent orthogonal shear stress and its location, and the life of the whole bearing is determined by combining the segments' individual survival probabilities.From a more general perspective, the Generalized Local Model (GLM) [7] allows to quantify the global failure probability of a component, irrespective of: 1) failure mode, governing Generalized Parameter (GP) and spatial distribution of GP, and 2) component size and shape.GLM intrinsically accounts for the scale difference between the characterization specimen and the analyzed domain (superficial or volumetric), so it is convenient for the post-processing of FE analyses.GLM has been applied successfully to predict fracture (brittle and cleavage-dominated) and fatigue crack initiation at the surface [7], but has not been applied to predict internal fatigue crack initiation caused by RCF.This may be explained by the high complexity of the determination of the stresses at the different points of a raceway.
Thus, the objective of this work is to develop and show a software for the probabilistic prediction of rolling contact fatigue in bearings, by combining the most advanced models for stress determination in the raceways and the Generalized Local Model.Section 2 presents the developed software (registered as iKonPro®), and Section 3 shows the software by its application to a wind turbine's pitch bearing.Section 4 discusses and correlates the predictions, and finally Section 5 closes the paper with the concluding remarks.
2. iKonPro®: approach, usage and methodology 2.1.Approach and usage iKonPro® is a Matlab application conceived to calculate the RCF-induced failure probability of multiple-row ball bearings, by considering only the raceways and excluding the rolling elements.The software is executed through a 12-line long script that defines the analysis options and data input directories.In fact, to perform a successful calculation, various detailed inputs have to be provided (Figure 1), namely, 1) material's fatigue properties, 2) balls' contact loads and angles, and 3) subsurface stress responses for different ball loads.The generation of those inputs requires the characterization and post-processing of experimental results or the development and analysis of FE models: • The fitting parameters for Castillo-Canteli's material model may be conveniently determined by ProFatigue software [8], based on an experimental SN curve.• The balls' contact-loads and -angles for a range of external loads that encompass the various operation loads can be determined by FE.When building the FE model of the whole bearing, it is generally interesting to simplify the rolling elements by non-linear springs that replace the balls [9], to reduce the size of the model and to improve the convergence.• The subsurface stresses produced by a range of different load levels that encompass the ball-loads observed in operation can be also determined by FE.This FE approach requires to build a model of the ball-raceway contact detail.Alternatively, analytical formulations such as that of Boussinesq-Cerruti [10] can be employed.
7th International Conference of Engineering Against Failure Journal of Physics: Conference Series 2692 (2024) 012037

Methodology
In order to perform a RCF calculation, the raceways are discretized into smaller domains (Figure 2), and the software calculates each and every domain.Internally, iKonPro®'s methodology (Figure 3) is based on the resolution of three problems: 1) ball kinematics and equilibrium, 2) stress states in analyzed raceway domains, and 3) local and failure probabilities.The first two problems are addressed following Portugal's strategy [5], while the Generalized Local Model [7] is used for the last one.
Below, the methodology is explained step by step: • Calculate ball locations: the azimuth locations (φ) of the balls are determined for all time instants, by assuming pure rolling against both rings.One of the rings is fixed and the other one rotates according to the target pitch orientation, so the advance of the balls is half of the rotating ring's advance.• Calculate ball contact loads and angles: the direction and magnitude of the loads transmitted by the balls are determined for all time instants, while the direction of the loads defines the polar location (θ) of the balls.Such calculation is done by linear interpolation, operating with the mentioned inputs for the the distribution of contact loads and angles for different applied loads.Note that good lubrication is assumed, so tangential loads are neglected at the contact.• Calculate raceway domains' location relative to closest ball: the relative locations of the analyzed domains (considering the centroid) are determined for all time instants with respect to the ball which is nearest in each time.The relative azimuth and polar locations are computed as the difference between the balls' and domains' locations, whereas the depth is constant and does not depend on ball location.• Calculate governing stress: the governing stresses are determined for all raceway domains and time instants, by linear interpolation of the inputs for stresses at different relative locations (azimuth, polar and depth) under various ball-loads.For the time being, the orthogonal shear stress (the shear in the depth-rolling plane) is taken as the governing parameter [3], although critical-plane criteria may be included in the future.
, where: • Calculate the raceway and bearing failure probability: finally, the global failure probability of the bearing (or the raceway, or a segment of the raceway) is computed based on the failure probabilities of the domains comprised, by the weakest link principle:

Results for an illustrative case
For demonstrative purposes, iKonPro® is employed for the prediction of RCF in a pitch bearing used for orienting wind turbine blades.The considered model is a four-point contact ball bearing made of 42CrMo4, with a pitch diameter of 2310 mm and two rows of rolling elements.Each row contains 102 balls (diameter = 65 mm, osculation = 0.53).The bearing is subject to a constant axial load of 8100 kN and the outer ring is oscillated following an alternating signal.The analyzed operating conditions (Table 1) combine different oscillation ranges ([3.2 • , 5.2 • , 60 • ]) and numbers of cycles ([60, 600, 600000]).Note that the smallest (3.2 • ) and intermediate (5.2 • ) oscillations allow, respectively, the balls to sweep 1) their initial region of influence and 2) the whole segment of ring corresponding to each ball.1).For all conditions, the damaged zone (P f,dom > 0) only exists in raceways that participate in the diagonal 1 load transmission (lower-left and upper-right).Such damaged zone is observed to be very confined to the load application location in the polar direction (θ), whereas its extension in the azimuth direction (φ) depends on the oscillation range.The most loaded region is always located slightly beneath the surface (only visible for 60 • oscillations).
As the number of cycles increases, the failure probability accumulates and the criticality increases, while the damaged zone gets bigger in the polar direction.As the oscillation range increases, the damaged zone gets successively bigger in the azimuth direction, comprehending the whole segment length for 60 • oscillations.
Table 2 also quantifies the maximum domain failure probability, which is found under all operating conditions at domain A (depth = 0.75 mm, φ = 0.06 • , θ = 49.19 • ).As observed, the values are identical for oscillation ranges of 3.2 • and 5.2 • , and significantly higher for 60 • oscillations.For example, the failure probability of domain A is predicted to be 0.0097 (0.97%) after 600,000 cycles of 60 • oscillations.
3.1.2.Segment (global) failure probabilities: Table 3 displays the failure probabilities of the segment (bearing sector corresponding to the first ball) under the analyzed operating conditions.Generally speaking, the increase of either the oscillation ranges or cycles produces an increase of the segment's failure probability, up to the saturated value of 1 for 600,000 cycles (regardless of the oscillation range).Contrary to the trend in local terms, the global failure probability of the segment is higher for higher oscillation ranges, even comparing 3.2 • and 5.2 • oscillations.Thus, the software correctly captures the size effect, that is, the growing failure probability produced by a damage zone that increases uniformly in the azimuth direction.

Discussion of the size effect based on domain stresses
This section explains the size effect based on the stresses determined at different representative domains (Table 4).The three domains considered (A, B, C) are located at the critical polar location (49.19 • ) and depth (0.75 mm), but at different azimuth locations (0.06 • , 1.03 • and 1.53 • , respectively).
Under the smallest oscillation (3.2 • ), domain A undergoes two cycles of orthogonal shear stress, each one of them with an amplitude of approximately 600 MPa.It starts and and ends with a high level of stress, which indicates that domain A is located within the ball's initial region of influence.On the contrary, domain B only suffers stresses considerably lower than the endurance limit of the material, and domain C is not even affected by the ball.Under the intermediate oscillation (5.2 • ), both domain A and B undergo two cycles of ±600 MPa at different moments of the oscillation, given their different azimuth locations.In contrast, domain C is barely affected by the ball contact.Comparing 3.2 • and 5.2 • oscillations, the maximum local failure probabilities are the same, because the local stress ranges are almost identical at the affected domains.However, for 5.2 • oscillations, the greater number of affected domains makes the overall failure probability of the segment also greater.
Under the large oscillation (60 • ), the three domains undergo several ±600 MPa cycles.In particular, domain A, B and C are subject to respectively 18, 17 and 16 cycles (fewer as further from ball's initial location).For 60 • oscillations, given the higher number of effective stress cycles, both the maximum local and global failure probabilities increase comparing to the other two operating conditions.

Correlation of failure results
Figure 4 shows a post portem photograph of a raceway after the experiment conducted under OP9 conditions (600,000 cycles of 60 • oscillations).The damaged zone generated by spalling is found to be huge, centered at the raceway in the polar direction (θ ≈ 45 predictions, although the depth of the rut is higher than that of the location with maximum local failure probability (0.75 mm).This discrepancy in depth may be explained by the additional erosive wear produced by the metallic particles that were detached at the real onset of spalling.In fact, the software predicts the failure of each bearing segment already by the time that 600 cycles are completed (Table 3).

Conclusions
This work develops a software for the probabilistic prediction of rolling contact fatigue in multiple-row ball bearings subject to any loads and rotations.iKonPro®, as it has been named, calculates the failure probability of the raceways and the bearing, taking as input the material fatigue properties, contact-load and -angle distributions discrete load cases, and the subsurface stress responses to different ball loads.The software has been shown by its application to a wind turbine's pitch bearing, concluding that: • The software correctly captures the size effect, because the failure probability is predicted to increase as the axysimmetrically loaded bearing is subject to bigger oscillations.• The predictions of the software are consistent with the experimental observations of a highly loaded bearing.Anyways, a probabilistic software like iKonPro® requires wider correlation, involving multiple bearing types and loads.

Figure 2 .
Figure 2. Illustration of how to discretize raceways.

•
Calculate stress ranges and cycles: the stress time-series of each raceway-domain is reduced to an equivalent set of constant amplitude stress reversals, by a built-in Matlab algorithm for Rainflow counting.• Calculate local failure probabilities: the local failure probabilities of all the raceway domains are determined by Equation (1), based on the volumes of the analyzed domain (S dom ) and the characterization specimen (S ref ), material properties (β, δ ref and λ), and parameter V .Parameter V is a normalized quantification of the damage produced by the shear-stress level (S xz ) and number of cycles (N ).

Table 1 .
Table 2 displays, in qualitative terms, the distributions of domain failure probabilities through the row R1 segment, under the analyzed List of analyzed operating conditions.