The existing engineering empirical life analysis models are not capable of considering the constitutive behavior of materials under contact loads; as a consequence, these methods may not be accurate to predict fatigue lives of rolling bearings. In addition, the contact stress of bearing in operation is cyclically pulsating, it also means that the bearing undergo non-symmetrical fatigue loadings. Since the mean stress has great effects on fatigue life, in this work, a novel fatigue life prediction model based on the modified SWT mean stress correction is proposed as a basis of which to estimate the fatigue life of rolling bearings, in which, takes sensitivity of materials and mean stress into account. A compensation factor is introduced to overcome the inaccurate predictions resulted from the Smith, Watson, and Topper (SWT) model that considers the mean stress effect and sensitivity while assuming the sensitivity coefficient of all materials to be 0.5. Moreover, the validation of the model is finalized by several practical experimental data and the comparison to the conventional SWT model. The results show the better performance of the proposed model, especially in the accuracy than the existing SWT model. This research will shed light on a new direction for predicting the fatigue life of rolling bearings.
1 Introduction
Complicated rotational machines and systems like machine tools [1, 2], wind turbines [3, 4], and robots [5, 6] having emerged during the last several decades. Rolling bearings are basic parts of such equipment and systems. Today’s complicated rotational machines and systems require rolling bearings work in harsh conditions, such as heavy load, high speed and high temperature, which brings about challenges to several crucial performances of rolling bearings like reliability and service life.
The fatigue life models of rolling bearings can be divided into three categories: engineering models, condition monitoring based models, and theoretical models. Engineering models including, not limited to, Lundberg–Palmgren (L–P) model, Ioannides–Harris (I–H) model, and Zaretsky model [7‐10]. Currently, condition monitoring-based life predictions tend to be a hot topic. For instance, Yakout [11] predicted the fatigue life of rolling elements according to vibration data. Wang et al. [12] proposed mixed effects models for fault prognostics of rolling element bearings. The models are able to simultaneously model different degradation process phases of rolling bearings. Cui et al. [13] established a Switching Unscented Kalman Filter (SKF) method for remaining useful life prediction of rolling bearings, and the effectiveness of the method is shown by comparing with the traditional SKF algorithm. Ahmad et al. [14] introduced a reliable technique for the health prognosis of rolling element bearings, which infers a bearing's health state through a dimensionless health indicator (HI) and estimates its remaining useful life (RUL) using dynamic regression models. Wang et al. [15] proposed a life prediction method for industrial rolling bearings based on state recognition and similarity analysis, which provides some theoretical guidance and basis for the safe operation and maintenance of rolling bearings.
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The mentioned models are mainly based on artificial intelligence techniques and statistical regression methods, and those require sufficient experimental data for model training. Furthermore, the above methods cannot describe the failure mechanism in the process of contact fatigue failure well. Accordingly, theoretical models based on the principles of mechanics are proposed. Warda et al. [16] introduced a fatigue life prediction method of radial cylindrical roller bearings, in which the influence of bearing geometric parameters were considered. Shi et al. [17] presented a relative fatigue life calculation method considering surface texture on high-speed and heavy-load ball bearing. Yang et al. [18] discussed the mechanical properties of double-row tapered roller bearings through expanding the mathematical model of three degrees of freedom, and analysed the contact load and fatigue life of bearings under different loads. Quagliato et al. [19] predicted the life of roller bearings by accelerated testing, and finite element (FE) models, was developed for crossed and tapered roller bearings. He et al. [20, 21] proposed a method to test the accelerated fatigue life using a small sample test, which provides an experimental reference for bearing design, and also provides the foundation for studies of the fatigue failure mechanisms of raceways.
Rolling bearings are subjected to alternating loads during operation. The load amplitude and mean stress continuously change with different working conditions. It is a consensus that the mean stress indeed affects the fatigue life of rolling bearings [22‐27]. For instance, Barbosa et al. [28] proposed an artificial neural network method considering the influence of mean stress on the fatigue life of metallic materials, which can estimate the safety region for high-cycle fatigue regimes. Zhang et al. [29] analyzed the influence of mean stress and phase angle on multiaxial fatigue behavior of TiAl alloy and established a life model with multiple loading variables, and the proposed method is of good accuracy compared with Matake method and McDiarmid method. Benedetti et al. [30] developed a new fatigue criterion based on strain-energy-density (SED) to illustrate the influence of mean stress and plasticity on the uniaxial fatigue strength. Kalombo et al. [31] used an artificial neural network to predict the fatigue life of an all-aluminum alloy 1055 MCM conductor, in which mean stresses are modeled. Li et al. [32] established a new fatigue model based on the effect of mean stress on high-cycle fatigue performance; the proposed model's prediction is closer to fatigue test data via comparing with Goodman, Gerber, Morrow, Soderberg, and Elliptic (ASME) models. Laszlo et al. [33] introduced a numerical fatigue assessment method for composite plates, which considered mean stress correction and multiaxial fatigue failure criterion inspection. Duan et al. [34] presented a fatigue life prediction method considering shrinkage cavity, secondary dendrite arm spacing (SDAS), and mean stress, which lays a solid foundation of the optimization design and lightweight design of aluminum alloy wheels. Rolling contact fatigue is a common failure mode of rolling bearings [35, 36]. The contact load and stress at each contact point of the bearing are cyclically pulsating and belong to asymmetric cyclic load. Therefore, the influence of mean stress needs to be considered when predicting the contact fatigue life of rolling bearings. Compared with the above mean stress correction models, the SWT model with a simple form and does not require additional material parameters. Moreover, it can reflect the sensitivity of the material. Hence, in this paper, we attempt to consider the sensitivity of different materials to mean stress and propose a modified life prediction model based on the SWT correction.
The rest of this paper is organized as follows. Section 2 proposes a life prediction model based on modified SWT correction that considers the mean stress effect and sensitivity. Section 3 develops the fatigue life prediction model of rolling bearings. Section 4 performs model validation using the experimental data of GH4133, 1Cr11Ni2W2MoV, and GCr15, and verifies the applicability of the proposed model. In Section 5, conclusions are drawn.
2 Modified Fatigue Life Prediction Model Based on Mean Stress Correction
2.1 Stress-life Based Prediction Methods
The fatigue life of rolling bearings can be evaluated using stress-life prediction methods that are theoretically based on S-N curve. The stress-life prediction method can be expressed by the Basquin formula, as:
$$\sigma N_{f}^{ - b} = A,$$
(1)
where Nf represents fatigue life, A denotes the fatigue strength constant, which is an inherent property of the material, b is the material constant.
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The specimen can withstand countless stress cycles without breaking under stress lower than a certain critical stress amplitude, and the fatigue life tends to be infinite. However, the Basquin formula fails to reflect the fatigue limit and its influence on fatigue. The full stress-fatigue life curve is shown in Figure 1. Considering the influence of the fatigue limit, the relationship between fatigue life, fatigue limit stress and stress range is established by Weibull [37], as:
where σac is the endurance-limit stress, Cf and β are material constants determined by experiments.
Figure 1
The full stress-fatigue life curve
×
Rolling bearings are subject to asymmetric cyclic loads during operation, see Figure 2. The load amplitude and mean stress are decisive factors of the fatigue life of rolling bearings. It should be pointed out that the Weibull formula is able to fit the fatigue life test results under a certain stress ratio or mean stress and not functioning in showing the effect of stress ratio or mean stress on fatigue life.
Figure 2
The asymmetric cyclic load
×
Walker correction reflects the mean stress sensitivity of materials by introducing the mean stress sensitivity coefficient γ, which can effectively improve the prediction accuracy [38], see Eq. (3). Moreover, Smith, Watson, and Topper [39] put forward a simple form of mean stress correction see Eq. (4) and which is used to modify Weibull formula to obtain the fatigue life prediction model considering the mean stress effect, see Eq. (5), as:
where σar is the equivalent stress amplitude, σmax is maximum stress, σa is stress amplitude, R denotes stress ratio. γ denotes the mean stress sensitivity coefficient, and its value is between [0 1], the larger γ, the less sensitive the material to mean stress, and vice versa. C0 and β0 are material constants.
2.2 The Proposed Model
SWT correction is a special form of Walker correction, and it determines the sensitivity of different materials to the mean stress to be 0.5, that is γ = 0.5. Dowling [40] found that γ is related to fatigue performance parameter yield limit of materials through a large number of fatigue tests of metal materials (alloys and steels), in detail, γ decreases as the yield limit increases. A compensation factor α is introduced to consider the sensitivity of the material to the mean stress. Substituting α into Eq. (5), a modified fatigue life prediction model based on SWT criterion is proposed, as:
where C1 and β1 are material constants. σb indicates the yield limit, σ0 represents the yield limit of similar materials when γ = 0.5.
3 Fatigue Life Prediction of Rolling Bearings Based on the Proposed Model
3.1 Load and Stress Distribution of Rolling Bearings
The main performance parameters of rolling bearings, including deformation, contact stress between rolling elements and rings, stiffness, and fatigue life, can only be calculated after the load distribution has been determined.
The load acting on the bearing is transmitted from one ring to the other through the rolling elements, so the bearing capacity is determined by the rolling element load. The rolling bearing under the action of radial load is shown in Figure 3. The load distribution of the rolling elements can be computed as [41]:
where Ψ denotes angular position, QΨ is the rolling element load at an angle Ψ. Qmax is the maximum rolling element load, and T demonstrates the load zone parameter.
Figure 3
Force distribution of rolling bearings under the radial load
where Z demonstrates the number of rolling elements. Ψ0 denotes the bearing range and cosΨ0 = 1‒2T.
When the bearing rotates at a low speed, the influence of centrifugal force Fc is small and can be ignored. At this time, the contact load Qij and Qoj are equal. Conversely, the radial force balance equation of the rolling element considering the centrifugal force Fc can be expressed as:
where Qij and Qoj represent the contact load between the inner ring and the rolling elements we well as between the outer ring and the rolling elements. Dm is the bearing circle diameter, Db is the rolling element diameter, ρ denotes the density of the rolling elements, ωb denotes the revolution angular velocity of the rolling element.
For a bearing, when two curved objects are pressed against each other under a load, a certain contact zone is generated at the contact point. Since the rolling contact between the raceway and rolling elements is a curved body, the Hertz elastomer contact theory can be used to calculate the contact stress and deformation in the rolling bearing.
For a ball bearing, the contact zone between a ball and a raceway is elliptical, according to the Hertz contact theory. The semi-major axis of the ellipse is represented by a, and the semi-minor axis is represented by b, as shown in Figure 4. The contact stress in the contact zone is distributed as an ellipsoid, see Figure 5. For a contact load Q, the contact stress at any point (x, y) in the contact zone can be expressed as follows [42]:
where σmax represents the maximum contact stress, μ and ν are elliptic integrals related to the curvature function F(ρ), E and λ are the elastic modulus and Poisson’s ratio of the material, respectively, and Q denotes the rolling element load. Σρ demonstrates the sum of the principal curvatures at the contact.
3.2 Fatigue Life Prediction Model of Rolling Bearings
The life prediction model proposed by Lundberg and Palmgren (L-P model) is used to predict the fatigue life of rolling bearings. But it cannot take the microstructure of the material into account, which restricts the universality of the model. However, it should be pointed out that the fatigue life prediction method proposed in this paper is based on the S-N curve of the material, which can describe the fatigue characteristics of the material well.
The number of stress cycles computed by Eq. (6) refers to the number of times that a certain point on the raceway of a bearing is subject to stress within a certain number of rotations. The contact load and stress of ring and rolling elements change when elements pass through different points in the load zone. The contact load and stress distribution at each point of the rotating ring are shown in Figure 6. The load and stress of each point of the stationary ring are unequal, and the contact load and stress at each load point show the same characteristics of pulsation cycle, but the amplitude values are different. The contact load and stress distribution at each point of the stationary ring are shown in Figure 7.
Figure 6
Contact load and stress distributions at each point of the rotating ring
Figure 7
Contact load and stress distributions at each point of the stationary ring
×
×
According to the contact stress distribution and the proposed modified model, the life (stress cycle number) of a bearing corresponding to each contact point can be calculated. The life unit of a rolling bearing is expressed in revolutions; hence, the number of stress cycles needs to be converted into revolutions.
When the inner ring rotates and the outer ring is stationary. Under the radial load, the contact stress of the outer ring raceway is the largest at the 0° angular position of the load zone, which is the place where the outer ring is most likely to fail. The number of stress cycles and the life of the outer ring at this point can be expressed as Eq. (17). The amplitude of the stress on a point on the inner ring raceway is periodically changing due to the rotation of the inner ring. According to Miner's damage accumulation theory, when the inner ring rotates for one revolution, the damage to the inner ring can be expressed as Eq. (18), and the life of the inner ring can be expressed as Eq. (19):
where Li and Le represent the life of the inner ring and outer ring, respectively. Ne denotes the number of stress cycles at the maximum stress position on the outer raceway, and Nij denotes the number of stress cycles of the inner raceway under the contact stress σj. ui and ue are the numbers of rolling elements passing through a certain point of the inner and outer rings when the inner ring rotates one revolution, which need to be obtained through the analysis of the kinematic characteristics of the bearing.
For a ball bearing with an outer ring speed of ne and an inner ring speed of ni, the contact angle is zero under pure radial force. When the cage rotates once concerning the inner or outer ring, Z rolling elements are passing through a certain point of the inner or outer ring. ui and ue can be expressed as Eq. (20) and Eq. (21), respectively.
$$u_{i} = \frac{{Zn_{ci} }}{{n_{i} }},$$
(20)
$$u_{e} = \frac{{Zn_{ec} }}{{n_{i} }},$$
(21)
where nci denotes the rotation speed of the cage relative to the inner ring, see Eqs. (22) and (23). nec denotes the rotation speed of the outer ring relative to the cage [43], see Eq. (24):
where nc is the rotation speed of the center of rolling element. Db is the diameter of the rolling element, and Dm is the mean diameter of the bearing.
Substituting Eqs. (20)−(24) into Eqs. (17)−(19), the fatigue life of the inner and outer rings of a rolling bearing can be obtained. Then, the overall life of ball bearing is computed by Eq. (25):
4.1 Validation of Modified Fatigue Life Prediction Model Based on SWT Correction
In this section, fatigue experimental data of materials GH4133, 1Cr11Ni2W2MoV, and GCr15 under different stress ratios are employed. GH4133 is a kind of alloy with high strength as well as good thermal stability and corrosion resistance. It is often used to manufacture turbine blades, turbine disks, and other aero-engine components. 1Cr11Ni2W2MoV is a kind of steel with good mechanical properties and has been widely used to manufacture engine discs and shafts that work below 600 °C. GCr15 is a high-carbon chromium bearing steel, which is a common material of rolling bearings. The properties of GH41331, Cr11Ni2W2MoV, and GCr15 are provided in Table 1. The life experimental data of GH4133 under R = 0.44 are listed in Table 2 [44]. Table 3 shows the life experimental data of 1Cr11Ni2W2MoV under the condition of R = −1 [44]. Tables 4 and 5 respectively show the life data of the contact fatigue test at R = 0 and the life data of the torsion fatigue test at R = −1 of GCr15 [45, 46].
Table 1
Material properties of GH4133, 1Cr11Ni2W2MoV, and GCr15
Material
E
(GPa)
Poisson’s
ratio
σb
(MPa)
Density
(g/cm3)
GH4133
223
0.36
878
8.21
1Cr11Ni2W2MoV
180
0.277
979
7.8
GCr15
207
0.3
1617
7.81
Table 2
Experimental data for GH4133 at R = 0.44
σmax (MPa)
Nt (cycles)
935
2535
933
2191
905
3026
905
2930
887
2508
865
2879
863
3720
862
4276
841
5320
806
5610
777
8461
756
17710
749
12855
744
12141
741
14352
Table 3
Experimental data for 1Cr11Ni2W2MoV at R = −1
σmax (MPa)
Nt (cycles)
1183
2590
1083
2912
987
4078
919
9402
839
23440
817
26124
Table 4
Contact fatigue experimental data for GCr15 at R = 0
σmax (MPa)
Nt (cycles)
5000
1.7629×107
5300
1.2790×107
5600
1.1836×107
5900
0.8910×107
6200
0.3655×107
Table 5
Torsion experimental data for GCr15 at R = −1
σmax (MPa)
Nt (cycles)
500
6.62×106
630
1.09×106
760
3.08×105
800
3.11×105
950
5.2×104
1000
3.78×104
The fatigue life of the three materials is calculated using the proposed model. In the meantime, the predicted life of the materials under the corresponding stress ratio conditions are also calculated using SWT model for comparison. The correlations of the materials’ tested lives and predicted lives derived from different models are shown in the Figures 8, 9, 10 and 11, respectively.
Figure 8
Predicted lives vs. tested lives of GH4133
×
In the figures, the solid black line indicates that the tested lives are exactly the same as the predicted lives. The red dashed lines indicate a life factor of ±2 to the tested lives. The closer predicted lives to the solid black line indicates the more accurate result.
From Figure 8, it is found that, for GH4133, nearly all predicted lives of the proposed model are within the ±2 scatter band, but the results predicted by the proposed model are obviously closer to the solid black line. This suggests that in comparison to SWT model, the proposed model is more reliable and accurate in predicting the fatigue life of the alloys.
From Figure 9, it is found that for 1Cr11Ni2W2MoV, it can be clearly observed that the prediction results of the proposed model are more consistent with test data than that of the SWT model. All predicted lives of the proposed model fall within a life factor of ±2 to the tested lives, which can be compared with that some predictions of the SWT model exceed the life factor of ±2. Similar conclusions are true to the GCr15, see in Figures 10 and 11.
Figure 9
Predicted lives vs. the tested lives of 1Cr11Ni2W2MoV
Figure 10
Predicted lives vs. tested lives of GCr15 at contact test
Figure 11
Predicted lives vs. tested lives of GCr15 at torsion test
×
×
×
In order to better demonstrate the advantages of the proposed model in predicting the fatigue life of materials, the predicted life deviation between the logarithmic predicted life and experimental life is used to describe the prediction error, see Eq. (26) [47, 48]. In detail, the mean value and the standard deviation of Perror respectively represent the precision and concentration of the model prediction. The smaller the mean value and standard deviation, the more accurate and reliable the prediction are
$$P_{error} = \lg (N_{p} ) - \lg (N_{t} ).$$
(26)
The prediction error of the proposed model and SWT model on the experimental data are shown in Figures 12, 13, 14 and 15. Moreover, with Eq. (26), the mean value and the standard deviation of Perror using the two models for the three materials are depicted in Table 6.
Figure 12
Prediction error of GH4133
Figure 13
Prediction error of 1Cr11Ni2W2MoV
Figure 14
Prediction error of GCr15 under contact test
Figure 15
Prediction error of GCr15 under torsion test
Table 6
Statistical analysis of model prediction errors
Material
Statistical parameters
Proposed model
SWT
1Cr11Ni2W2MoV
Mean
0.0098
0.2880
Standard deviation
0.0860
0.1113
GH4133
Mean
0.0280
0.1976
Standard deviation
0.0746
0.0758
GCr15 (contact)
Mean
0.0230
0.1613
Standard deviation
0.1163
0.1176
GCr15 (torsion)
Mean
0.0865
0.2711
Standard deviation
0.1288
0.1436
×
×
×
×
Overall, it is obviously seen from Table 6 that both the mean value and standard deviation of Perror of the proposed model are the smallest among the results of SWT model. This implies that the proposed model is better among the two models in predicting the fatigue life for alloys and steels.
4.2 Life Prediction of the Proposed Model for Rolling Bearings
In this section, a deep groove ball bearing (code 6206), made by GCr15, is used in the experiment, see Table 7. Under the conditions of radial force of 5 kN and a rotation speed of 12000 r/min, the distribution of rolling element load is obtained through quasi-static analysis. Then, according to the Hertz contact theory, the contact stress of the rolling bearing is analyzed.
Table 7
Specification of the tested bearing
Parameter
Value
Ball nominal diameter Db (mm)
9.525
Pitch diameter of the bearing Dm (mm)
46
Coefficient of inner raceway groove curvature radius fi
0.515
Coefficient of outer raceway groove curvature radius fe
0.52
Number of balls Z
9
The rolling element load at different angular positions is shown in Figure 16, and the contact stress of inner raceway distribution at different angular positions is shown in Figure 17. In detail, at the current conditions, the maximum rolling element load Qmax is 2509 N, the maximum contact stress of the inner raceway and the outer raceway, respectively, are 3102 MPa and 2636 MPa.
Figure 16
Rolling element load at the different angular positions
Figure 17
Contact stress distribution of inner raceway at the different angular positions
×
×
According to Eqs. (17)‒(19), the fatigue lives of inner ring and outer ring are calculated. Finally, through Eq. (25), the predicted life of rolling bearing is obtained.
Likewise, using the proposed model and SWT model, the predicted life of the bearing is calculated and compared. The comparison between the bearing’s tested lives and predicted ones derived from different models are shown in Figure 18 [49].
Figure 18
Predicted life vs. the tested life of the rolling bearing
×
From Figure 18, it is worth mentioning that two predictions computed from the SWT model fall within a life factor of ±2 to the tested lives, and one is outside. However, all predictions of the proposed model fall within a life factor of ±2 to the tested lives. Similarly, a statistical analysis is conducted, and its mean value and the standard deviation of Perror is shown in Figure 19 and Table 8.
Figure 19
Prediction error of rolling bearing
Table 8
Statistical analysis of model prediction errors
Statistical parameters
Proposed model
SWT
Mean
0.0311
0.3052
Standard deviation
0.2215
0.2215
×
As it can be seen in Figure 19 and Table 8, the proposed model with a smaller mean value of Perror provides better predictions of bearing than the conventional SWT model. In general, through the proposed model, the fatigue life of rolling bearings made of steels can be predicted with higher precision and simple calculation process.
5 Conclusions
In this paper, a modified fatigue life model based on the SWT correction that considers the mean stress effect and sensitivity is proposed to predict fatigue life of bearings. Experimental data of materials is used to verify the proposed method. The conclusions are drawn as follows:
(1)
Considering the sensitivity of different materials to the mean stress, a modified model is established based on the SWT correction to estimate the fatigue life. The experimental data of three materials: GH4133, 1Cr11Ni2W2MoV, and GCr15 are used for model verification. Compared with the SWT model, the applicability of the proposed model is better, and the life prediction results of the three materials are more accurate, which shows that the proposed model is suitable for life prediction of alloys and steels.
(2)
Based on the proposed model, the life prediction model of rolling bearings is established. Moreover, by using the proposed model and the SWT model, the fatigue life of ball bearing is predicted under the working conditions on 5 kN radial force and 12000 r/min, and then the predicted results are compared with the tested ones. The comparison shows that the proposed model is consistent to the experimental results, and it is feasible to apply the proposed model to the fatigue life prediction of rolling bearings.
Acknowledgements
Not applicable.
Competing interests
The authors declare no competing financial interests.
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