1 Introduction

The automotive wheel bearing is an integral component of automobiles that distributes the weight of the vehicle while aiding the rotation of the wheels (Fig. 1). Wheel bearings have been used to reduce vehicle weight since the development of front-wheel-drive vehicles in the 1960s. The structure of wheel bearings has evolved from Generation 1 (Gen. 1), which comprises a single bearing, to Gen. 2 and Gen. 3, which have a complex structure comprising a flange attached to the hub and an outer ring (Fig. 2). The need for accurate life prediction and evaluation techniques arose as bearings became lighter, smaller, and more complex. Recently, the demand for lighter and smaller automobiles has increased because of fuel efficiency and CO2 emission regulations; consequently, the need for evaluating wheel bearing life has grown [1, 2].

Fig. 1
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Suspension system and automotive wheel bearing

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Fig. 2
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Examples of automotive wheel bearings: Gen. 1, Gen. 2, and Gen. 3 [3]

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The life of automotive wheel bearings can be categorized into the raceway and flange lives. The raceway life is based on the spalling occurring in the raceway and can be determined analytically according to ISO 281:2007 [4] using the calculation method for the rolling bearing basic rating life proposed by Lundberg and Palmgren [1, 5, 6]. The flange life is based on the failure occurring in the wheel bearing stress concentration area containing the flange and can be numerically predicted by fatigue analysis using the finite element method [7,8,9]. Lee et al. [7] performed an analysis of the failure of high-speed rotating cylindrical vessels based on fracture mechanics. However, the rotating stress for the high-speed rotating cylindrical vessels was calculated by an analytical formula. Jeon et al. [8] evaluated the durability of a torsion beam axle. By replacing the structural analysis process of the quasi-static fatigue analysis with a multi-body dynamics analysis, the analysis time was shortened while maintaining reliability. Lee et al. [9] performed the fatigue life prediction of bellows under random loading, such as high frequency, using the vibration fatigue analysis technique. In the reported studies [7,8,9], the rotation of the structure was not numerically considered in the durability analysis. However, the rotation of rotational structures, such as bearings, must be considered for accurate analysis results.

In this study, we examined the fatigue analysis of the automotive wheel bearing flange considering rotation. Generally, fatigue analysis can be divided into two stages: stress and fatigue analyses. We conducted the stress analysis using MSC.Marc [10] and fatigue analysis using FEMFAT [11]. To obtain the required fatigue material properties, a rotary bending fatigue test was performed. We calculated the stress histories of the heat treated and non-heat treated regions by simulating the effect of rotation. To verify the reliability of the analysis, an endurance test was performed and the test and analysis results were compared.

2 Stress Analysis

The structure of the wheel bearing can be divided into the hub, inner ring, outer ring, and rolling elements as shown in Fig. 3. As the raceway has rolling contact with the rolling elements, it requires a higher hardness than the other parts and is typically heat-treated to enhance hardness. We modeled the respective heat and non-heat treated regions of the hub and outer ring as sperate parts to facilitate analysis. Figure 3 shows the geometry and configuration of the modeled bearing.

Fig. 3
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Geometry and configuration of modeled bearing

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We used element 7 (a 3-dimensional arbitrarily distorted brick element) and element 134 (a 3-dimensional four-node tetrahedral element) of MSC.Marc to conduct the stress analysis. Elements 7 and 134 have eight and four nodes per element, respectively, and three degrees of freedom at each node. Hypermesh [12] was used to create the mesh, and the results are shown in Fig. 4. There were 73,728 nodes (54,182 and 19,546 for elements 7 and 134, respectively) and 268,522 elements (164,360 and 104,162 for elements 7 and 134, respectively). The material properties are shown in Table 1.

Fig. 4
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Generated mesh

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Table 1 Material properties
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The boundary and loading conditions are shown in Fig. 5. All the degrees of freedom of the hub bolts were constrained. From the vehicle information, wheel bearing loads, the axial force, Fa and radial force, Fr were obtained and applied to the outer ring bolts using the rigid body elements in MSC.Marc. Contact was simulated for the areas in contact between the hub and inner ring. Convergence of the analysis could not be guaranteed when the contact between the rolling elements and the raceways was modelled. However, replacing the rolling elements with spring elements can guarantee the convergence and be more efficient because of an associated reduction in the analysis time. The spring stiffness can be calculated by bearing theory, and a nonlinear compression-only spring was used as the rolling elements transmit loads only in compression [13,14,15,16].

Fig. 5
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Boundary and loading conditions

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Figure 6 shows the stress analysis result. The main failure regions are indicated by the labels A, B, and C, and the maximum principal stresses for each region were 721 MPa, − 23 MPa, and 301 MPa, respectively. Point A is located in a heat treated region, whereas points B and C are located in non-heat treated regions.

Fig. 6
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Maximum principal stress results of the stress analysis

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3 Fatigue Analysis

As fatigue material properties are required to perform the fatigue analysis, we performed a rotary bending fatigue test on the bearing materials to obtain these properties. The H-7 rotary bending fatigue testing machine from Shimadzu [17] was used for this test, and the test specimen was fabricated according to KS B ISO 1143 [18]. The test was performed at a stress ratio of R =  − 1, fully reversed loading, 300 RPM, and room temperature. Figure 7 shows the resulting S–N curves obtained from the test.

Fig. 7
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S–N curves for 1055 and 1055 hardened materials

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Rotation should be considered in the fatigue analysis to accurately predict the bearing life. As the hub and inner ring rotate inside the bearing and the load is applied to the outer ring, the rotation of the hub and inner ring must be considered. From the results shown in Fig. 6, only the hub was considered because the inner ring does not exhibit a main failure region, and the stress in the inner ring is low. As the hub was symmetrical at a pitch of 36°, the geometries for conducting the stress analysis were modeled by rotating them at intervals of 36° for a total of 10 cases. Figure 8 shows the models for Cases 1 and 2.

Fig. 8
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Models for fatigue analysis, Case 1 is the geometry at 0° and the hub and inner ring are rotated by 36° for Case 2

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As the hardness of specific regions has been increased by heat treatment, fatigue analysis was performed by dividing the model into heat treated and non-heat treated regions, and the stress history of each region was analyzed.

3.1 Stress History of Heat Treated Region

The maximum principal stress for Case 1 was 721 MPa at the point A1, as shown in Fig. 9a, corresponding to point A in Fig. 6. To obtain the stress history at point A as affected by rotation, the stress at A2 was obtained for Case 2, where the point A1 was rotated by 36°. When the stress was obtained in the same manner for Cases 3–10, the stress history at point A was obtained, as shown in Fig. 9b. The maximum stress is 721 MPa, and the minimum stress is − 2 MPa.

Fig. 9
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Maximum principal stress history of point A in the heat treated region of the hub

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To obtain the stress history for Cases 2–10, the same procedure was followed as detailed for Case 1. The maximum principal stresses for Cases 2–10 were obtained, and the stress histories corresponding to the different rotational starting angles were obtained. The maximum principal stresses were identical and shifted by a pitch of 36° in all 10 cases. Moreover, in each case, the location of the maximum principal stress was identical to point A. Figure 10 shows the stress histories as affected by the different rotation starting angles of Cases 1, 2, and 3. Cases 4–10 are omitted because they are identical to Cases 1, 2, and 3 and separated by a pitch of 36°.

Fig. 10
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Maximum principal stress histories of point A in the heat treated region of the hub for the different starting rotation angles of Cases 1, 2, and 3

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3.2 Stress History of Non-heat Treated Region

To obtain the stress history for the non-heat treated region, the same procedure was followed as detailed for the heat treated region. The maximum principal stresses for Cases 1–10 were obtained, and the stress history for point B was obtained. Unlike the results for the heat treated region, two different stress histories were obtained. One history is presented for Cases 1, 3, 5, 7, and 9, and another for Cases 2, 4, 6, 8, and 10.

The maximum principal stress for Case 1 was − 23 MPa at the point B1, as shown in Fig. 11a, corresponding to point B in Fig. 6. The stress history of point B is shown in Fig. 11b for the starting rotation angles corresponding to the odd-numbered cases. The maximum stress is 310 MPa, and the minimum stress is − 23 MPa.

Fig. 11
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Maximum principal stress history of point B in the non-heat treated region of the hub

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The maximum principal stress for Case 2 was − 9 MPa at point B2, as shown in Fig. 12a. The stress history of point B is shown in Fig. 12b for the starting rotation angles corresponding to the even-numbered cases. The maximum stress is 306 MPa, and the minimum stress is − 9 MPa. Comparison of the stress histories of Cases 1 and 2 shows that the stresses for Case 1 are slightly higher than those for Case 2. The reason is that the maximum principal stress for Case 1 is located at the center of the flange reinforcement at 180°, whereas the maximum principal stress for Case 2 is located at both ends of the flange reinforcement at 216° and 252°.

Fig. 12
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Maximum principal stress history of point B in the non-heat treated region of the hub

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In Cases 1, 3, 5, 7, and 9, the maximum principal stress values were identical and shifted by a pitch of 72°. Figure 13a shows the stress histories for Cases 1, 3, and 5. Cases 7, 8, and 9 are omitted because of their similarity to Cases 1, 3, and 5. Similarly, the maximum principal stresses for Cases 2, 4, 6, 8, and 10 are shifted by a pitch of 72°, and the resulting stress histories are shown in Fig. 13b. As the stress history of point C is similar to that of point B, the development of the stress history for point C is not presented in this paper.

Fig. 13
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Maximum principal stress history of point B in the heat treated region of the hub for Cases 1–6

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3.3 Fatigue Analysis

The stress amplitude and mean stress were calculated from the maximum principal stress histories obtained in Sects. 3.1 and 3.2. The cumulative damage and life were calculated according to the Miner rule using the commercial software FEMFAT [11]. We used the “modified equivalent stress in critical plane” option for the stress section as provided in FEMFAT. The calculated life for points A, B, and C are 2,563,000, 3,067,000, and 3,402,000 cycles, respectively, as shown in Fig. 14a.

Fig. 14
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Fatigue analysis results and endurance test results

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We conducted an endurance test to verify the results of the fatigue analysis, and the test results were compared with the fatigue analysis results. The endurance test results showed that failure occurred at point A after 2,122,560 cycles, as shown in Fig. 14b. Therefore, the failure location and expected life were consistent with the results of the fatigue analysis, demonstrating the validity of the fatigue analysis process proposed in this study.

4 Conclusions

We examined the fatigue analysis of an automotive wheel bearing flange by considering the effect of rotation. We performed sequential stress and fatigue analyses. We modeled the respective heat treated and non-heat treated regions of the flange and outer ring separately for the stress analysis. To model the rolling elements, we replaced them with spring elements to circumvent the convergence problem. From the stress analysis, the maximum principal stresses were 721 MPa, − 23 MPa, and 301 MPa in three main failure regions.

A rotary bending fatigue test was performed on the bearing materials to obtain the fatigue material properties. We calculated the stress histories of the heat treated and non-heat treated regions by simulating the effect of rotation. The cumulative damage and life were calculated according to the Miner rule using the commercial software FEMFAT. We conducted an endurance test to verify the results of the fatigue analysis, and the test results were compared with the fatigue analysis results. The failure location and expected life were consistent with the fatigue analysis results, verifying the validity of the fatigue analysis proposed in this study.