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Abstract

This article provides insights into the localized stress in a defect zone of the rolling bearings when the rolling elements pass through the defect. A dynamic finite element model of a rolling bearing with an artificial round defect in its outer raceway is solved numerically using the explicit dynamic finite element software package, ABAQUS. The maximum Mises stress and maximum contact pressure in the defect zone are obtained in the simulation. The effects of radial load, rotation speed, and initial defect size on the stress level are investigated. The results show that higher stresses are generated during the rolling balls passing through the defect. The radial load and defect size have significant effects on the stress level of defect, while low-level rotation speed has neglected effects.

Keywords

Local stress rolling bearing explicit dynamic analysis contact pressure defect

Introduction

Ball bearings are widely used in rotating machinery as the key supporting elements, especially in high-speed situation, for example, aero-engines, machine tools, spacecraft, computer, and electric motors.^1–3 The damage and failure of bearings are the major sources of mechanical system breakdown, which often lead to great economic losses and even result in severe accidents. In fact, the bearing failure is an accumulated process instead of an unexpected event. When a defect exists in a rolling bearing, the stress level around the defect will change, and the change of the stress level will result in quick propagation and growth of the defect. Therefore, it is important for engineers to take continuous effort to carry out local stress analysis of a defect rolling bearing.

Considerable studies have been devoted to investigate the dynamic response of bearings with defects. For example, Singh et al.² simulated the contact force and vibration response for a defective rolling element bearing using a two-dimensional (2D) finite element model. Liu et al.^4–7 used a piecewise response function to describe the localized defect with different sizes and studied the vibration response. The effects of the defect shape, defect edge discontinuities, radial load and shaft speed on the pulse waveform characteristics were also reported in their work. Furthermore, the relationship between time-varying contact stiffness and defect sizes was also considered in their numerical models. Shao et al.⁸ developed a new dynamic analysis algorithm to model a localized surface defect and investigate the effects of the defect width, depth, and types on the vibration response of a cylindrical roller bearing. Kulkarni and Sahasrabudhe⁹ presented a dynamic model for predicting the vibration behavior of a ball bearing with localized defect and simulated the pulse generated by the ball striking the defect. The blending functions of the cubic hermite spline were used in their study. Petersen et al.¹⁰ developed a method to calculate and analyze the quasi-static load distribution and stiffness variations for a radially loaded double row rolling element bearing with a raceway defect. Nakhaeinejad and Bryant¹¹ modeled the multi-body dynamics of faulty rolling element bearings using vector bond graphs and studied the effects of type, size, and shape of faults on the vibration response in rolling element bearings. Wang et al.¹² proposed a multi-body dynamic model to investigate the vibration responses of a cylindrical roller bearing with localized surface defect. He also discussed the effects of defect types, depth, width, and length on the dynamic responses. Patel and Upadhyay¹³ developed a mathematical model to investigate the non-linear dynamic behavior of a rotor-bearing system due to localized defects in both inner race and outer race.

The aforementioned work generally concerned about the vibration response of defective bearing, rarely taking the stress level of defect zone into account. Literature retrieval shows that quite a few works have been fulfilled to study the stress levels of defective ball bearings. Finite element method (FEM) was often used to investigate the stress levels or dynamic responses of bearing components.^14–22 For example, Murer et al.¹⁶ built a static finite element model to analyze the stresses and contact forces of rolling bearing. Pandkar et al.¹⁷ used an elastic–plastic FEM to study the microstructure-sensitive accumulation of plastic strain of bearing steel. Their study found that carbide particles acted as stress concentrators and introduced a shear stress cycle with a non-zero mean stress at the scale of the carbide microstructure. Morales-Espejel and Gabelli¹⁸ simulated the progression of surface rolling contact fatigue damage of rolling bearings with artificial dents, the stress distribution around the dents was obtained in the simulation. Biboulet et al.^19,20 studied the additional pressure, the subsurface stress, and the risk integral of rolling element bearing due to indents as a function of their geometry and the operating conditions. They combined these risk integrals to calculate reduction factor of a debris life. Zhu et al.²¹ studied the stress and deformation of a deep groove ball bearing based on a static finite element model, but no defect was introduced in their work. Yin et al.²² studied the Mises stress of the ball bearing with point defects and the non-linear dynamic characteristics of ball bearing with defects. To the best of our knowledge, few works have been done to study the stress level of defect zone in rolling bearings.

The focus of this work is on the analysis of localized stress of a ball bearing with an artificial round defect on its outer raceway. In fact, the operation of a rolling bearing is highly non-linear, with three major characteristics, complex rolling contact, high-speed dynamic events, and short duration impact. The modular Abaqus/Explicit is particularly well suited for analyzing the aforementioned problem. Therefore, the finite element software, ABAQUS, was used to fulfill the research objective. In this article, an explicit dynamic finite element model was established to investigate the local stress around an artificial round defect in outer raceway of a ball bearing. The effects of radial load, rotation speed, and initial defect size on the stress level were analyzed.

Modeling approach

Finite element model

A finite element model of a three-dimensional (3D) half section deep groove ball bearing (i.e. SKF 6205-2RS) was built, as shown in Figure 1. The finite element model consists of the following components: outer ring, inner ring, cage, and rolling balls. As shown in Figure 1, the defected zone is highlighted in the rectangular box. The defect location was directly in the center of loaded zone. As in this location, the rolling ball was subjected to the largest force, which was under the most severe situation. The parameters of the bearing are shown in Table 1. The radial clearance of the bearing model is 4 µm, and the clearance between cage slot and rolling ball is 0.1 mm. The defect size design was motivated by the test data from Case Western Reserve University bearing data center. The researchers set several defect specifications in their test bearings (SKF 6205-2RS) to obtain vibration data. One of the defect specifications is 0.014 in (0.3556 mm) in diameter and 0.011 in (0.2794 mm) in depth. Considering this, we used the same defect depth. The defect diameters have three specifications. One is the same (0.3556 mm) as above. The other two are twice (0.7112 mm) and thrice (1.0668 mm) as large as above.

Figure 1.

Schematic of the finite element model of a 3D section bearing.

Table 1.

Parameters of the FE bearing model.

Parameter name	value
Outer diameter (D_o, mm)	52.0
Inner diameter (D_i, mm)	25.0
Ball diameter (D, mm)	7.940
Pitch diameter (d_m, mm)	39.04
Breadth (B, mm)	15.0
Inner ring osculation (r_i, mm)	4.089
Outer ring osculation (r_o, mm)	4.208
Number of rolling balls	9

FE: finite element.

The fatigue load limit of SKF 6205-2RS bearing is 330 N, and the basic static load is 7800 N. The reason we chose 1000, 1500, and 2000 N as the radial load in the simulation is out of the following two considerations. Firstly, these three levels of loads are between the fatigue load limit and basic static load. Secondly, they are also close to the common load of this kind of bearing. The limiting speed of SKF 6205-2RS bearing is 8500 r/min. We chose 600, 1200, and 1800 r/min as rotation speed in the simulation. Because these three levels of rotation speeds were not only below the limiting speed but also close to the common speed of this kind of bearing. For instance, the researchers in Case Western Reserve University used 1730, 1750, 1772, and 1797 r/min as rotation speed in their test.

An artificial round defect was introduced to the outer raceway. As shown in Figure 2, the defects were cylindrical pits with the same depth of 0.2794 mm and three different diameters of 0.3556, 0.7112, and 1.0668 mm, located directly in the loaded zone.

Figure 2.

Defect zone in outer raceway: (a) mesh refinement around a defect zone and (b) the diameter and degree span of a defect zone.

The materials of all the components were assumed to be homogeneous and linear elastic. The density is ρ = 7800 kg/m³, the modulus of elasticity is E = 210 GPa, and the Poisson’s ratio is υ = 0.3. As a result, the plastic strain of the model was neglected and the materials remained undamaged even if the stress exceeds the materials’ elastic limits.

All the components in the finite element model were meshed using 8-node linear brick reduced integration elements (C3D8R). In order to simulate the contact between balls and raceway more precisely, the size of the mesh elements of the contact surfaces should be set as small as possible. However, the central processing unit (CPU) running time increases exponentially with the element dimension decreases. Therefore, different mesh element sizes were chosen in the defect zone, loaded zone, and unloaded zone to balance both CPU running time and simulation accuracy. The mesh element sizes of 0.042, 0.2, and 0.4 mm were set in the defect zone, loaded zone, and unloaded zone in the circumferential direction of outer ring, respectively. In order to simulate the stress level more accurately, increased mesh density around defect was set. The average mesh element size around defect was 0.023 mm.

Interactions, loads, and boundary conditions

It should be noted that the cage was mainly used to separate and guide the rolling elements; hence, its stress and deformation were not considered. In order to save the running time, cage was simplified as a rigid body. The mating surface with the inner ring is coupled with a reference point located at its geometric center.

A frictional contact with a coefficient of sliding friction, 0.05, was defined for all contact interfaces, which were rolling balls–outer ring raceway, rolling balls–inner ring raceway, and rolling balls–cage. The general contact (explicit) algorithm was utilized throughout the simulation.

For a deep groove ball bearing only subjected to radial load, it is reasonable to apply symmetry constraint to the symmetric section in the finite element model. The mating surface of outer ring was totally fixed. Other components are constrained with frictional contact.

Three levels of radial load, rotation speed, and defect size were built in the model. The study factors and their levels are presented in Table 2. Since this study only investigates the effects of each single variable factor, a controlling variable method was used in the simulation.

Table 2.

Study factors and levels.

Model no.	Level of factors
Model no.	Radial load (F_r, N)	Rotation speed (ω, r/min)	Defect diameter (D_d, mm)	Degree span of defect (rad)
1	1000	1200	0.7112	0.03028
2	1500	1200	0.7112	0.03028
3	2000	1200	0.7112	0.03028
4	1000	600	0.7112	0.03028
5	1000	1800	0.7112	0.03028
6	1000	1200	0.3556	0.01514
7	1000	1200	1.0668	0.04542

To study the effects of radial load, the rotation speed and defect size was set to 1200 r/min and 0.7112 mm, respectively. The radial load was applied to the reference point of inner ring mating surface. During rotation, the loading direction was fixed, pointing toward to the defect in outer raceway.

To study the effects of rotation speed, the radial load and defect was set to 1000 N and 0.7112 mm, respectively. The inner ring was constantly rotated with three levels of rotation speeds in an anticlockwise direction.

To study the effects of defect size, the radial load and rotation speed were set to 1000 N and 1200 r/min, respectively. The defects were cylindrical pits with the same depth and three different diameters, located directly in the load zone.

Results and discussion

The Mises stress and contact pressure levels were monitored throughout the simulation. In order to verify the accuracy of the finite element model, the output of the energy balance was also monitored.

Energy balance

Energy output is an important part of an ABAQUS/Explicit analysis. Comparisons between various energy components can be used to evaluate whether an analysis is yielding an appropriate response. In the finite element analysis, the artificial strain energy for the whole model (ALLAE) includes energy stored in hourglass resistances, transverse shear in shell and beam elements. It is generally recommended that the artificial strain energy should be less than 10% of its internal energy for the whole model (ALLIE). Large values of artificial strain energy indicate that significant elemental distortions (i.e. defined as hourglassing phenomena in finite element analysis) have occurred and a mesh refinement is necessary to be done.

Seven finite models with different combinations of radial load, rotation speed, and defect size were built. The ratio of ALLAE to ALLIE was less than 10% for all the models, which showed that the simulation model were reasonable. See details in Table 3.

Table 3.

Ratio of artificial strain energy to internal energy.

Model no.	ALLAE	ALLIE	Ratio (%)
1	0.1674	7.506	2.230
2	0.3421	14.78	2.315
3	1.3190	23.74	5.556
4	0.1367	7.353	1.859
5	0.1794	7.783	2.305
6	0.1491	7.466	1.997
7	0.1505	7.505	2.005

ALLAE: artificial strain energy for whole model; ALLIE: internal energy for whole model.

The process of rolling balls passing through defect

In this article, the process of rolling balls passing through defect zone was divided into five stages. Figure 3 shows the five stages of Model no. 1.

Figure 3.

The process of rolling balls passing through defect. The process has five stages: (a) before entering the defect, (b) on entering, (c) on the middle of the defect, (d) on exiting the defect, and (e) after exiting the defect.

The first stage (see Figure 3(a)) is at the moment just before the rolling ball entering the defect. It showed that the stress level had no difference from that produced by contact with an undamaged raceway. The maximum Mises stress is in the subsurface of raceway.

The second stage (Figure 3(b)) is at the moment the rolling ball is entering the defect, contacting with the front edge of the defect. The stress level increases at this stage. The maximum Mises stress is on the front edge of the defect.

The third stage (Figure 3(c)) is at the moment the rolling ball is on the center of the defect. The stress level reaches peak at this stage for Model no. 1. The maximum Mises stress is on the middle edge of the defect.

The fourth stage (Figure 3(d)) is at the moment the rolling ball is leaving the defect, contacting with the back edge of the defect. The stress level approximately falls back to the value produced at the second stage. The maximum Mises stress is on the back edge of the defect.

The fifth stage (Figure 3(e)) is at the moment just after the rolling ball leaving the defect zone. The stress level falls back to the value of the first stage. The maximum Mises stress is in the subsurface of raceway.

It can be found that the stress distributions were slightly discontinued along axial direction. It may be caused by the larger mesh sizes in axial direction. However, the larger mesh elements were far away from the defect zone, it has little influence on the stress level around defect zone. So, the calculated results were credible.

Effects of radial load

As shown in Table 2, Model nos 1, 2, and 3 were used to study the effects of radial load on stress level around the defect zone. Three levels of radial load, 1000, 1500, and 2000 N, were tested in the simulation. The rotation speed and defect size of the three models were set to be 1200 r/min and 0.7112 mm, respectively.

The maximum Mises stress and maximum contact pressure are shown in Figure 4. Before the rolling ball enters the defect (see Figure 3(a)), the maximum contact pressure can be calculated through Hertzian contact theory. The calculated results and simulated results are shown in Table 4. All the relative errors for the three levels of radial load are less than 5%, which indicates that the simulated maximum contact pressure is reasonable. The maximum Mises stress and contact pressure increased with the increase in radial load, which indicates that a high-level radial load can promote the defect extension and growth.

Figure 4.

Effects of radial loads on stress level of defect zone: (a) maximum Mises stress and (b) maximum contact stress.

Table 4.

Comparison of calculated contact pressure and simulated contact pressure.

Radial loads	Maximum contact pressure (MPa)
Radial loads	Calculated results	Simulated results	Relative error (%)
1000	2408	2456	1.99
1500	2757	2886	4.68
2000	3035	3179	4.74

Effects of rotation speed

As shown in Table 2, Model nos 1, 4, and 5 were used to study the effects of rotation speed on stress level around defect zone. Three levels of rotation speed, 600, 1200, and 1800 r/min, were tested in the simulation. The radial load and defects size of the three models were set to be 1000 N and 0.7112 mm, respectively.

The maximum Mises stress and maximum contact pressure are shown in Figure 5. It can be concluded that the three levels of rotation speed had neglected effects on the stress level of defect zone. This is because the rotation speed is too slow to produce adequately large centrifugal force. Taking the rotation speed of 1800 r/min for example

$F_{c} = \frac{1}{2} m d_{m} ω_{c}^{2}$ (1)

$m = \frac{1}{12} ρ π D^{3}$ (2)

$ω_{c} = \frac{π n_{i}}{60} (1 - \frac{D}{d_{m}})$ (3)

where F_c is the centrifugal force of a rolling ball, m is the mass of a rolling ball, and ω_c is the angular speed of cage. F_c can be calculated by

$F_{c} = \frac{1}{2} m d_{m} ω_{c}^{2} \approx 0.5 N$ (4)

Figure 5.

Effects of rotation speed on stress level of defect zone: (a) maximum Mises stress and (b) maximum contact stress.

The centrifugal force is much less than the radial load. Thus, the low levels of rotation speed had little effects on the stress level of defect zone. Much higher rotation speed needs to be tested in the future work.

Effects of defect size

As shown in Table 2, Model nos 1, 6, and 7 were used to study the effects of defect size on stress level around defect zone. Three defect diameters of 0.356, 0.711, and 1.067 mm (which were equivalent to degree span of 0.0151, 0.0303, and 0.0454 rad) were tested in the simulation. The radial load and rotation speed of the three models were set to be 1000 N and 1200 r/min, respectively.

The maximum Mises stress and maximum contact pressure are shown in Figure 6. Both of the two parameters demonstrate the same change tendency. The stress level increased with the increase in defect size. This is due to the decrease in contact area. Larger defect diameter made the contact area between rolling ball and defect zone get smaller. However, if the defect diameter becomes larger than the major axis of the contact ellipse, the rolling ball will lose supporting capacity when it enters the defect zone. In this case, the stress level may decrease but it will be much higher when rolling ball is on entering or exiting the defect zone. Thus, larger defect size needs to be tested.

Figure 6.

The effects of defect size on stress level of defect zone: (a) maximum Mises stress and (b) maximum contact stress.

It is also interesting to note that the maximum contact pressure level produced when rolling ball is entering the defect zone is a little higher than that produced when rolling ball is exiting the defect zone. Figures 4(b), 5(b), and 6(b) have shown such a phenomenon.

Conclusion and future work

This article developed a finite element model of deep groove ball bearing with an artificial round defect on outer raceway. The simulation was performed based on explicit dynamic finite element software, ABAQUS. The process of rolling balls passing through defect zone was divided into five stages. The effects of radial load, rotation speed, and initial defect size on the stress level were analyzed. This work complements the research about stress concentration phenomenon in rolling bearings. The conclusions drawn from the study can be summarized as below:

Higher Mises stress and contact pressure were generated during the rolling balls passing through the defect.

The maximum Mises stress and contact pressure increased with the increase in radial load. The simulated maximum contact pressure of defect-free zone agreed with the calculated results very well.

For the three small defect sizes in this study, the maximum Mises stress and contact pressure increased with the increase in defect size. The change tendency of stress level depends on the defect size. Much larger defect size needs to be tested in the future work in order to simulate groove peeling, for example, 1.422, 1.788, 2.134, and 2.845 mm.

Footnotes

Academic Editor: Yongming Liu

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research,authorship,and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research,authorship,and/or publication of this article: The author(s) are grateful to the National Natural Science Foundation of China (Grant Nos 51205247 and 51575340) and the Research Project of State Key Laboratory of Mechanical System and Vibration (MSVZD201401).

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Local stress analysis of a defective rolling bearing using an explicit dynamic method

Abstract

Keywords

Introduction

Modeling approach

Finite element model

Interactions, loads, and boundary conditions

Results and discussion

Energy balance

The process of rolling balls passing through defect

Effects of radial load

Effects of rotation speed

Effects of defect size

Conclusion and future work

Footnotes

Declaration of conflicting interests

Funding

References