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International Journal of Innovative Research in Engineering & Science ISSN 2319-5665

(July 2014, issue 3 volume 7)

8

CONTACT STRESS ANALYSIS OF DEEP GROOVE BALL BEARING 6210

USING HERTZIAN CONTACT THEORY

Shailendra Pipaniya1, Akhilesh Lodwal

2

1Research Scholar

Department of Mechanical Engineering, Institute of Engineering and

Technology Devi

Ahilya Vishwavidyalaya, Indore-452017(M.P.), India

2Assistant Professor

Department of Mechanical Engineering, Institute of Engineering and

Technology, Devi

Ahilya Vishwavidyalaya, Indore-452017(M.P.), India

Abstract

When curved surfaces are in contact, the theoretical contact area of two spheres is a point

and the theoretical contact area of two parallel cylinders is a line. As a result, the pressure

between two curved surfaces should be infinite. The infinite pressure at the contact should

cause immediate yielding of both surfaces. In reality, a small contact area is being created

through elastic deformation, thereby limiting the stresses considerably. These contact stresses

are called Hertz contact stresses. Example of curved surfaces in contact is ball bearing.

Where static loads are encountered with bearings that are not rotating, resistance to plastic

deformation becomes important. A limiting static capacity is commonly defined as the load

corresponds to a hertzian contact pressure of approximately 4200 N/mm2 at the center of the

contact area. This paper determines the contact stress between the inner race and ball of

Single row Deep groove ball bearing using analytical and finite element analysis. The

calculation procedure consists of calculation of maximum contact pressure at different loads

and comparison of results with finite element analysis to predict the contact pressure between

inner race and ball of Single row Deep groove ball bearing.

Keywords: Hertzian contact pressure. Deep groove ball bearing, inner race, ball.

1. Introduction

Ball bearings are commonly used machine elements. They are employed to permit rotary

motions of, or about, shafts in simple commercial devices and also used in complex

engineering mechanisms. Deep groove ball bearings are the most widely used bearings in

industry and their market share is about 80% of industrial rolling element bearings. A deep

groove ball bearing can support a thrust load of about 70% of its radial load. [1] The radial

load and axial load capacity increases with the bearing size and the number of balls.

The contact stress refers to the localized stress that develop as two curved surfaces come in

contact under the effect of radial load as in case of ball and raceways of ball bearing. Hertz

theory considers the elastic deformation and stress distribution near the contact of the rolling

elements and races. Under load, due to an elastic deformation, the line or point contact

becomes a contact area, this area is very small, resulting in a very high maximum contact



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pressure of the order of 15 GPa. The calculations of the maximum contact pressure and deformation at the contact area of the rolling clement and raceways are according to Hertzs contact theory. A Contact stress analysis of ball bearing diameter > 400 mm was carried out

by Pandiyarajan et all.[2] The comparison of the total deformation of thrust ball bearing &

contact stress b/w ball & raceways & its effect on fatigue life of thrust ball bearing done by

Prabhat Singh et all. [3]

2. Rolling Contact Stresses

Calculation of stresses and corresponding deformation at a contact point between rolling

elements and raceways employs elasticity relations established by Hertz in 1881.Hertzs theory considers the contact of two bodies with curved surfaces under force W. There is a

theoretical point or line contact between the rolling elements and races. But due to elastic

deformation, the contact area between two curved bodies in point contact has an elliptical

shape. In machinery that involves severe shocks and vibrations, the contact stresses can be

very high.

In the United States, the standard bearing material is SAE 52100 steel hardened to 60 RC.

This steel has a high content of carbon and chromium. It is manufactured by an induction

vacuum melting process, which minimizes porosity due to gas released during the casting

process. The allowed limit of rolling contact stress for SAE 52100 is 4.2 Gpa.[4] Failure from

contact stresses generally falls into two categories:

Localized deformations by yielding or distortions, and

Fracture by progressive spreading of a crack (fatigue).[5] These failures results in undesirable noise and vibrations in rotating ball bearing.

3. Mathematical modelling

The contact area between two curved bodies in point contact has an elliptical shape. (Fig.1)

Figure 1. Contact areas in ball bearing

According to Hertzs theory, the equation of the pressure distribution in an ellipsoidal contact area in a ball bearing is

P = ( 1- x2/a

2 y2/b2 )Pmax



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Here, a and b are the small radius and the large radius, respectively, of the ellipsoidal contact

area, as shown in figure 2. The maximum pressure (or stress) at the centre of an ellipsoidal

contact area is given by the following Hertz equation.

Pmax = 3 W/ 2 a b

Figure 2. Pressure distribution in an elliptical contact area

Here, W = Wmax is the maximum load on one spherical rolling element. The maximum

pressure is proportional to the load, and it is lower when the contact area is larger. For ball

bearings, the maximum load, Wmax, on most heavily loaded ball can be estimated by the

equation.

Wmax = 5W/ no. of ball [6]

3.1 Geometry of Ball Bearing

Figure 3. Geometry of ball Bearing.



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4. Tables

Table 1. Geometric parameters of ball bearing used in analysis.[7]

Ball bearing model 6210 Deep Groove ball

bearing

Ball diameter 12.7 mm

Bore diameter (d) 50 mm

Inner raceway

diameter (d1)

57.291 mm

Outer raceway

diameter (D2)

82.709 mm

Outside diameter (D) 90 mm

No. of balls (n) 10

Inner race groove

radius (rgi)

6.6 mm

Outer race groove

radius (rgo)

6.6 mm

Width (B) 20 mm

Table 2. Ball bearing material used in analysis.

Ball bearing material AISI 52100 chrome

alloy steel

Modulus of elasticity 203300 Mpa

Poissons ratio 0.3

Density 7833.413 kg/m3

5. Methodology

Maximum pressure (Pmax) at the centre of ellipsoidal contact area is calculated by Hertzian

elliptical contact theory using different loads (W) whose results given in Table 3.



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Table 3. Analytic results

Load (Newton) Pmax ( Mpa)

500 N 1979.3449

1000 N 2493.049

2000 N 3141.1175

3000 N 3595.349

4000 N 3956.4198

5000 N 4261.9308

6000 N 4528.2940

These results are compared with the finite element analysis of ball and inner raceway. The 3-

Dimensional Modelling has been done through modelling software Pro-e the commercial

software ANSYS Workbench used as a FEA tool in this analysis work. The following figures

followed by meshing of model are images of contact stress analysis of ball and raceways at

different loads.

Figure 4. Meshing of model

1. AT 500 N

Figure 5. Contact stress at 500 N



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2. AT 1000 N

Figure 6. Contact stress at 1000N

3. AT 2000N


4. AT 3000 N




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5. AT 4000N


6. AT 5000 N


7. AT 6000N




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6. Results

The calculated values of contact stress (Pmax) should be less than allowable stress 4.2 Gpa for

AISI 52100 Chrome alloy steel for proper functioning of ball bearing. Maximum contact

stress based on Hertzian contact theory is calculated at different loads corresponding FEA is

performed in order to justify the results of calculated stresses. The comparison of calculated

stresses with FEA results are given in Table 4 followed by a graph.

Table 4 Comparison of both results

Load (N) Analytic

contact

stress (Pmax) in Mpa

FEA

contact

stress

(Pmax) in

Mpa

Percentage

error %

500 1979.3449 1747.7 13.25

1000 2493.049 2354.8 5.87

2000 3141.1175 3398.3 8.19

3000 3595.349 3794 5.5

4000 3956.4198 3916.5 1.02

5000 4261.9308 4499.9 5.58

6000 4528.2940 4620.3 2.03

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

500

2000

4000

6000

contact stress in Mpa

Load

in

N

FEA results

Analyticresults

Both the results are compared and plotted which coincides satisfactorily.

7. Conclusions

Results indicate that at 5000 N contact stresses at contact of inner raceway and ball crosses

the limit of allowable limit contact stress. This analysis is used to study the failure behavior to

increase the service life of ball bearing. The future work of this paper will be the calculation

of contact stress using different ball bearing materials and different sizes of bearings.



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Acknowledgements

I am very much thankful to all the faculty and staff members of Mechanical Engineering

Department of IET DAVV, Indore. I am also thankful for the department to provide

experimental facilities in the department.

References

[1] Michael M. Khonsari, E. Richard Booser, Applied Tribology Bearing Design and Lubrication, John Wiley and Sons Ltd , England 2008.

[2] Pandiyarajan. R, Starvin.M.S, Ganesh.K.C, Contact Stress Distribution of Large Diameter Ball Bearing Using Hertzian Elliptical Contact Theory Procedia Engineering 38 (2012) 264-269.

[3] Prabhat Singh, Prof. Upendra Kumar Joshi, Fatigue Life Analysis of Thrust Ball Bearing Using Ansys, International Journal of Engineering Sciences and Research Technology, 2277-9655 January 2014.

[4] TEDRIC A. HARRIS, Rolling Bearing Analysis, 4th edition , John Wiley and Sons Inc., New York, 2001.

[5] Bernard J. Hamrock, William J. Anderson, Rolling Element Bearings, Lewis Research Centre, NSA Reference Publication, June 1983.

[6] Avraham Harnoy, Bearing Design in Machinery, 2003 by Marcel Dekker, Inc.

[7] DYNAROLL Corporation, 12840 Bradley Avenue Sylmar, CA 91342

[8] Yongming Liu, Brant Stratman, and Sankaran Mahadevan, Fatigue crack initiation life prediction of railroad wheels, International Journal of Fatigue 28, 2006, p 747-756.

[9] Tatjana Lazovic, Mileta Ristivojevic, Radivoje Mitrovic, Mathematical Model of Load Distribution in Rolling Bearing. FME Transactions (2008) 36, 189-196.

[10] V B Bhandari, Design of Machine Elements, 2nd edition Tata McGraw Hill Publication, chapter 15.

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