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International Journal of Advanced Research in Engineering and Technology

(IJARET) Volume 6, Issue 11, Nov 2015, pp. 80-90, Article ID: IJARET_06_11_008

Available online at

http://www.iaeme.com/IJARET/issues.asp?JType=IJARET&VType=6&IType=11

ISSN Print: 0976-6480 and ISSN Online: 0976-6499

© IAEME Publication

___________________________________________________________________________

THERMAL STRESS ANALYSIS OF A BALL

BEARING BY FINITE ELEMENT METHOD

M. Chandra Sekhar Reddy

Associate Professor,

Department of Mechanical Engineering,

UCE(A), Osmania University.

ABSTRACT

High cutting speeds and feeds are essential requirements of a machine tool

structure to accomplish its basic function which is to produce a work piece of

the required geometric form with an acceptable surface finish at as high a rate

of production as is economically possible. Since bearings in high speed

spindle units are the main source of heat generation. Friction in bearings

causes an increase of the temperature inside the bearing. If the heat produced

cannot be adequately removed from the bearing, the temperature might exceed

a certain limit, and as a result the bearing would fail. To analyse the heat flow

in a bearing system, a typical ball bearing and its environment has been

modelled and analysed using the finite element method. The maximum

temperature in the bearing has been calculated as a function of heat

generation with the rotational speed as a parameter. The goal of this analysis

is to see how fast the temperature changes in the bearing system with respect

to rotational speeds. In this thesis, at high speed range, a steady state thermal-

stress simulation is performed by using FEA method to investigate

temperature distribution of the bearing and the result shows that the

temperature increases gradually with increase in rotational speed and it is

validated by analytical formulation done. Further the increase of rotation

speed the inner ring centrifugal displacement increases which causes larger

contact deformation and stress. The dynamic stiffness of the variable preload

bearing is analysed analytically and it is found that the radial stiffness

decreases with increase in rotational speeds.

Key words: Bearing, Stress, FEM, Stiffness.

Cite this Article: M. Chandra Sekhar Reddy. Thermal Stress Analysis of A

Ball Bearing by Finite Element Method. International Journal of Advanced

Research in Engineering and Technology, 6(11), 2015, pp. 80-90.

http://www.iaeme.com/IJARET/issues.asp?JType=IJARET&VType=6&IType=11

Thermal Stress Analysis of A Ball Bearing by Finite Element Method


1. INTRODUCTION

High speed machining is a promising technology to drastically increase productivity

and reduce production costs. The technology of high speed machining is still

relatively new. Although theories of high-speed metal cutting were reported in the

1930s, machine tools capable of achieving these cutting speeds did not exist until the

1980s. Only recently, industry has started experimenting with the use of high speed

machining in production. The aircraft industry was first, with the automotive industry

and mould and die makers now following. Because of little experience in this new

field, there are still many problems to be solved in the application of high speed

machining. Current problems include issues of tooling, balancing, thermal and

dynamic behaviours, and reliability of machine tools.

The demand for high speed machine tools and three coordinate measuring

machines are rapidly increasing in response to the development of production

technology such precision machining which requires high-precision parts and high

productivity. Research on high speed machine tooling can be approached on the main

spindle and feed system. A high speed/precision feed system reduces non-

cutting/operating time and tool replacement time, making production more

economical.

The high speed precision ball bearing is a main part in high speed/precision feed

system, and there are many joints existing in the ball bearing, such as the interfaces

between the bearing and the shaft, the bearing and the bearing support and so on.

When two surfaces are in contact, the presence of surface roughness produces

imperfect friction at the joint, no matter how much the pressure between the surfaces

is. The friction in ball bearings entails a sudden and violent heating of balls that can

have very detrimental effects. The increase of temperature generated by these

phenomena can involve mechanical micro-deformations and an overheating of

cooling fluid (especially when dealing with cryogenic fluids). Such temperature

heating of ball bearing plays a significant role in thermal characteristics of the feed

system, causing serious thermal deformation that subsequently degrades the accuracy

of machine tool and other mechatronics instrument where the precision ball bearing is

used.

After the invention of the wheel, it was learned that less effort was required to

move an object on rollers than to slide the object over the same surface. Even after

lubrication was discovered to reduce the work required in sliding, rolling motions till

required less work when it could be used. For example, archaeological evidence

shows that the Egyptians 2400 BC, employed lubrication, most likely water, to reduce

the manpower required dragging sledges carrying huge stones and statues. The

Assyrians, ca. 1100 BC, however, employed rollers under the sledges to achieve a

similar result with less manpower. It was therefore inevitable that bearings using

rolling motion would be developed for use in complex machinery and mechanisms. In

a simplistic manner, the evolution of rolling bearings, Dowson provides a

comprehensive presentation on the history of bearings and lubrication in general; his

coverage on ball and roller bearings is extensive. Though the concept of rolling

motion was known and used forty thousands of years, and simple forms of rolling

bearings civilization, the general use industrial revolution. Leonardo da Vinci (1452-

1519) AD conceive the basic construction of the modern rolling bearing.

The universal acceptance of rolling bearings by design engineers was initially

impeded by the inability of manufacturers to supply rolling bearings that could

compete in endurance with hydrodynamic sliding bearings. This situation, however,



has been favourably altered during the twentieth century, and particularly since 1960,

by development of superior rolling bearing steels and constant improvement in

manufacturing, providing extremely accurate geometry, long-lived rolling bearing

assemblies. Initially this development was triggered by the bearing requirements for

high speed aircraft gas turbines; however, competition between ball and roller bearing

manufacturers for worldwide markets increased substantially during the 1970s, and

this has served to provide consumers with low-cost, standard design bearings of

outstanding endurance. The term rolling bearing includes all forms of bearings that

utilize the rolling action of balls or rollers to permit rotation to shaft on, constrained

motion of one body relative to another. Most rolling bearings are employed to permit

rotation of a shaft relative to some fixed structure. Some rolling bearings, however,

permit translation, that is, relative linear motion, of a fixture in the direction provided

by a stationary shaft, and a few rolling bearing designs permit a combination of

relative linear and rotary motion between two bodies.

The term “rolling bearing” includes all forms of roller and ball bearing which

permit rotary motion of a shaft. Normally a whole unit of bearing is sold in the

market, which includes inner ring, outer ring, rolling element (balls or rollers) and the

cage which separates the rolling element from each other.

Rolling bearings are high precision, low cost but commonly used in all kinds of

rotary machine. It takes long time for the human being to develop the bearing from

the initial idea to the modern rolling bearing which can be seen from figure.1, The

reason why bearing is used is that first it can transfer moment or force. Secondly and

maybe more important is that it can be interchanged easily and conveniently when it’s

broken. It has less possibility for the shaft or housing to be worn out. Usually the

bearing first cracks and then the shaft or housing is broken. If the above situation

happens it is really easy to figure it out: just buy a new bearing from the market with

the same parameter and replace it. That’s why bearings are so often used.

Figure 1 Single row angular contact ball bearing

1.1. Four-Point-Contact Ball Bearing

The four-point-contact ball bearing is a single row angular contact bearings designed

to handle two-directional axial loading. These bearings require a minimum amount of

space and can be custom designed to meet unique loading parameters. And the four-

point-contact ball bearings can carry either pure axial loading or combined loading

provided the axial component is more significant. For heavy industrial requirements,

this style of bearing can be designed as a large slewing-ring bearing with bolted down

races and internal or external integral gearing. Figure 2 shows a four point contact ball

nbearing.

Thermal Stress Analysis

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Figure

A mathematical model based on a five degrees of freedom dynamic system

been presented by Alfares

contact ball bearings on the vibration behaviour of a grinding machine spindle

ended by saying the larger the initial preload applied, the less vibration

generated and as the initial preload increases,

that makes dominant frequencies of the

systems equipped with angular contact ball bearings are developed

[2] by to examined axial forces

[3] calculated a mathematical formulation of inner ring centrifugal displacement

based upon elastic theory.

basic equations of angular contact ball bearing are set up, effect of inner ring

centrifugal displacement on the dynamic characteristics of high

ball bearings are studied. Gao et al

order to simulate the contact shape, size and stress of thrust ball bearings

element analysis. Jin et al

calculate the heat generation rate of supporting in a kind of feed system which is used

widely in machine tools.

Guo et al. [6] carried out

ball bearing using ANSYS and validated with the mathematical formulations based on

hertz contact theory. Wang et al

contact subsurface stress field of hybrid c

a motorized high speed spindle with the bearing system to measure the temperature

distribution in the bearing system and other components.

contact problem of deep grove ball bearing ba

developed a thermo mechanical model of high speed spindle system.

2. MATHEMATICAL MODEL F

BEARING

Roller bearings are usually used for applications requiring exceptionally large load

supporting capability. Roller bearings are usually much stiffer structures and provide

greater fatigue endurance.

produce a un wanted disturbance while in working conditions. So a care should be

taken in determining the micro geometry of the ball bearing.

Although ball and roller bearings appear to be simple mechanisms, their internal

geometries are quite complex. The

form as shown in figure 3.

depicts the diametral clearance and various diameters. The

mean of the inner- and outer



Figure 2 Four point contact ball bearing

model based on a five degrees of freedom dynamic system

Alfares et al. [1] to study the effects of axial preloading of angular

contact ball bearings on the vibration behaviour of a grinding machine spindle

the larger the initial preload applied, the less vibration

initial preload increases, the stiffness of the bearing incr

dominant frequencies of the system shift to higher values. S

h angular contact ball bearings are developed Jedrzejewski et al

axial forces produced in high-speed bearing system.

calculated a mathematical formulation of inner ring centrifugal displacement

. In consideration of inner ring centrifugal displacement the


centrifugal displacement on the dynamic characteristics of high-speed angular contact

Gao et al. [4] an analytical simulation has been conducted in

contact shape, size and stress of thrust ball bearings

Jin et al. [5] in their study, an analytical method was carried out to


. [6] carried out finite element analysis on hybrid ceramic angular contact


Wang et al. [7] developed an accurate method for calculating the

contact subsurface stress field of hybrid ceramic ball bearing. Yu et al


distribution in the bearing system and other components. Yin et al. [9]

contact problem of deep grove ball bearing based on Ansys. Zahedi et al

developed a thermo mechanical model of high speed spindle system.

MATHEMATICAL MODEL FOR ANGULAR CONTACT B

Roller bearings are usually used for applications requiring exceptionally large load

Roller bearings are usually much stiffer structures and provide

. Even if the geometry of a ball bearing is perfect, i

wanted disturbance while in working conditions. So a care should be

taken in determining the micro geometry of the ball bearing.


geometries are quite complex. The ball bearing can be illustrated in its most simple

ure 3. The radial cross section of a single-row ball bearing

diametral clearance and various diameters. The pitch diameter

and outer-race diameters, di and do, respectively, and is given by

y Finite Element Method

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model based on a five degrees of freedom dynamic system has

to study the effects of axial preloading of angular

contact ball bearings on the vibration behaviour of a grinding machine spindle. They

the larger the initial preload applied, the less vibration amplitudes are

bearing increases

Spindle bearing

Jedrzejewski et al.

speed bearing system. Wang et al.

calculated a mathematical formulation of inner ring centrifugal displacement

In consideration of inner ring centrifugal displacement the


speed angular contact

an analytical simulation has been conducted in

contact shape, size and stress of thrust ball bearings using finite

n their study, an analytical method was carried out to


analysis on hybrid ceramic angular contact


n accurate method for calculating the

Yu et al. [8] developed


. [9] simulated a

Zahedi et al. [10]

OR ANGULAR CONTACT BALL

Roller bearings are usually used for applications requiring exceptionally large load-

Roller bearings are usually much stiffer structures and provide

Even if the geometry of a ball bearing is perfect, it will still

wanted disturbance while in working conditions. So a care should be


bearing can be illustrated in its most simple

row ball bearing

pitch diameter, dm, is the

, respectively, and is given by



dmdi do( )

2

Figure 3 Macro Geometry of Bearing

Generally, ball bearings and other radial rolling bearings such as cylindrical roller

bearings are designed with clearance. From figure 3, diametral clearance is as follows

pd do di 2 D.( )

Race conformity is a measure of the geometrical conformity of the race and the

ball in a plane passing through the bearing axis (also named centre line or rotation

axis), which is a line passing through the centre of the bearing perpendicular to its

plane and transverse to the race. Figure 4, depicts a cross section of a ball bearing

showing race conformity, expressed as

fr

D

Figure 4 Cross section of a ball and an outer race showing race conformity

3. PROBLEM DEFINITION

The thermo-mechanical model for the spindle needs to include all interacting effects

inside the spindle relevant to the objective. The model should account for all heat

sources, heat transfer, heat sinks and relative thermal expansion within the system. In

this case bearings in the model are one of the problems. Friction in bearings causes an

increase of the temperature inside the bearing. If the heat produced cannot be



adequately removed from the bea

and as a result the bearing would fail.

Proposed possible way to solve the problem

• Literature study of different approaches to model roller bearings

• Set-up of a 3-D reference FE

• Identification of the main sources for the nonlinearity of roller bearings

• Define of a simplified bear

• Validation and recommendations for the use of simplified bearing models

4. METHODOLOGY

To analyse the heat flow in a bearing system, a typical ball bearing and its

environment has been modelled and analysed using the finite element method. The

maximum temperature in the bearing has been calcul

generation and with the rotational speed

Figure

The flow chart of methodology is shown in fig. 5.

created is imported to A

undergoes steady state thermal

temperature distribution in

Now the structural boundary conditions

properties to undergo steady state structural

deformation of the bearing at various points

5. HEAT GENERATION IN B

SOLUTION

The major heat generation of the system is caused by the cutting process

friction between the balls and races of the bearings

cutting heat is taken away by coolant and chips, the heat generated by bearings is the



adequately removed from the bearing, the temperature might exceed a certain limit,

and as a result the bearing would fail.

Proposed possible way to solve the problem

Literature study of different approaches to model roller bearings

D reference FE-model

the main sources for the nonlinearity of roller bearings

Define of a simplified bearing model

Validation and recommendations for the use of simplified bearing models

METHODOLOGY



maximum temperature in the bearing has been calculated as a function of heat

e rotational speed as a parameter.

Figure 5 Flow Chart of Methodology

The flow chart of methodology is shown in fig. 5. The design model w

created is imported to Ansys by giving all process parameters which the mo

undergoes steady state thermal analysis by giving all the boundary condition. Here the

temperature distribution in the model that occurred in the bearing comes out as

structural boundary conditions are applied to the bearing by updating

steady state structural analysis which finally gives

deformation of the bearing at various points.

HEAT GENERATION IN BEARING – ANALYTICAL

The major heat generation of the system is caused by the cutting process

friction between the balls and races of the bearings. Assumed that the majority of



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ring, the temperature might exceed a certain limit,

the main sources for the nonlinearity of roller bearings

Validation and recommendations for the use of simplified bearing models



ated as a function of heat

The design model which is

nsys by giving all process parameters which the model

analysis by giving all the boundary condition. Here the

that occurred in the bearing comes out as output.

by updating the

analysis which finally gives the

ANALYTICAL

The major heat generation of the system is caused by the cutting process and the

that the majority of




dominant cause of temperature change. In angular contact ball bearings heat is

generated mainly by three sources. One is the rolling of imperfect mechanical bodies

load known has load related heat generation, another source of heat generation is

viscous shear of lubricants between the solid bodies, assuming that the balls are

purely rolling on the race way of the surfaces; this refers to viscosity related heat

generation. Finally, heat is generated by the balls due to spinning motion named has

spin related heat generation. Considering this harries [4]

derived the analytical

formulation for heat generation that occur in bearing. The heat generated by a bearing

can be computed as

Hf = 1.047x10-4

nM

Where Hf is the heat generated power (W), n is the rotating speed of the bearing

(rpm), M is the total frictional torque of the bearing (N mm). The total frictional

torque M consists of two parts, one is the torque M1 due to applied load and the other

one is the torque M2 due to viscosity of lubricant. That is

M = M1 + M2

The torque due to applied load can be empirically approximated by the following:

M1 = f1 Fβ dm

In which f1 is the factor depending upon bearing design and relative load. For ball

bearing

f1 = z (Fs/Cs) Y

Where Fs is the static equivalent load and is given by

Fs=XoFr+Yo Fa

Xo, Yo are the values taken from the table A4, and Fr is the radial force acting on the

bearing which is consider has zero, Fa is the axial force on bearing and the values are

given has at 150

angle Xo = 0.5, Yo = 0.46 and the axial force Fa = 945 N.

6. MODELING AND ANALYSIS

The bearing is composed of a retainer and ceramic balls having the mass 0.075kg. Its

dimensions are D=47 mm, d=25 mm, r1, 2= 2 mm, r3,4=4mm, B=12mm, ao=150 and

Z=14 balls. The bearing is supposed to operate under maximum basic dynamic load

rating of C=9.56kN and static load rating of C0=5.6kN and at rotate speed of 5000-

55000rpm.

Figure 6 Model of Ball Bearing



After the model is done, then the next will be the simulation. Importing the solid

model to “new simulation” in ANSYS Workbench and the analysis procedure is

described below.

SOLID186 is hexahedron, 3-D, 20-node solid element which has quadratic

displacement behaviour as is shown in Fig. 7. Every node has three degrees of

freedom: translations in the nodal x, y, and z directions. When the uniform reduced

integration is used, it is helpful to prevent volumetric mesh locking in nearly

incompressible cases.

Figure 7 SOLID 186 geometry

The figure 8, below shows the mesh of the model

Figure 8 Messed model

6.1. Thermal-Stress Analysis

For conducting the thermal-stress analysis, Ansys Workbench provides a very flexible

environment. By creating the geometry in the first physical environment, and using it

with any following coupled environments, the geometry is kept constant. For our

case, we will create the geometry in the Thermal Environment, where the thermal

effects will be applied.

7. RESULTS AND DISCUSSIONS

The CAD model which is used for analysis undergoes a steady-state thermal analysis.

In this steady-state numerical analysis temperature distribution in the bearing is

measured with respect to different rotational speeds. Here the heat generation value

plays the major role in the thermal analysis which is calculated and discussed. The

heat generation is mainly due to the torque developed in the bearing, here two types of

torque are considered, torque due to applied load and torque due to viscous shear of

the lubricant. This obtained heat generation value is inputted to the analysis with

preferred boundary conditions and the temperature distribution, total heat flux in

entire modal is measured.



With continuing to the structural analysis, the obtained temperature load data from

thermal analysis is given has the input by updating the boundary conditions. Finally

the total deformation of entire bearing modal and the maximum stress distribution at

the contact points are measured.

The modal which undergoes the steady-state thermal-stress analysis, results are

plotted shown in below figure 9. Here temperature distribution, total heat flux,

deformation of the bearing and maximum stress in the bearing with respect to

rotational velocity is shown.

Figure 9 Temperature Distrubtion, Total Heat Flux, Deformation and Equivalent Stress



The steady state thermal

finite element method and all the results are plotted with respect to rotational velocity.

The change in heat generation value with respect to the rotational velocity is plotted in

figure 10. At high speed, thermal effects on the dynamic

must be considered. When bearing speed increases, there is an increase of heat

generation at the bearing contact locations which generates thermal load to the

bearing. The torque in the bearing which generally developed due to rota

and loads are the main source of heat generation. The thermal load affects bearing

stiffness and hence the dynamic response, and subsequently will change the heat

generation at the bearing contact location. So the heat generation in bearing i

the major causes for complex thermal expansion in the thermo

bearing system.

Figure 11 shows the variation in temperatures of the bearing rings at a range

(5000-55000 rpm) of rotational speeds

inner ring contact forces and their corresponding heat generation rate higher than

those of the outer ring. This shows in the plot that the bearing inner ring temperature

is higher than the outer ring temperature.

shown in Figure 12.

Figure

Figure 10 Bearing heat generation

function of rotational speed



The steady state thermal-stress analysis is performed on the model by means of



At high speed, thermal effects on the dynamic response significant and



bearing. The torque in the bearing which generally developed due to rota



generation at the bearing contact location. So the heat generation in bearing i

the major causes for complex thermal expansion in the thermo-dynamic spindle

shows the variation in temperatures of the bearing rings at a range

of rotational speeds. In a rotating bearing, centrifugal forces make


the outer ring. This shows in the plot that the bearing inner ring temperature

an the outer ring temperature. Maximum stress at various speeds is as

Figure 12 maximum stress at various speeds

eat generation rate as

function of rotational speed Figure 11 Bearing rings temperatures for

different speeds


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model by means of



response significant and



bearing. The torque in the bearing which generally developed due to rotational speeds



generation at the bearing contact location. So the heat generation in bearing is one of

dynamic spindle

shows the variation in temperatures of the bearing rings at a range

. In a rotating bearing, centrifugal forces make


the outer ring. This shows in the plot that the bearing inner ring temperature

Maximum stress at various speeds is as

gs temperatures for

different speeds



8. CONCLUSION A thermal model was developed to study the heat generation rate, temperature

distribution, deformation and thermal stress occurred in the bearing system at various

stages with rotational speed as parameter and preload applied to a feed system. The

thermal stress simulation was conducted, and it was observed from the simulation

that the temperature in the bearing increases with increase in heat generation

developed by bearing and also it was found that, at the rotational speed of 5000rpm,

the maximum temperature in inner ring is 41.90

C and temperature at outer ring is

40.750

C. The effects of bearing stiffness with respect to variable preload for different

bearing speeds have been studied and it has been concluded that the rotational speeds

have more effect on radial stiffness of the bearing which tends to decrease with

increase of speeds.

REFERENCES

[1] Mohammed A. Alfares., Abdallah and A. Elsharkawy., 2003, “Effects of

axial preloading of angular contact ball bearings on the dynamics of a

grinding machine spindle system”, Journal of Materials Processing

Technology 136, pp.48–59

[2] J. Jedrzejewski and W. Kwasny., 2010, “Modelling of angular contact ball

bearings and axial displacements for high-speed spindles”, Manufacturing

Technology 59, pp.377–382

[3] WANG Bao-min., MEI Xue-song and HU Chi-bing., 2010, “Effect of inner

ring centrifugal displacement on the dynamic characteristics of high-speed

angular contact ball bearing”, International Conference on Mechatronics and

Automation

[4] Ji Gao and Rui Zhang., 2010, “Contact simulation of thrust ball bearing based

on Ansys”, Advanced Materials Research Vols. 154-155, pp.1629-1633

[5] Chao Jin., Bo Wu and Youmin Hu., 2011, “heat generation modelling of ball

bearing based on internal load distribution”, Tribology International

[6] B. Guo., Y.Q. Han and W.J. Lei., 2011, “Finite element analysis of hybrid

ceramic ball bearing contact”, Key Engineering Materials, Vols.474-476, pp.

2064-2070

[7] Cheng Wang., Wei Yu1 and Chengzu Ren., 2011, “An accurate method for

calculating the contact subsurface stress field of hybrid ceramic ball bearing”,

Solid State Phenomena Vol. 175, pp.215-218

[8] Dong-man Yu., Chang-pei Shangb., Di Wang and Zhi-hu Gao.,2011,

“Bearing loads study for high-speed motorized spindle”, Key Engineering

Materials Vols.480-481, pp.1511-1515.

[9] M. Chandra Sekhar Reddy and Talluri Ravi Teja. New Approach to Casting

Defects Classification and Optimization by Magma Soft. International

Mechanical Engineering and Technology, 5(6), 2014, pp. 25-35.

[10] Baojian Yin and Xintao Xia., 2011, “Key technology of contact problem of

deep groove ball bearing based on Ansys”, Advanced Materials Research,

Vols. 230-232, pp.1067-1071.

[11] Dr. Fathi Al-Shammaa and Khawla A .Al-Zubaidy. The Effect of Dynamic

Impact Loading with Combined Buckling Stresses on The Dynamic Surface

Crack Propagation In Plates Subjected To Thermal Stresses. International

Journal of Mechanical Engineering and Technology, 4(6), 2013, pp. 122-

137.

[12] A.Zahedi., M.R. Movahhedy., 2012, “Thermo mechanical modelling of high

speed spindle”, Scientia Iranica, Scient-278, pp.1-12.

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