THERMAL STRESS ANALYSIS OF A BALL BEARING BY FINITE ELEMENT METHOD

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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
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ISSN Print: 0976-6480 and ISSN Online: 0976-6499
© IAEME Publication
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THERMAL STRESS ANALYSIS OFA 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.
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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

dm
di 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
f
r
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.

THERMAL STRESS ANALYSIS OF A BALL BEARING BY FINITE ELEMENT METHOD

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