Application of the finiteelement method for determining the stiffness of rolling bearings
ABSTRACT The paper presents the results of numerical tests performed with use of the FEM method, the aim of which was determining the stiffness of the outer raceway  rolling element  inner raceway system of bearing 6307. The characterization obtained has been compared with a characterization determined with analytical methods and in the next stage, it will be used to determine the total stiffness of the bearing, variable in working time. Correct modelling of bearing stiffness is one of important conditions for obtaining correct results of simulation calculations. Obtained results will allow the determination of possibilities of limiting vibroactivity of toothed gears, commonly used in transport.
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Article: Fatigue life prediction of the radial roller bearing with the correction of roller generators
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ABSTRACT: The paper presents the methodology of fatigue life prediction of radial cylindrical roller bearings, which allows to take into account in the fatigue life calculation geometric parameters of the bearing, including radial clearance and the profiles of rollers. In addition, the methodology takes into account the effect of combined load and misalignment of the bearing rings on the fatigue life. The stress distributions which are necessary to calculate the predicted fatigue life were determined by solving numerically the Boussinesq problem for elastic halfspace. The Lundberg and Palmgren model was used for the calculation of the predicted fatigue life of the bearing. The paper focuses on determining the effect of roller profiles on the bearing fatigue life. Pressure distributions obtained by the described methodology were compared to the distributions determined according to the finite element method. The calculated fatigue life of cylindrical roller bearing was compared with the experimental results.International Journal of Mechanical Sciences 12/2014; 89:299–310. · 2.06 Impact Factor
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TRANSPORT PROBLEMS 2008
PROBLEMY TRANSPORTU Tom 3 Zeszyt 3
Bogusław ŁAZARZ, Grzegorz PERUŃ*, Sławomir BUCKI
Silesian University of Technology, Faculty of Transport
Krasińskiego St. 8, 40019 Katowice, Poland
*Corresponding author. Email: grzegorz.perun@polsl.pl
APPLICATION OF THE FINITEELEMENT METHOD FOR DETERMINING
THE STIFFNESS OF ROLLING BEARINGS
Summary. The paper presents the results of numerical tests performed with use of the
FEM method, the aim of which was determining the stiffness of the outer raceway 
rolling element  inner raceway system of bearing 6307. The characterization obtained
has been compared with a characterization determined with analytical methods and in the
next stage, it will be used to determine the total stiffness of the bearing, variable in
working time. Correct modelling of bearing stiffness is one of important conditions for
obtaining correct results of simulation calculations. Obtained results will allow the
determination of possibilities of limiting vibroactivity of toothed gears, commonly used
in transport.
ZASTOSOWANIE METODY ELEMENTÓW SKOŃCZONYCH DO
WYZNACZANIA SZTYWNOŚCI ŁOśYSK TOCZNYCH
Streszczenie. W artykule przedstawiono
numerycznych z uŜyciem metody MES, których celem było określenie sztywności
układu bieŜnia zewnętrzna  element toczny  bieŜnia wewnętrzna łoŜyska 6307.
Otrzymana charakterystyka została porównana z charakterystyką wyznaczoną metodami
analitycznymi i w kolejnym etapie zostanie uŜyta do wyznaczenia całkowitej, zmiennej
w czasie pracy, sztywności łoŜyska. Prawidłowe zamodelowanie sztywności łoŜysk, jest
jednym z warunków uzyskania poprawnych wyników obliczeń symulacyjnych,
prowadzonych w celu znalezienia moŜliwości ograniczenia wibroaktywności przekładni
zębatych, powszechnie stosowanych w transporcie.
1. INTRODUCTION
Applying the FEM method for modelling a system comprised of an inner and outer raceway of the
bearing and the rolling element between them is the next step of works on the method of calculating
stiffness of bearings. This method is being developed for the needs of a model stand for testing toothed
gears operating in a circulating power system [1]. Correct modelling of bearing stiffness is one of
important conditions for obtaining correct results of simulation calculations and so it will allow
reduction of the number of laboratory tests. It is expected that the results obtained with use of the
model developed will allow the determination of possibilities of limiting vibroactivity of toothed
gears, being the major element of power transmission systems.
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34 B. Łazarz, G. Peruń, S. Bucki
In the discussed calculation method, it was assumed that the bearing stiffness depends on the
stiffness of the outer raceway  rolling element – inner raceway systems being under load [7]. These
stiffnesses are nonlinear functions of load imposed on the rolling element, thereby dependent on its
position in relation to the direction of the force. Such approach allows taking into consideration
disfunctions of the vibration signal resulting from the changeable bearing stiffness caused by both,
changes in the position of rolling elements in relation to the direction of the force and damage and
wear and tear of the interacting bearing elements.
2. METHODS OF DETERMINING THE STIFFNESS OF BEARINGS
Descriptions of many methods of calculating bearing stiffness can be found in professional
literature. The ones more precise require much data concerning the elements of a bearing [2], which
most often are not available in manufacturers’ catalogues. For this reason, approximate methods are
used most frequently, which allow determining the values of bearing deformation, depending on the
value of load imposed on the bearing.
On the modelled stand, eight singlerow ordinary ball bearings 6307 are mounted. Radial
dislocation of the journal of such bearing, determined as a function of maximal load of the rolling part
and of the rolling element’s diameter, is described by dependence (1) [2, 4]. If, additionally, the
number of rolling elements is known, dependence (2) can be used [2, 3].
[]
m
D
Q
t
r
µ
α
δ
,
cos
44
1
, 0
3
3
max
2
⋅
⋅
=
(1)
where: Qmax − maximal load of the rolling part [N], Dt – rolling element diameter [mm], α – bearing
operation angle [rad].
Q
r
1 , 0
⋅
where: dk – diameter of the bearing ball [mm], R − radial load of bearing [N], e – number of rolling
elements in the bearing.
For small values of the bearing operation angle, whose value is used in formula (1), considerable
occurrence of stiffness characteristics can be obtained for a number of rolling elements equal to 5,
however, the 6307 bearings installed on the stand, depending on the manufacturer, have 7 or 8 rolling
elements.
The method suggested in this paper [7] requires determining the number of rolling elements under
load as well as the stiffness characterization of one system: outer raceway – rolling element  inner
raceway. In order to determine the number of rolling elements under load, the knowledge is necessary
concerning the size of radial clearance of the bearing (for the 6307, the adequate values were taken
from [5]) and the load distribution angle, ψε – Table 1.
In order to determine the load distribution on individual rolling elements and the maximal load
δmax of the rolling part, a notion of the load distribution angle coefficient is introduced [4]:
δ
ε
222
max
+
g
[]
e
R
2
Qm
d
k
;, 96, 0
3
2
=⋅=
µδ
(2)
()
ε
ψ
δδ
cos1
2
1
1
1
max
max
−=
+
−==
g
g
d
(3)
Page 3
Application of the finite element method for determining the stiffness of rolling bearings 35
Tab. 1
Load distribution angle ψε for various values of radial clearance g
and deformation δmax of bearing 6307
C2 clearance [µ µ µ µm]
normal clearance [µ µ µ µm]
min max min
1 11 6
87,3 69,2 76,7
88,6 77,5 82,5
89,1 81,1 84,8
89,3
83,1 86,0
89,4 84,3 86,8
89,5 85,2 87,3
89,6 85,8 87,6
89,6 86,3 87,9
89,7 86,7 88,2
89,7 87,0 88,3
89,7 87,3 88,5
89,8
87,5 88,6
89,8 87,7 88,7
89,8 87,8 88,8
89,8 88,0 88,9
89,8 88,1 88,9
It arises from here that the load distribution area, ψε , depends on the maximal deformation δmax and
clearance g in the bearing [4]:
C3 clearance [µ µ µ µm]
min
15
64,6
74,2
78,5
80,9
82,5
83,6
84,4
85,1
85,6
86,0
86,3
86,6
86,9
87,1
87,3
87,4
ψ ψ ψ ψε [°]
[°] [°]
[°]
max
20
max
33
δ δ δ δmax
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
160
60,0
70,5
75,5
78,5
80,4
81,8
82,8
83,6
84,3
84,8
85,2
85,6
85,9
86,2
86,4
86,6
51,5
63,1
69,2
73,0
75,6
77,5
79,0
80,2
81,1
81,9
82,5
83,1
83,5
83,9
84,3
84,6
)
2
arccos(
max
g
g
+
=
δ
ψε
(4)
Load distribution on the rolling elements in the bearing, depending on the value of the load
distribution angle is shown in Fig. 1 [4].
εd=1
εd=0,5
ψε=180°
ψε=90°
0 <εd< 0,5
0 <ψε≤ 90°
Fig. 1. Load distribution on rolling elements in the bearing, depending on the load distribution angle
Rys. 1. Rozkład obciąŜenia na części toczne w łoŜysku w zaleŜności od wartości kąta rozkładu obciąŜenia
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36 B. Łazarz, G. Peruń, S. Bucki
An aspect of the bearings’ modelling method described in [7], more difficult to solve, is correct
determination of stiffness of the raceways – rolling element system. A characterization of stiffness of
a rolling element – raceways system, determined by analytical methods based on geometrical
dependencies in the bearing and with use of the above formulas, is shown in Fig. 2 [7].
25000
Radial load of the rolling element, N
0
5000
10000
15000
20000
020406080100120140160
Total deflection of a rolling element and raceway, µm
Fig. 2. Stiffness of a rolling element – bearing’s raceways system determined analytically
Rys. 2. Wyznaczona analitycznie sztywność układu element toczny  bieŜnie łoŜyska
3. DETERMINING THE STIFFNESS OF A ROLLING ELEMENT – BEARING RACEWAYS
SYSTEM BY FEM
In order to verify the stiffness characterization obtained analytically, the researchers decided to
model the rolling element – bearing raceways system and determine the stiffness with the FEM
method. Correct modelling of this system is a determinant of preparing a correct model of the whole
bearing, in which the determined characterization may be one of the input data. Stiffness
characterization of such system is nonlinear and, since the connection transmits only compression,
unilateral [6].
Since the FEM models allow free representation of the raceway surface, it is possible to [6]:
• determine the real characterization of force – deflection,
• determine the dependencies between the force and the state of stress in the rolling element and
in raceways,
• determine plastic deformations of the raceway,
• taking into account the rolling expansion (plastic deformation) and wear of the raceway.
When determining stiffness of the outer raceway – rolling element – inner raceway system, it is
possible to use a discrete 2D circular symmetric model through linearization from the radial direction
to the circumferential direction of the bearing [6]. In the study, however, the authors decided to apply
threedimensional models, which allow direct usage of the results obtained. For their creation,
a geometric model was used, consisting of a fragment of bearing circumference with a symmetrically
situated (between the raceways) rolling element (Fig. 3).
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Application of the finite element method for determining the stiffness of rolling bearings 37
Fig. 3. Geometric model of the system: rolling element – bearing raceways
Rys. 3. Model geometryczny układu element toczny  bieŜnie łoŜyska
A grid of finite elements was created in HyperMesh programme, with use of C3D8I and C3D6
hexahedral elements from the finite elements library of the Abaqus application and taking into
consideration two symmetry planes of the model. In the contact place of the rolling element with the
raceway, the researches decided to compact the lattice (Fig. 4).
Fig. 4. FEM model of the system: rolling element – bearing raceways with a finite element grid
Rys. 4. Model MES układu element toczny  bieŜnie łoŜyska z nałoŜoną siatką elementów skończonych
The amount of all elements was 41 132 with 42 789 grid points. The points located on the outer
surface of one of the raceways were taken all degrees of freedom away. Taking the symmetry planes
of the model into consideration, tantamount to simplifying the FEM method to one fourth of the
original geometric model, allowed significant shortening of the calculation time, but, simultaneously
extorted the introduction of slip planes. In the YZ plane, the tested bearing element was deprived of
the possibility of moving towards X, while in the XY plane, towards Z. For the declared dislocations
of the inner raceway towards the outer raceway along axis Y, the load values were read out, i.e. the
reaction forces in the place of fastening the bearing.