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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, 40-019 Katowice, Poland

*Corresponding author. E-mail: grzegorz.perun@polsl.pl

APPLICATION OF THE FINITE-ELEMENT 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.

wyniki przeprowadzonych badań

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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 single-row 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)

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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

0 2040 6080100 120140 160

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 non-linear 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 2-D 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

three-dimensional 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.

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38 B. Łazarz, G. Peruń, S. Bucki

In the model created by means of FEM, individual parts of the bearing interact. This necessitated

entering additional contact elements in the Abaqus programme’s solver. They are defined through an

additional contact surface superimposed on the external surfaces of solid elements. In the case under

consideration, those elements were placed on the inner sides of the raceway and on the rolling element

(Fig. 5). The direction of the contact elements was defined as well as the type of interaction between

the elements (by introducing the friction coefficient into the simulation) and the interacting contact

surfaces.

Fig. 5. Contact surface of raceway and ball

Rys. 5. Powierzchnia kontaktowa bieŜni i kulki

Material properties were set for the model made of bearing steel described with Young modulus,

E=210 000 MPa, and Poisson’s ratio, ν=0.3. The calculations were performed with use of four models

of a bearing, differing in the value of congruence coefficient w, whose values amounted to 0.95, 0.96,

0.97 and 0.98, respectively. This coefficient, according to [8], is defined as a ratio of the ball radius to

the raceway radius.

4. CALCULATION RESULTS

As a result of the calculations, values of stresses and loads were obtained as a function of

displacement of the tested system’s elements. Fig. 6 presents contour lines of reduced stresses for the

model whose congruence coefficient equalled 0.98. A diagram of characteristics of the joint stiffness

for various values of congruence coefficient is shown in Fig. 7.

Fig. 6. Results of FEM analysis – contour lines of reduced stresses for congruence coefficient of 0.98

Rys. 6. Wyniki analizy metodą elementów skończonych – warstwice napręŜeń zredukowanych dla

współczynnika przystawania 0,98

Contact elements

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Application of the finite element method for determining the stiffness of rolling bearings 39

Radial load of the rolling element, kN

0

5

10

15

20

25

30

35

40

020406080100120140160180200

wyznaczona analitycznie

w=0,98

w=0,97

w=0,96

w=0,95

Total deflection of a rolling element and raceway, µm

Fig. 7. Stiffness characteristics of the outer raceway - rolling element - inner raceway system determined with

use of FEM for various congruence coefficients and characterization determined analytically

Rys. 7. Charakterystyki sztywności układu bieŜnia zewnętrzna - element toczny - bieŜnia wewnętrzna określone

z uŜyciem MES dla róŜnych współczynników przystawania oraz charakterystyka wyznaczona analitycznie

It can be noted from the diagram that the characterization obtained for the congruence coefficient

w=0.97 turned out to be the most similar to the results of analytical calculations.

5. CONCLUSIONS

It appears from the tests that the congruence coefficient has a significant influence on the stiffness

values of the modelled system obtained by means of numerical analysis. For this reason, it is necessary

to accurately reflect the bearing’s geometry in the model in order to obtain correct results. Therefore,

precise knowledge of the dimensions, including first of all the raceway radius and the rolling element

diameter, is required. Further calculations have also shown that it is possible to create a correct FEM

model of a bearing, allowing the determination of its stiffness in taking into consideration any possible

damage and wear of its elements.

Scientific work financed from funds earmarked for science in the years 2006-2009 as a research

project.

Bibliography

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w układzie mocy krąŜącej. Zeszyty Naukowe Politechniki Śląskiej, seria Transport, z. 63, Gliwice,

2006, s. 163-172.

2. Łazarz B.: Zidentyfikowany model dynamiczny przekładni zębatej jako podstawa projektowania.

Instytut Technologii Eksploatacji, Katowice – Radom, 2001.

3. Müller L.: Przekładnie zębate. Dynamika. Wydawnictwa Naukowo-Techniczne, Warszawa, 1986.

4. Krzemiński - Freda H. ŁoŜyska toczne. Państwowe Wydawnictwo Naukowe, Warszawa, 1985.

determined analytically

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40 B. Łazarz, G. Peruń, S. Bucki

5. SKF Katalog główny. SKF, 2007.

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7. Łazarz B., Peruń G.: Modelowanie łoŜysk tocznych w układach napędowych z przekładnią zębatą.

XXXV Jubileuszowe Ogólnopolskie Sympozjum Diagnostyka Maszyn, Węgierska Górka, 2008.

8. Smolnicki T., Rusiński E., Malcher K.: Modele dyskretne łoŜysk wieńcowych w maszynach

podstawowych górnictwa odkrywkowego. III Konwersatorium Bezpieczeństwo oraz degradacja

maszyn, Wrocław – Szklarska Poręba, 1997.

Received 4.02.2008; accepted in revised form 23.09.2008