Ball Bearings Static

7/28/2019 Ball Bearings Static

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BALL BEARINGS STATICBEHAVIOR AND LIFETIME

Milan ZELJKOVIAleksandar IVKOVI

Ljubomir BOROJEV

Abstract: In the paper an analysis of the previous

research results related to the static behavior and life of

ball bearings is presented. In addition, theoretical basis

for the determination of deformation, stiffness, and

change the contact angle and life of ball bearings with

angular contact are shown. Based on that a nonlinear

mathematical model for analysis of the static behavior of

ball bearings has been developed. In this paper only

some of the outcomes of previous research are presented.

Presented results are related to the special ball bearings

with angular contact as follows: single-ball bearing formain spindle assembly of machine tools and two rows ball

bearings (HUB unit bearings) for wheel cars.

Key words: Bearings with angular contact, HUB unit

bearings for wheel cars, Static behavior

1. INTRODUCTIONRoller bearings and/or bearing assemblies are now widelydistributed from cars, machines of all types, to the a large

number of other products. Although these bearings hasexperienced its peak decades ago, is still, as in many otherareas, cannot say that there are no outstanding issues, ie.interesting area for research. The need for differentconstruction roller bearings, with the development ofmachining techniques, there is more to the fore, especiallysince for the past few years, the internal structure is notsignificantly changed, and the increasing demands forspeed, stiffness and bearing life are significantlyincreased. This increase is particularly related to the

precision ball bearings with angular contact of supportmain spindle machine tools, as well as special integratedwheel car bearings (HUB unit bearings).

Rapid development of the care and machine tool industryaccelerated the standardization and mass production ofroller bearings. A variety of applications forced the

producers of roller bearings on extensive research

procedures, directed at bearings themselves, exploitationconditions and specific machine requirements.

2. PREVIOUS RESEARCH REVIEWBy carefully observing the basic mechanism of roller

bearing, it is evident that it is based on the exploitation ofimportant mechanical properties of materials. It almost

seems that there is not a similar machine where theelement has so difficult exploitation conditions. The entireload is transferred through several roller elements, whichrealize the ring contact to the point or along the line.Even at moderate loads, this causes extremely highconcentration of contact force / stress. Point of contact isconstantly moving with the turn of the rings, so that thematerial is exposed and that the extreme conditions thathave to express the dynamic character of the load. Underthe influence of such loads, coupled parts can deformationand based on Hertz theory, contact surface has the shapeof an ellipse [2], [7], [8], [11]. If the value of the racewayradius is approximately equal to ball radius, the load

bearing increases, therefore reduce the maximum speed,and vice versa [10]. In addition, there is rolling correction

profile raceway, so that the raceway of balls rollingcontact is realized in three or four points [20].It should also be noted that the phenomenon of crossingover the current roller elements from unload to the loadstate is is present, followed by an intense pulse loads.Roller bearing load is transferred through the rollerelements of the inner to the outer ring or vice versa. Thesize, layout and transmitted to the load and stiffness foreach individual rolling body depends on the internalgeometry of bearings. Credible analysis of the distributionof load and stiffness roller bearings should include non-linear load and the connection between the contactdeformation and load. For the analysis and calculation of

bearing a different mathematical models are used.Analysis of the developed models for the study of static

behavior of bearing has been shown that from the point ofwiev of design phase these models can be classified intotwo groups as follows: the previous calculation (sizing and optimization of

basic geometric size); final calculation (check bearing behavior

identification).In addition a review of previous research related to the

bearing behaviour identification are shown.Lin [9] analyzes the displacement and the coefficient ofstiffness for radial ball bearings with angular contact fromthe production of SKF. For the analysis John Harris'smethod based on the theory of Hertz-contact has beenused. In order to compare the results obtained by thementioned method and Palmgren's empirical relations thefinite element method has been used.

Mullick [11] researchs radial stiffness of radial bearingsand boll bearings with angular contact using John Harris'smethod and the finite element method. For solvingsystems of nonlinear equations, using Newton Raphson'smethod, while in contact analysis uses finite elements

method. The influence of sliding and gyroscopic momentis neglected. The results showes that the relativedisplacement and stiffness rings of bearings depend on theradial, axial, the combined load and centrifugal force.


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Antoine et al. [2], [3] propose two new approximate,methods for determination the angle of contact on theouter and inner bearing depending on the preloaded andspeed for special cases of elastic preloaded. Methods

based on the theory of Hertz-contact. They start fromassumptions that the force of preload constant and doesnot affect on the speed and change the angle of contact. Insolving the system of equations, it is considered that acertain speed, for a preload comes only to the cancellationof the axial clearance, and that there is no axialdeformationSince Hertz's contact theory based on a number of tabulardata, and it provides the basic input data for John Harris'smethod, Kang et al. [8] modify this method. Using finiteelement method and the empirical relationship (exponent)

between stress and strain in the theory of Hertz-contacthas been modified.

Bourdon A. at al. [4], [5] propose a general methodologyfor modeling nonlinear behavior of ball and roller

bearings. Models og the stiffness matrix of complex

mechanical systems were developed, in order to predictthe static behavior, load and stress distribution. Themethod was applied to determine the deformation of thering gear bearings for cars and helicopters. For allconsidered cases of deformation of the ring bearings aresignificantly influenced by the change of contact angleand load distribution.Sun M. K. at al. [15] investigated the relationship

between contact deformation, clearance and change ofstiffness bearing. The results obtained by analytical andexperimental method suggest that to determine the elasticdeformation of bearing elements must be taken intoaccount axial and radial clearances in the bearing.

Wei L. at al.. [21] investigated the influence of preload,centrifugal force and gyroscopic moment on the bearingstiffness. They concluded that if the value of contact angleof ball and raceway exceed 8.9 , the value of the radial

bearing stiffness decreases with increasing speed.Experiments have shown that by contact angle 40 andrevs of 15,000 rpm radial stiffness decreases more than20%. They also noted that an increase in temperatureaffects the increase in preload of bearing and increase thenatural frequency of oscillation.

Abele E., and V. Fredler [1], behavior of roller bearingbody at different speeds of the main spindle have beeninvestigated by analytical method. By increasing the

speed, centrifugal force reduces the contact angle on theoutside of raceway. On the other hand, on the inner ringappears reduction of contact force with increasing contactangle. Increasing the angle difference between the innerand outer contact ring bearing stiffness decreases. Atmaximum revs radial stiffness of the front bearing isreduced to 1 / 3 of the initial value. In these conditions,the rear bearing stiffness is reduced up to 40%. Stiffnessreduction is partially compesated by increase of thetemperature with rise speed (internal temperatureincreases until the outer ring, for cooling, it remains thesame).An important requirement for the assembly of the main

spindle, in modern machine tools is to achieve highspeeds. In conventional bearings contact angle on theinner and outer ring exists a large deviation of theincrease speed due to centrifugal force. Axial

displacement of the inner ring (elastic mounted bearing)and increasing the normal force in the area of contact onthe outer ring are typical effects caused by increasinginternal load bearing and reduction in bearing life Basedon these findings, Weck E. M. et al. [20] investigated the

bearings with the new internal geometry. Instead twocontact zones , these bearings have three or four zonecontact in order to ensure constant contact angle anddecrease the normal force load on the inner ring.Wang L. at al. [19] investigated roller bearings withceramic elements which have been realized in the pastdecade. Based on these studies it can be concluded thatthe ceramic roller elements of the material (Si3N4), can

be used in extreme conditions. Compared with steel rollerelements, hybrid bearings have significant advantages interms of life. The smaller density of the material greatlyreduces the dynamic load on the body and the racewaydue to the smaller centrifugal force, especially inmachines where the high speeds are required.Elastic displacements in bearing consist of: a) elastic

displacement between the body and roller bearing ringsand b) contact displacement on the surface of the innerring fitting on the sleeve and the outer ring in the hausing.In previous works are taken into account only the elasticdisplacement between the roller body and the ring interms of Hertz's assumption of an ideal form of the roller

body and rings, with the clearance. The influence ofcentrifugal force, gyroscopic moment, and temperature onthe change of angle of contact and nonlinear bearingstiffness have been less investigated. On the other hand,the impacts of positive or negative clearance in the elasticdisplacement are largely represented. Influence ofdeformation on the surface fitting of the inner ring and the

sleeve, and the outer ring and a housing on elasticstiffness of support has been introduced by Sun. M. K[15]. For approximate determination of the deformationdue to elastic deformation of contact between the roller

body and rings, without knowledge of the geometry ofraceway for some types of bearing, are the terms proposed

by several authors Brandlein, J. [6], Tedric, A. H.Michael, N. K. [16], Tedric, A. H. [17] as well as bycertain manufacturers of bearings - SKF [ 14]Zaretsky E. V. at al. [22], on the basis of Ludenberg-Palmgren theory, analyzed bearing life with ceramic andsteel balls. Under normal operating conditions the resultof life radial ball bearings and ball bearing with contact

angle have been shown. Bearings with steel balls underthe same load have longer life than the equivalent hybridbearings under the same conditions, according to theserelations. However, the experimental and experientialdata, the authors state that hybrid bearings have a muchhigher life than the life obtained with Ludenberg-Palmgren relation. This error appears from theassumptions Ludenberg-Palmgren's no osculation of ringsand angles of contact are unchanged, and the destructioncaused by crack occurring under the surface contact at adepth that corresponds to the maximum tangential stress[10]. Inaccuracy of the first assumption can be shownusing the exact model of elastic-deformation of rolling

bearing on which is possible to determine the actualcontact load, or the equivalent dynamic load [16]. Thesecond assumption was justified for materials andconstruction of roller elements which are used 30 to 40's


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of last century, at the time when the above mentionedtheory emergences. Today, however as a result ofsignificant improvement of material, from which theelements are made, bearing and increased accuracy of

production parts, usually bearing failures do not occur dueto the subsurface destruction, but due to the of surfacedestruction and wear [12].

3. A MATHEMATICAL MODEL OF BALLBEARINGS

In the contact areas of the rolling body and raceway, butthe normal forces created stresses in the main directions,which are far above normal in the other mechanicalelements. Static terms, support raceway-rolling body -raceway, the uncertain system. Such a system is difficultto solve by the usual methods, and becomes very complexwhen taking into account the effects of clearance in the

bearing and the contact angle changes due to effects ofstatic force and centrifugal force and gyroscopic moment.

In order to fully determine the static characteristicsbearing such as load in the bearing, elastic deformation,stiffness and change the contact angle must be set of

balance equations that are nonlinear. Solving the set ofequations that requires knowledge of the internalgeometry of bearings.

3.1. The parameters of the contact surfaceTouch the two curved body is completely defined byHertz's contact theory [7], [8] [9], [16], [17] (Fig. 1). Forthe calculation of two body contact and surface pressuresoccur at the same time, an important role radius ofcurvature (rI1, rI2, rII1, rII2) that the bearing is always

consistent with the main plane of curvature (Fig. 2). If weobserve the thrust bearing section, we can see that the

profile of body is more convex than roller body the profileof concave curved paths, which means that the roller bodyand achievedraceway have contact at one point. This typeof contact exists at all ball bearings.

Fig. 1.Geometry ofcontacting bodies [16]

Fig. 2. Ball bearing

geometry [17]

Assuming that both bodies have a common point ofcontact have a common tangent plane and a commonnormal line in which is load, contact between the roller

body and raceway the ball bearings with angular contactis defined by

1.Curvature radius sum:

1 1 24

1uk s

d r

=

(1)

2. Curvature difference:

( )

1 2

11 2

41

s

u

s

rF

r

=

(2)

0cosk

m

d

d

= (3)

where is: dk-diameter of ball, ru i rs radius of inner, outerraceway respectively, dm medium diameter of bearing, 0initial of contact angle.

3.2. Connection between deformations and loadOn the basis of Hertz's contact theory [16] can be definedcontact load bearing:Q= Knn

1,5 (4)

Normally displacement between raceways that are underload is the sum of displacement between the roller bodyand racewaysFrom it follows [7]:

n u s = + (5)

and the rigidity of bearing along the lines of contact, Kn,in the function of contact stiffness of roller body andraceway:

3/ 22 / 3 2 / 31 1

n

s u

KK K

= +

(6)

In the above expression, Ks and Ku are the contactstiffness between roller bodies, external or internalraceway.Two of the contact stiffness is a function of geometry andmechanical properties of materials bearing and can bedetermined from the relation [7]:

( )( )

11/ 2

/ // 3/ 2*

//

1 11,6568 s u k s us u

k s us u

KE E

= +

(7)

3.3. Determination of axial deformation, and thecontact angle of bearing due to axial loading

By ball bearings under the influence of axial load, theload is distributed equally to all roller elements [16]. So:

sina

FQ

Z = ; where Z number of roller body (8)

If we neglect the effect of centrifugal force angle ofcontact between roller body, external and internalraceway is the same. Therefore, it is higher after preload

(Figure 2). Preload causes axial displacement a. Axialdisplacement is a component of normal displacement nroller body along the lines of contact (Figure 2) [16].

0cos 1cosn k

B d

=

(9)

where B is the total curvature raceway.

Fig. 3. Angular-contact ball bearing under thrust load [7]


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Taking into account the relationship (4) and (9) thecontact load could be obtained as [8] [17]:

3/ 2 0cos 1cosn

Q K A

=

(10)

Substituting relations (9) in relation (10), is obtained:

03 / 2

cossin 1

cosa

n

F

ZK A

=

(11)

Stability and convergence of the function largely dependson the initial angle. In order to predicted angle be equal tonominal contact angle, relation (11) must be expressedthrough cos:

12 / 3

0 3 / 2 2cos cos 1

1 cosa

n

F

ZK A

= +

(12)

Relation (12) could be solved by numerical method forthe condition (0) = 0. Previous relationship is valid forthe case when a preload is known. Axial shift a is relatedto n and the Figure 3 is determined as:

( )0

sin sina n

A A = + (13)

or( )0sin

cosaA

= (14)

In the case of two rows ball bearings with angular contact,as is the case for an HUB unit bearing, on both racewayswill undergo axial deformation, due to preload that

provides contact between the roller body and theraceways. Increasing the external load bearingrelationship between the deformation decreases, and thus

pre aims to reduce the deformation of the bearing theadditional external load. In this situation the total axial

deformation of the bearing 1 is:1 p a = + (15)

and the bearing 2:

2 0

p a a

p a

> =

(16)

If we take into account the external axial load acting onthe bearing, then from relations (11):

0 01 23 / 2

1 2

cos cossin 1 sin 1

cos cosa

n

F

ZK A

=

(17)

or if the previous relationship is expresed by cos2function, we obtain:

( )

( )

1/ 2

22

2 21

1/ 2 21

1 coscos

1 cos

cosa

n

F

ZK A

= +

(18)

combination of relations (15) and (16), is obtained:

1 2 2 p + = (19)

Replacing relations (15) for the 1 and relationships (16)in relation to 2 (14) gives:

( ) ( )1 0 2 0

1 2

2 sin sin

cos cosp

A

= (20)

Relationships (18) and (20) are solved for the 1 and 2 bynumerical methods. Axial deformation p and the contactangle p due to preload can be obtained from relations(12) and (11) where in this case p= a i cos= cosp.

3.4. Determination of deformation and thecontact angle of the bearing due to thecombined load

In many cases, on the bearing (especially in the HUB unitbearings) act combined loads (axial and radial), whichsubstantialy change load, deformation and contact stresson the roller bodies. When on the roller body acts contactload under certain angle, center of curvature of racewaysare fixed versus the appropriate raceways, while thedistance between the centers of raceway increases (Figure4)

Fig. 4. Ballraceway contact : a) before applaying load,

b) after applaying load [16]

n ss A = + + (21)

n u ss A = + = (22)

Based on Figure 5 and 6 can be determined the position ofthe center of the inner and outer raceway in radialdirection in the uncharged condition.

Fig. 5. Loci of raceway

groove curvature radii

centers before applying

load [17]

Fig. 6. Ball bearing

showing ballraceway

contact due to axial shift of

inner and outer rings [17]

0cos2 2m k

u u

d dR r

= +

(23)

while:

0coss uR R A = (24)where: dm-medium diameter of the bearing, ru-radius ofinner raceway, dk- diameter of ball, 0 initial of thecontact angle, A-distance between the center of the outerand inner radius of raceway.If the inner ring not rotates, then the outer ring and thecenter of the outer raceway move. The distance between

the center of curvature (s) inner and outer raceway isdetermined by [3]:

( ) ( )1/ 22 2

0 0sin cos cosa rs A A = + + +

(25)


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where the: /a a A = i /r r A

= . In the previousexpression: a and r are axial and radial deformation ofthe bearing, -angle of the zone load.Substituting relations (25) in (22) gives the bearingstiffness in the direction normal to the raceway [17]:

( ) ( )1/ 22 2

0 0sin cos cos 1n a rA = + + +

(26)

Based on Hertz's contact theory for the relationshipbetween deformation and load, you get a contact load ofthe roller body in any position [7], [8], [9]:

3/ 2n n

Q K= (27)

Whereas Kn is the stiffness bearing along the lines ofcontact, obtained on the basis of Hertz's contact theory.Therefore:

( ) ( )3/ 21/ 22 23/ 2

0 0sin cos cos 1n a rQ K A = + + +

(28)

For any position of ball contact angle is determined

from [7]:

( ) ( )

01/ 22 2

0 0

sinsin

sin cos cos

a

a r

+=

+ + +

(29)

or

( ) ( )

01/ 22 2

0 0

cos coscos

sin cos cos

r

a r

+=

+ + +

(30)

If the normal load on the roller body resolved into axialand radial component over the contact angle is obtained:

sinaQ Q = (31)cosrQ Q = (32)

Axial and radial load bearing is equal to the sum of thecomponents of the normal load, and [17]:

0

sina

F Q

=

=

= (33)

0

cosrF Q

=

=

= (34)

From the conditions of static equilibrium is obtained:

( ) ( ) ( )

( ) ( )

3/ 21/ 22 2

0 0 0

1/ 22 200 0

sin cos cos 1 cos cos cos

0sin cos cos

a r r

r n

a r

F K A

=

=

+ + + + = + + +

(35)

and

( ) ( ) ( )

( ) ( )

3/ 21/ 22 2

0 0 0

1/ 22 20

0 0

sin cos cos 1 sin

0sin cos cos

a r a

a n

a r

F K A

=

=

+ + + + = + + +

(36)

Relationships (35) and (36) are a nonlinear system ofequations with unknown a and r. Relationships can besolved by numerical method. After determining thedeformation can be determined maximum load of therollerbody for = 0 from the relation:

( ) ( )3/ 21/ 22 23/ 2

max. 0 0sin cos 1n a rQ K A = + + +

(37)

Substituting a and r in relation (29) or (30) determinesthe change of contact angle depending on the externalload.

4. DETERMINATION OF THE BALLBEARING LIFE

Based on the analysis of modern solutions of the bearingand bearing assemblies, for further detailed analysisthemselves of the bearing was chosen HUB unit bearingas a special case of the two rows ball bearings with

angular contact. As these types of the bearings areworking with variable load, to calculate the equivalentdynamic load the expression [14 ] should be used:

31 1 2 2 3 3I I IP t P t P t P= + + (38)

where PI1-the equivalent of the load bearing, when thevehicle is moving in a straight path, PI2 - the equivalent ofthe load bearing, when the vehicle turns left; PI3 theequivalent of the load bearing, when the vehicle turnsright and t - the proportion of straight or curve driving.Bearing life is determinated by the new theory [14], bythe relation:

1 23

p

hna t CL a a f hP

=

, (39)

where:C [kN] - dynamic capacity; P [kN] - the equivalentdynamic capacity; p=3 for the ball bearing; ft

temperature factor; a1 - a factor bearing failure

probability; factor a23 (a23 by FAG-, or aSKF - by SKF)contains in itself interdependent factors influence material(a2) and working conditions (a3).Dynamic bearing capacity significantly depends on thegeometry of raceway and bearing contact angle, ie the

preload and external load. Angle contact bearings arechanged during operation due to the rotation roller body

and the load. For these reasons, for accuratedetermination of the bearing life, dynamic capacity isnecessary to determine over relationship [4]:

( )0,30,7 3,33 3,33

C i C C

= + (40)

where C- dynamic capacity of the rotating ring and isdetermined from [4]:

( )

( )( )

1,390,410,7 2/3 1,8

1/31

1298,1 cos

1r

JR rC i Z D

D r R J

= +

(41)

while C- dynamic capacity of the non rotating ring andis determined from [4]

( )

( ) ( )

1,390,410,7 2/3 1,8

1/32

12

98,1 cos1

rJR r

C i Z DD r R J

= + (42)

where: R-radius of the corresponding raceways, r- radiusraceway; D-diameter ball; coefficient obtained from therelation (3); i - number of rows of roller body; -contactangle. In the case of single rows ball bearings withangular contact, the contact angle is determined from therelation (12), if it is axial loading. In the case of combinedload contact angle is determined from the relation (29) or(30). When it comes to two rows ball bearings withangular contact the contact angle is determined from therelation (18) and (20) in case of axial load or combinedload case from the relation (29) or (30), assuming no

effects of centrifugal force and gyroscopic moment, andthe angles of both raceway identical. Z-number of roller

body in a raceway, Jr., J1 and J2 integrals for bearing loadand the corresponding ring.


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5. REVIEW OF RESEARCH RESULTSThe mathematical model (R.M.M.) for the analysis ofstatic behavior and determination of the ball bearings lifewas developed in MATLAB programming system. Forsolving nonlinear equations Newton method has beenused. The model is verified on two special types of

bearings as follows: single rows ball bearings withangular contact 7011 and three preload sizes (small,medium and large) and the HUB unit bearing, as a specialcase of two rows ball bearings with angular contact. Someresults of deformation due to changes in static contactangle, the ball bearings with angular contact, have beencompared with the results obtained by finite elementmethod (FEM) and Palmergan's empirical relations (P.)Results of bearing life were compared with resultsobtained by preliminary experimental tests.

5.1. Change the angle of contact and deformationdue to external forces and preload

In Figure 7 is shown change of the contact angle of ballbearings with angular contact type 7011 due to the effectsof preload. In Figure 8 and 9 are shown changes in axialdeformation and stiffness depending on the preload forthe same type of bearings.

Fig. 7. Change the angle of contact due to the action of

preload

Fig. 8. Change of axial deformation, depending on the

preload

Fig. 9. Change of axial bearing stiffness depending on the

preload

The figures 10, 11, 12 show changes in deformation,stiffness and the contact angle due to the effects ofexternal axial load for bearing 7011, and for three preloadsizes (small, medium and large).

Fig. 10. Change of axial deformation, depending on the

external axial load for different preload case

Fig. 11. Change of axial stiffness depending on the


Fig. 12. Change of contact angle, depending on the


In Figure 13 changes of radial and axial deformationdepending on the radial load at a constant axial load forthe HUB unit bearing 32x58 are shown. Figure 14 showsthe changes of radial and axial deformation depending onthe axial load at a constant radial load for the same type of

bearings.

Presented load conditions correspond to exploitationconditions for a specific type of bearing vehicles. Changeof the contact angle, depending on the angle of the loadzone of individual roller body for different values of axialload is shown in Figure 15

Fig. 13. Change of axial and radial deformation,

depending on changes in radial load


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Fig. 14. Change of axial and radial deformation,

depending on changes in axial load

Fig. 15. Change the contact angle, changes depending on

the angle of load zone for different values of axial load

5.2. Bearing life

To determine HUB unit bearing life, using the developedmathematical model, the relation (39), (40), (41) and (42)have been used, with the determination of contact angleused relation (18) and (20). At this stage of theinvestigation is supposed that change of the contact angleoccurs because of preload, or that the vehicle is movingstraight. From this, it follows that the bearing assemblyoperates only after the radial load, which does not causechange in contact angle. This assumption is aconsequence of the conditions of experimental studies inorder to the terms of the mathematical modeling be closeras much as possible to the experimental conditions of the

preliminary examination.Based on experimental tests was calculated failure

probability distribution of the bearing and the actual(testing) bearing life (L90) with 10% probability of failure(reliability 90%) based on the known Weibull relations.Table T.1 presents the results of the bearing life obtained

by experimental testing and of the bearing life obtainedby mathematical modeling.

Table 1. Comparison results of bearing life, obtain

experimental and computer method

Bearing life [h]Experimental L90 Matematical L

789 819

From the previous table it can be concluded that there isno great difference between the experimental testing andmathematical modeling to determine the bearing life. Past

results should be viewed with particular caution becausethe experimental conditions do not correspond toexploitation. The analysis of the results obtained bymathematical modeling to conclude that the exploitation

conditions (mixed load) bearing life significantlydecreases to 240 [h] or 120.000 [km]. It should be notedthat the projected lifetime is usually somewhere betweenthese bearings 70.000-100.000 [km] depending on vehicletype.

6. CONCLUSIONThe paper presents a cross-section study of static behaviorof bearings as well as works related to the determinationof the bearing life. Additionally the theoretical basis fordefining mathematical models for determiningdeformation, stiffness, changes in contact angle and thedetermination of bearing life with angular contact areshown . Also, some research results of the static behaviorof bearings, and verification of mathematical models of

bearings for special applications and single-ball bearingwith angular contact, for support the main spindle ofmachine tools and two rows the ball bearing with angularcontact for support a car wheel are presented. From theresults it can be concluded that the mathematical model

satisfactory describes the static behavior of ball bearingsfrom the point of deformation and stiffness. Based on theresults of ball bearings life, it could be concluded that it isnecessary to design and develop experimental stand fortesting HUB unit bearing that would fully meet itsexploitation conditions.

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ACKNOWLEDGEMENT

This paper is part of the research on projects of "Researchand Development of Roller and Bearing Assemblies TheirComponents," TR 14048, financed by the Ministry ofScience and Technological Development of the

Government of the Republic of Serbia.

CORRESPONDENCE

Milan ZELJKOVI, Prof. PhD.University of Novi SadFaculty of Technical SciencesTrg Dositeja Obradovia 6

21000 Novi Sad, [email protected]

Aleksandar IVKOVI, Assistant MSc.University of Novi SadFaculty of Technical SciencesTrg Dositeja Obradovia 621000 Novi Sad, [email protected]

Ljubomir BOROJEV, Prof. PhD.University of Novi Sad

Faculty of Technical SciencesTrg Dositeja Obradovia 621000 Novi Sad, Serbia

[email protected]

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