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1 machine design, Vol.6(2014) No.3, ISSN pp CALCULATION OF RADIAL STIFFNESS FOR SINGLE-ROW BALL BEARING WITH FINITE ELEMENT ANALYSIS Ivana ATANASOVSKA 1 * - Radivoje MITROVIC 2 - Sonja STEFANOVIC 1 - Natasa SOLDAT 2 - Žako MIŠKOVIĆ 2 1 Institut Kiilo Savic, Belgade, Sebia 2 Univesity of Belgade, Faculty of Mechanical Engineeing, Belgade, Sebia Peliminay note Received ( ); Revised ( ); Accepted ( ) Abstact: In this pape the adial stiffness fo single-ow ball beaing is eseach. The adial stiffness paticipates in the nonlinea behavio of the ball beaing and the machine equipment, so the pecise detemination of this value is vey impotant. The theoetical consideation of the ball beaing adial stiffness is used fo the constuction of expeimental device fo the measuement of defomations necessay fo adial stiffness calculation. The expeiment is pefomed fo one paticula type of single-ow ball beaing with adial contact. The obtained analytical and expeimental esults ae used fo veification of the Finite Element Model developed fo numeical calculation of the adial stiffness fo the same ball beaing type and the same load conditions. Key wods: ball beaing, stiffness, contact, Finite Element Analysis 1. INTRODUCTION With expanding applications of olling beaings, its technical equiements ae inceased also. Beside high accuacy of otation and wok at high speeds, beaings demand longe sevice life and compact constuction and deceasing of noise and vibation what have been geneated by thei wok. Demands of consideations factos that have influence on coect and eliable wok of beaings has appeaed as esult of wide use in almost all types of otating machines. The coelation between analytical, expeimental and numeical detemination of the adial stiffness and load distibution on olling elements, that depend on the intenal adial cleaance of olling beaings is pesented in this pape. 2. DEFINITIONS FOR BALL BEARING RADIAL STIFFNESS Defomation chaacteistics of olling beaing depend upon elastic popeties of mateial, geomety of beaing pats, chaacte and intensity of extenal load, total numbe of olling elements and othe factos. These chaacteistics of beaing can be descibed by beaing stiffness, [1, 2]. Stiffness of machine elements is ability of opposing elastic defomation unde the effect of extenal load and its assessment is the most impotant in analysis of static and dynamic defomations of the whole system. Beaing s stiffness in wok conditions is not the constant value, as it is povided by standads. It depends on volumetic and contact stiffness of the olling elements and beaing ings. Volumetic stiffness depends on geometical chaacteistics of contact sufaces of olling beaings pats and elastic popeties of a mateial. Contact stiffness, except given chaacteistics, depends on the load [1, 2, 3]. One of the main causes fo changing of beaing stiffness is changing of intensity and diection of contact foces and contact defomations between olling elements and beaing s ings duing thei wok. The point of contact between beaing s ings and olling elements is moved with changing of olling elements position and ings otation too. Futhemoe, load distibution between olling elements and intensity and diection of contact foces and contact defomations between beaing s ings and olling elements ae changed also, and poduces beaing stiffness changing. Duing beaings wok the elastic defomation that can be descibed with Hetz s theoy appea in contact of final numbe of olling elements with beaing ings as esult of the contact load. Hetz ceated solutions fo two spheical bodies in contact loaded by foce F whee the elastic defomation occus, [4]. In olling beaing, olling elements touch oute and inne ace at the same time, so that the total elastic contact defomation is obtained as sum of contact defomations which occu between olling elements in contact. Hence, based on the elastic defomations that appea in beaing elements contact (H.R.El-Sayed), fo beaing without intenal cleaance, shaft axis will move to i o (1) whee: δ i - the elastic contact defomation between the most loaded olling element and inne aceway, δ o - the elastic contact defomation between the most loaded olling element and oute aceway Contact defomation δ is esult of acting of the oute load F, and then the adial stiffness of a beaing is: *Coespondence Autho s Addess: Institut Kiilo Savic, Vojvode Stepe 5, Belgade, Sebia, iviatanasov@yahoo.com
2 Ivana Atanasovska, Radivoje Mitović, Sonja Stefanović, Nataša Soldat, Žako Mišković: Calculation of Radial Stiffness fo Single-Row Ball Beaing with Finite Element Analysis; Machine Design, Vol.6(2014) No.3, ISSN ; pp C 86 F (2) Unde acting oute adial foce, lagest displacements in beaing ae in adial diection. Also, adial displacement beaing axis is popotional to total elastic contact defomation on the place of the most loaded olling element in diection of extenal adial foce. On the base of expession fo single-ow adial ball beaing developed in [1, 2]: 32 F Fimax Fo C 0 k z (3) the elation between extenal adial foces and coesponding adial displacement of beaing axis is obtained by following expession: Cz F k 32 By diffeencing of last expession it can be obtained expession fo adial stiffness of single-ow adial ball beaing in function of geomety and beaing pats mateial and oute load distibution: df 3 Cz C d 2 k F 13 whee: C δ - constant, dependent fom the beaing intenal geomety, module of elasticity and Poissons facto of mateial beaing pats, z- total numbe of olling element in the beaing, k facto of load distibution, F - oute adial load. Radial stiffness of olling beaings is pogessive consideing chaacte of changing value of elastic displacement of beaing axis unde the influence of extenal load, as shown in Figue 1. Fig.1. Shape of the adial stiffness function Inceasing of eal contact suface with contact stain inceasing has influence on contact stiffness inceasing. While volumetic stiffness of mechanical pats and assemblies diectly depends on geometic chaacteistics and mateial, contact stiffness which occued when contact exists depends on defomations (load), geomety and elastic popeties of mateial elements in contact. Theefoe, it has the main influence on nonlinea behavio of ball beaings as a pat of machine assemblies. (4) (5) Based on Stibeck-Palmgen equation [5], total adial displacement of beaing axis is given with following expesion: F 1, 26 (6) z D 13 k whee the connection between foce on the place of the most loaded ball and extenal adial load is given with following expession: F Fo 4,37 z (7) Then, the expession fo adial stiffness of single-ow ball beaing with adial contact and without adial cleaance has the following fom: ,19 k C z D F (8) whee: C - stiffness, D K - ball diamete, F - extenal adial load. 3. ANALITICAL AND EXPERIMENTAL RESULTS FOR RADIAL STIFFNESS OF BALL BEARING Analytical esults Based on expession (5) и (8) fo single-ow ball beaing with adial contact and without adial cleaance, calculation of adial stiffness fo ball beaing type 6206 has been conducted. With espect on chaacte of elastic contact defomations changing unde extenal adial load and popotionality between adial displacement of beaing axis and the biggest contact defomations, it is clea that value of total adial displacement of beaing axis depends on chaacte of adial stiffness changing. With inceasing of extenal adial foce, total adial displacement of beaing axis will be deceasing, as the esult of adial stiffness of beaing pats inceasing. No unifom distibution of extenal load on balls in contact is defined by the facto k and depends on total numbe of balls and size of intenal adial cleaance. The eal facto of no unifom extenal load distibution k expesses the influence of adial cleaance on adial stiffness value. The Fig. 2 shows that with inceasing of adial cleaance е (facto of no unifom extenal load distibution k), adial stiffness is educed, and adial displacement of beaing axis is inceased unde extenal load. The ball beaing adial stiffness depends fom intenal beaing geomety and also fom beaing type and measuement seies. Theefoe, fo given shaft diamete and given extenal load, with changing beaings measuement seies, chaacteistic of stiffness is changing, too Expeimental esults Expeimental detemination of beaing adial stiffness was pefomed on device which is adapted to the easy installation on vetical pulsato SCHENCK PVQ 0082,
3 Ivana Atanasovska, Radivoje Mitović, Sonja Stefanović, Nataša Soldat, Žako Mišković: Calculation of Radial Stiffness fo Single-Row Ball Beaing with Finite Element Analysis; Machine Design, Vol.6(2014) No.3, ISSN ; pp with load in ange of zeo to N, shown in Fig.3 and Fig.4, [6]. Test samples ae single-ow ball beaings 6206, which ae divided in two goups, beaings with cleaance е=15μm and е=20μm. Results ae witten in fom of diagam of displacement changing with foce changing. Based on known size of diagams values of foce - displacement of chaacteistic points which wee detemined analytically it was easy to compae the esults. Based on analytical expessions, a detailed analysis of the expeimental data is conducted. In Fig. 5, Fig.6 and Fig.7 a compaative eview of analytical and expeimental solutions ae shown. With incease of extenal load the diffeence as compaed to theoetical values of the esults is inceased also. It can be seen that contact defomation of olling elements in stat point of contact is highe than the theoetical value fo lage extenal loads. analytical calculation expeiment - specimen 1 expeiment - specimen 2 Fig.2. Beaing adial stiffness in function of the value of intenal adial cleaance and loads Fig.5. Changing of the total contact defomation of beaing pats in stat point of contact on the most loaded ball with extenal load vaiation, beaing 6206 with intenal adial cleaance е=20μm With analyzing the Fig. 6 it can be concluded that with incease of extenal load, the most loaded ball caies the bigge pat of the load than it was obtained analytically. Hence, fo given beaing, the situation is wose than it was expected theoetically. It is esult of lage contact defomations that have occued in stat point of olling elements contact. Fig.3. Expeimental device with acquisition system analytical calculation expeiment - specimen 1 expeiment - specimen 2 Fig.6. Changing of the edial load on the most loaded ball with extenal load vaiation, beaing 6206 with intenal adial cleaance е=20μm Fig.4. Expeimental device with installed a ball beaing specimen In Fig. 7 dependence of adial stiffness fom beaing vesus load fo beaing 6206 with intenal adial cleaance 87
4 Ivana Atanasovska, Radivoje Mitović, Sonja Stefanović, Nataša Soldat, Žako Mišković: Calculation of Radial Stiffness fo Single-Row Ball Beaing with Finite Element Analysis; Machine Design, Vol.6(2014) No.3, ISSN ; pp е=20μm is shown. The lagest diffeence between theoetical and expeimental solutions is 13.6%. This gives that adial stiffness in highe oute load zone is bigge than theoetical value, which is esult of lage contact defomations that have occued at stat point of olling elements beaings contact. In zone of oute load up to 7000 N, diffeences in solutions ae less than 10%. Results of Finite Element Analysis fo equivalent defomations, fo beaing 6206 with intenal adial cleaance е=20μm and F =4000 N ae shown with contou plot in Fig.9, and the esults of Finite Element Analysis fo defomations in adial diection ae shown in Fig.10. Fo calculation of numeical esults fo the adial stiffness fo analyzed cases of ball beaing, the value of total adial displacement of beaing in adial diection is used. Fig. 11 shows the eading of the value of total adial displacement of beaing in adial diection, fo beaing 6206 with intenal adial cleaance е=20μm and F =4000 N. Also, the value of total contact defomation of beaing pats in stat point of contact on the most loaded ball 0 is used fo compaation with analytical and expeimental esults and fo the veification of FEA. Fig. 12 shows the eading of the value of total contact defomation of beaing pats in stat point of contact on the most loaded ball 0, fo beaing 6206 with intenal adial cleaance е=20μm and F =4000 N. Fig.7. Changing of the adial stiffness with extenal load vaiation, beaing 6206 with intenal adial cleaance е=20μm 4. FINITE ELEMENT ANALYSIS FOR DETERMINATION OF RADIAL STIFFNESS OF BALL BEARING 6206 Fo the Finite Element Analysis of the stess and defomation of the single-ow ball beaing type 6206 the softwae Ansys is used. 2D finite element model is developed and shown in Fig.8 fo the case of intenal adial cleaance е=20μm. Fo ball beaing modeling the solid 2D elements with fou nodes ae used and fo contact between balls and beaing ings the symmetic flexible contact behavio is modeled. The extenal load is defined in the cente of shaft which is in contact with inne ing of the beaing. The Finite Element Analysis is pefomed fo the extenal load of 1000N, 2000 N, 4000 N and 5000 N. Fig.9. Results of Finite Element Analysis fo equivalent defomations, fo beaing 6206 with intenal adial cleaance е=20μm and F =4000 N 88 Fig.8. Finite Element Model fo beaing 6206 with intenal adial cleaance е=20μm Fig.10. Results of Finite Element Analysis fo defomations in adial diection, fo beaing 6206 with intenal adial cleaance е=20μm and F =4000 N
5 Ivana Atanasovska, Radivoje Mitović, Sonja Stefanović, Nataša Soldat, Žako Mišković: Calculation of Radial Stiffness fo Single-Row Ball Beaing with Finite Element Analysis; Machine Design, Vol.6(2014) No.3, ISSN ; pp Fig.11. The Finite Element Analysis esult fo the value of total adial displacement of beaing in adial diection, fo beaing 6206 with intenal adial cleaance е=20μm and F =4000 N Fig.13. FEA esults fo the changing of the total contact defomation of beaing pats in stat point of contact on the most loaded ball with extenal load vaiation, beaing 6206 with intenal adial cleaance е=20μm Fig.12. The Finite Element Analysis esult fo the value of total contact defomation of beaing pats in stat point of contact on the most loaded ball 0, fo beaing 6206 with intenal adial cleaance е=20μm and F =4000 N The FEA esults fo the changing of the total contact defomation of beaing pats in stat point of contact on the most loaded ball with extenal load vaiation fom 1000 N to 5000 N, fo beaing 6206 with intenal adial cleaance е=20μm is shown in Fig. 13. The FEA esults fo the changing of the adial load on the most loaded ball with extenal load vaiation fom 1000 N to 5000 N, fo beaing 6206 with intenal adial cleaance е=20μm is shown in Fig. 14 and the FEA esults fo the changing of the adial stiffness with extenal load vaiation fom 1000 N to 5000 N, fo the same beaing case is shown in Fig. 15. The esults obtained with FEA shown in Fig.13, Fig.14 and Fig.15 ae compaing with esults obtained fo the same beaing case by analytical calculation and expeiment (Fig. 5, Fig. 6 and Fig.7). Vey good supeposition of esults is obtained. In this way the veification of Finite Element Model is pefomed. Fig.14. FEA esults fo the changing of the adial load on the most loaded ball with extenal load vaiation, beaing 6206 with intenal adial cleaance е=20μm Fig.15. FEA esults fo the changing of the adial stiffness with extenal load vaiation, beaing 6206 with intenal adial cleaance е=20μm 89
6 Ivana Atanasovska, Radivoje Mitović, Sonja Stefanović, Nataša Soldat, Žako Mišković: Calculation of Radial Stiffness fo Single-Row Ball Beaing with Finite Element Analysis; Machine Design, Vol.6(2014) No.3, ISSN ; pp CONCLUSIONS The adial stiffness of single-ow ball beaing is vey impotant value because paticipates in the nonlinea behavio of the ball beaing and the machine equipment which the pat is. The expeimental detemination of the adial stiffness is pefomed fo one type of single-ow ball beaing with adial contact. The esults of expeiment give excellent supeposition with analytical esults. So, obtained diagams ae used fo veification of the Finite Element Model developed fo numeical calculation of the adial stiffness fo the same ball beaing type and the same load conditions. The veified Finite Element Model could be pefomed fo any single-ow ball beaing type fo obtaining the adial stiffness that is equied fo dynamic analysis of machine equipment with single-ow adial ball beaings. ACKNOWLEDGMENTS REFERENCES [1] Mitović, R., Msc Thesis. (1988). Analysis of the influence of elastic defomation and intenal adial cleaance of ball beaing on the load distibution and stess capacity, Faculty of Mechanical Engineeing, Univesity in Belgade (in Sebian) [2] Ristivojević, M.; Mitović, R. (2002). Load distibution geas and beaings, Belgade (in Sebian) [3] Recetov, D.N. Detalji mašina [4] Eschmann; Hasbagen; Weigand. (1953). Die Walzlagepaxis, Munchen [5] Hais,T. A. (1984). Rolling Beaing Analysis, New Yok [6] Stefanovic S., Phd Thesis. (2005). Influence of elastic defomations and intenal adial cleaance on static load capacity of olling element beaings, Kagujevac (in Sebian) The wok has been funded by the Ministy of Education and Science of Republic of Sebia, Gant TR 35029: Development of the methodology fo woking capacity, eliability and enegy efficiency impovement of enegetic mechanical systems. 90
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