ANALYTICAL BEARING MODEL FOR ANALYSIS OF INNER LOAD DISTRIBUTION AND ESTIMATION OF OPERATIONAL LUBRICATION REGIME

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1 58 th ICMD September 017, Prague, Czech Republic ANALYTICAL BEARING MODEL FOR ANALYSIS OF INNER LOAD DISTRIBUTION AND ESTIMATION OF OPERATIONAL LUBRICATION REGIME Jakub CHMELAŘ 1, Votěch DYNYBYL 1 Department of Deigning and Machine Component, FME, CTU in Prague Department of Deigning and Machine Component, FME, CTU in Prague Abtract A mathematical model of roller bearing i preented in thi paper. Calculation of load ditribution and diplacement i baed on current tandard ISO/TS Bearing operating lubrication regime analyi i involved. It i baed on lambda parameter coniting of EHD layer thickne value that i calculated from equation for wide elliptical contact and a compoite urface roughne value. Model wa uccefully verified againt commercial KISSoft oftware. Key word: Bearing, Roller Bearing; Lubrication; EHD; Bearing mathematical model. INTRODUCTION Identification of bearing propertie during operation i very important tep for analyi and implementation. Since well-known claical engineering equation for calculation of bearing life doe not provide a detail inight a demand for a more complex model arie. The complex model hall provide information about actual load ditribution between roller, which i baed on internal geometry uch a radial clearance or roller profile modification. Additionally information regarding lubrication quality baed on lubricant rheological parameter, mating urface texture and operating condition are of interet. Such model baed on current theoretical finding i preented in thi paper. MATERIALS AND METHODS Analyi of load ditribution on rolling element within the roller bearing wa carried out uing method decribed in tandard (International Organizatipon for Standardization [ISO], 008). For roller bearing, method aume Hertz line contact. Calculated load ditribution i valid for low to medium peed and quai-tatic loading. Dynamic effect uch a centrifugal and inertial force are neglected. Another aumption concern the deformation of outer ring/race where tiff upporting tructure i conidered and calculated deformation are only within the area of contact. For preented analyi, unidirectional radial load with mall moment load i applied. Roller load ditribution Deflection of -th rolling element wa calculated from radial deflection or inner ring according to (1) r co( ) (1) Where i a total operational radial clearance, uually obtained from catalogue and modified in order to cover it change due to temperature deformation. Coordinate ϕ (rad) i an angular coordinate of -th roller. For an analyi, roller wa divided into 41 equally length lamina. Thi allow including a roller profile modification into calculation and load ditribution over the roller. For current analyi, a logarithmic profile modification decribed a function of coordinate x k (mm) from roller center wa ued (). The modification i depicted on the Fig. 5. xk P( x ) Dweln1/(1 ( ) ) () k Lwe 118

2 58 th ICMD September 017, Prague, Czech Republic The deformation of each lamina including roller profile modification i calculated according to (3). If the reult i negative, zero ubtitute it. x tan P( x ) (3) k k Fig. 1 Coordinate of bearing model (ISO, 008) Radial load F r (N) and/or moment M z (Nm) are upported by rolling element. The fraction of load per lamina of each rolling element i calculated according to equation (4). In the analyi, roller are ubtituted by 1D pring. Each pring ha a tiffne covering the roller itelf and it contact with both race. Q (4) c 10 / 9,k The pring tiffne c L ditributed between lamina become c. For roller and race made of teel c L i expreed in eq. (5), where Lwe (mm) i length of body and n i number of lamina. Other poible value of c L (N / mm) can be found in the literature (Harri & Kotzala, 006). Exact value for conidered geometry and material can be alo obtained by mean FEM analyi. c c 8 / 9 L WE (5) n L n Lat tep of an analyi i an iterative olution of tatic force and momentum equilibrium equation (6) and (7) for δ r (mm) and ψ (rad) repectively. Z n Z n 10 / 9 Fr c co( J ), 0 (6) 9 k M co( ) 10 / Z c J xk, 0 (7) k 1 k 1 1 k 1 Contact Preure calculation Contact preure ditribution on roller wa baed on Hertz contact theory for line contact (Harri & Kotzala, 006). For each lamina and roller repectively, contact half width b (mm) and contact preure P (MPa) were calculated according to equation (8) and (9), where R and E are defined by eq. (14) and (15); w i length of lamina. b 8 ( Q / w ) R E x (8) P ( Q / w ) Rx b (9) 119

3 58 th ICMD September 017, Prague, Czech Republic Lubrication layer thickne Minimal thickne of lubrication layer i calculated from equation (10) publihed by Dawon and Higginon. U, G and W are dimenionle group; R x (mm) i mutual curvature of bodie in contact, perpendicular to rolling direction. Radiu Ry (mm) for roller bearing with profile modification could be etimated by circle curve fitting to profile modification. h0 (mm) i minimal lubrication layer thickne in contact (Wheeler, et al., 016). Following equation i modified for wide elliptical contact, where Rx / Ry > 3. From aymptotic analyi it could be proved, that the lat bracket with exponent in (10) become 1 for Rx / Ry > 0 h ( Rx / Ry) 3.63 R U G W (1 e ) (10) y Equation (10) wa obtained by curve fitting of numerical olution of Reynold equation for elliptical contact in wide range of condition. Preented form i modified for wide contact with ellipticity parameter Rx / Ry >.5. For detail, ee (Wheeler, et al., 016). Main aumption for uch equation are: model of mooth contact, fully flooded lubrication regime, contact preure maintained below GPa and iothermal condition within lubrication layer. It alo aume Newtonian behavior of lubricant. Wheeler et. al found and decribed in tudy (Wheeler, et al., 016), that uch emi-analytical equation, tend to overetimate minimal lubrication layer thickne about a 6 % comparing to full numerical olution. Therefore, in the model, minimal thickne wa reduced by 6 %. In addition, mot emi-analytical equation tend to deviate in operational condition where high load and low peed occur. Therefore, it i neceary to pay attention when aeing a lubrication regime laying in area where tranition from boundary lubrication to EHD come about. For accurate reult, full numerical olution of Reynold equation i recommended (Wheeler, et al., 016). The application of (10) for line contact i further limited to pure radial loading or very light off-axi moment loading that doe not coniderably influence the load ditribution over the length of element. For one-ided kewed load ditribution on the roller, lubrication layer height would have to be verified by complete numerical model of contact. Dimenionle group ued in equation conit of following parameter: U Speed parameter: U 0 (11) ER y Material parameter: Load parameter: W G E (1) * \ W (13) ER y R x (mm) i mutual curvature of bodie in: (14) Ry R1 y R y E (N/mm ) i a mutual elatic modulu: 11 1 E E E 1 (15) Where, η 0 i a lubricant dynamic vicoity at atmopheric preure (MPa.); α * i a vicoity-preure coefficient (1/MPa), u U i entraining peed of lubricant to the contact: 1 u U (mm/) (16) For roller bearing, with aumption of pure rolling, entraining peed could be calculated from kinematic analyi (Spike, 015). The cage/retainer revolution peed, the unknown, i ubtracted from the rotation of inner and outer ring, o the rolling element remain teady tate and only rotate around their axi. Thi tranformation allow writing two equation for roller peripheral peed: at inner race and outer race. The unknown retainer peed i then removed by combination of equation and the peripheral peed of roller can be eaily calculated. Entraining peed (16) i for roller bearing calculated acc.to (18) 10

4 58 th ICMD September 017, Prague, Czech Republic U Ri i Ri Dwe Ri Dwe / (17) Lambda parameter Generally, in roller bearing, lubrication film thickne i of imilar order a urface roughne. Therefore there i defined a lambda parameter, that compare minimal lubrication thickne to compoite urface roughne. Thi i provided by equation (18), where x i root mean quare (RMS) value of roller and race urface roughne, h0 i minimal lubrication layer thickne calculated according to (10).The RMS value x can be etimated from Ra urface parameter a 1.5Ra. h 0 (18) race roller Baed on extenive experimental reearch following value of lambda were found to correpond with lubrication regime: Λ < 1 the load i upported mainly by urface aperitie, both bodie are in direct contact, aperitie are deformed, urface are loaded by hear tre if liding occur. For 1 < Λ < 3 the mixed lubrication regime i preent, meaning that applied load i yet partly upported by lubricant thin film, but till direct contact between bodie exit. When Λ > 3, the lubrication layer eparate both contact bodie, the load i upported only by lubricant and local deformation of the bodie influence the lubrication layer thickne. The load i except minor traction hear mainly in normal direction (Harri & Kotzala, 006). RESULTS AND DISCUSSION Bearing model The verification of bearing model wa baed on comparion reult provided by oftware KISSoft (KISSoft, 013) accompanied by a module for bearing calculation according to tandard ISO (008). For a purpoe of verification, a radial roller bearing N306 wa ued. The geometry i ummarized in the Tab 1. Tab. 1 Summary of N306 parameter Parameter Deignation Value Dimenion Inner bore diameter d 30 mm Outer race diameter D 7 mm Width of bearing B 19 mm Number of roller Z 1 Roller element diameter Dwe 11 mm Roller element active width Lwe 11 mm Inner ring race diameter Di 43,5 mm Radial internal clearance (SKF Group, 013) Pd 0,035 mm Baic dynamic load rating C N Young modulu E N/mm Poion ratio ν 0,3 The verification of radial deflection and moment load wa done for three radial load cae defined a P/C = 0,05; 0,1 and 0,. Light moment loading wa alo applied, in order to have ame load cae a 11

5 58 th ICMD September 017, Prague, Czech Republic defined in KISSoft for teted cae. A you can ee in Tab. the deviation between radial deflection reult obtained by the code implemented in thi paper and the reult obtained by KISSoft wa below 0.1 %. Similar ituation i for mialignment aement, where the error topped 6 %. The error might be introduced by different conidered roller geometry in the KISSoft calculation that would influence the angle. Tab. Bearing model verification radial and angular diplacement Radial Moment KISSoft P/C model Error [%] load [N] load [Nm] [μm] [μm] model [mrad] KISSoft [mrad] Error [%] 0,05 95 N ,6 8, , ,73 34, , ,198 45, Reult of load ditribution verification are tated in Tab. 3. Due to ymmetrical loading of element, only half of loaded element were compared. Actual ditribution i depicted in a Fig.. Preure ditribution per lamina on the maximally loaded roller i depicted on the Fig. 3, numerical value of maximal preure on inner race and outer race are evaluated in Tab. 4 and compared with reult obtained from KISSoft a verification. Abolute error indicate almot perfect fit. Tab. 3 Bearing load ditribution verification Load P/C=0. KISSoft Error Roller model [N] [%] #No [N] Tab. 4 Max contact preure verification Load P/C=0. KISSoft Error Property model [N/mm [μm] [%] ] Inner race Outer race Fig. Roller load ditribution Fig. 3 Roller contact preure ditribution Lubrication Lambda parameter i calculated for every element race contact. The reult i graphically preented on the Fig.4. It i obviou, that le favourable lubrication condition occur on the inner race contact, where lambda parameter for peed 500 rpm indicate boundary lubrication Λ =.. Lubrication regime on the outer race i analyed in the boundary regime too, but due to lighter load, the lambda for mot loaded element reache Λ =.8. 1

6 58 th ICMD September 017, Prague, Czech Republic Tab. 5 Summary of parameter ued for lubrication calculation Parameter Deignation Value Lubricant dynamic vicoity at atm. preure η 0 (60 C) (3) MPa. (cp) Vicoity-preure coefficient α * (60 C) MPa -1 Race (inner / outer) urface roughne Ra 0,08 μm Roller urface roughne Ra 0,03 μm Equivalent radiu of profile modification Ry 1414 mm Bearing inner race peed n 500 rpm Fig. 4 Lambda parameter ditribution Fig. 5 Roller profile modification CONCLUSIONS Paper preent a mathematical model of a roller element bearing baed on a method decribed in the tandard (ISO, 008) extended by calculation of lubrication layer thickne for every loaded element. Model calculate load ditribution on roller, contact preure ditribution on the roller. It i poible to apply a radial and moment load. Model wa uccefully verified againt commercial CAE oftware KISSoft. Lubrication i calculated according to emi-analytic equation for wide elliptical contact preented by Hamrock and Dowon (10). Lambda parameter (18) for etimation of lubrication regime by comparing lubrication layer thickne with urface compoite roughne i calculated and reult are preented a polar figure Fig 4. Lubrication calculation i limited by ymmetrical load ditribution on the roller element. Preented model will be further extended to cover other operating condition. REFERENCES 1. International Organizatipon for Standardization. (008). Rolling bearing Method for calculating the modified reference rating life for univerally loaded bearing (ISO/TS 1681:008).. Harri, T. A., & Kotzala, M. N. (006). Eential concept of bearing technology. CRC pre. 3. Wheeler, J. D., Vergne, P., Fillot, N., & Philippon, D. (016). On the relevance of analytical film thickne EHD equation for iothermal point contact: Qualitative or quantitative prediction?. Friction, 4(4), Spike, H. (015). Baic of EHL for practical application. Lubrication Science, 7(1), KISSoft, A. G. (013). KISSSoft Releae 03/013 Uer Manual. Bubikon, Switzerland. 6. SKF Group, (013) Rolling Bearing, Catalogue, S. G.. PUB BU/P / EN. SKF. Correponding author: Ing. Jakub Chmelar, Dep. of Deigning and Machine Component, FME, Czech Technical Univerity in Prague, Technicka 4, Praha 6, Prague, , Czech Republic, phone: , Jakub.Chmelar@f.cvut.cz 13