Analysis of Frictional Torque in Raceway Contacts of Tapered Roller Bearings

Transcription

1 Analysis of Frictional Torque in Raceway Contacts of Tapered Roller Bearings H. MATSUYAMA * S. KAMAMOTO ** * Bearing Research & Development Department, Research & Development Center **Mechatronic Systems Research & Development Department, Research & Development Center To obtain the simplified formula of viscous rolling resistance in roller-raceway contacts of tapered roller bearings, full EHL analyses in line contacts have been carried out for a wide range of dimensionless EHL parameters. The frictional torque in raceway contacts taken into account for the simplified formula has been compared to the frictional torque obtained from experiments. Furthermore, the frictional torque formula for oil bath lubrication has been proposed. Key Words: tapered roller bearings, frictional torque, viscous rolling resistance, EHL. Introduction Tapered roller bearings are widely used in industrial applications such as automobiles. They have advantages in load carrying capacity and rigidity in comparison with ball bearings. But, frictional torque of tapered roller bearings is much greater than that of ball bearings and heat generation due to frictional loss is also greater. Recently, as less fuel consumption, smaller sie, higher speed are expected for automobiles, bearings are also expected to show lower frictional torque and less heat generation )~3). Therefore, it is necessary to predict the frictional torque of tapered roller bearings in the design stage. The formulae of frictional torque for tapered roller bearings have been proposed by many researchers 4)~7). Frictional torque of tapered roller bearings under adequate lubrication is mainly generated by contacts between rollers and inner/outer raceways (hereinafter referred to as the raceway contacts) 7)~8), and the frictional torque in the raceway contacts is caused by viscous rolling resistance. Viscous rolling resistance can be obtained using the theory of EHL (elastohydrodynamic lubrication), and the simplified formulae for viscous rolling resistance have been also proposed 7), 9)~). However, there are some problems for using these formulae in a wide range of bearing operating conditions because they were derived under limited conditions 3)~4). In this paper, full EHL analysis in line contacts has been carried out for a wide range of bearing operating condition, and a simplified formula for viscous rolling resistance under full flooded condition has been derived. Also, the frictional torque generated in the raceway contacts of tapered roller bearings has been experimentally obtained and the result has been compared with that predicted from EHL theory. As the result of theoretical and experimental analyses, a new formula for calculating frictional torque of tapered roller bearings has been proposed.. Frictional Torque of Tapered Roller Bearings. Composition of Torque Generally, frictional torque of tapered roller bearings comprises of the following four factors: a friction between roller and inner/outer raceway s friction between roller-end and rib, d friction between roller and cage pocket, and f churning resistance of lubricant. At normal rotational speeds, factors d and f are relatively small; frictional torque of tapered roller bearings is mainly influenced by factors a and s. Friction between rollers and raceway contact surfaces comes from elastic hysteresis loss and viscous rolling resistance, but viscous rolling resistance is much greater than elastic hysteresis loss. On the other hand, friction between the end faces of the rollers and the rib is sliding frictional resistance consisting of Coulomb friction and fluid friction. The relationship between frictional torque and rotational speed of tapered roller bearings is shown in Fig.. At low speeds, sliding frictional resistance in roller-ends and rib contact (hereinafter referred to as the rib contacts) is the controlling factor. Because torque at the rib contacts decreases as rotational speed increases and torque at the raceway contacts (which is a cause of viscous rolling resistance) increases, viscous rolling resistance has a great influence on total bearing torque at practical operating speeds. Frictional torque Total bearing torque Raceway contacts Rib contacts Rotational speed Fig. Frictional torque of tapered roller bearing KOYO Engineering Journal English Edition No.59E (00) 53

2 A b h a c. Basic Equation In this paper, a single row tapered roller bearing under axial load shown in Fig. is discussed. Spin moment in rib contacts, friction between rollers and cage, and churning resistance of lubricant are not taken into account in the following analysis. The following equation is obtained from static balance of moment that acts on a roller in Fig. 3. Fi+ Fr ( + l sinc e ) m i mo = 0 a where F 0, F i and F r are frictional forces that act on the roller from the inner ring, outer ring and rib respectively; m i and m 0 are frictional moments (viscous rolling resistance) caused by eccentricity of pressure distribution in EHL contact. Frictional torque m' which acts on the inner ring from a roller is given as follows: m' = r ifi + Fr r i + l sin b + e cos h + m i s Thus the sum of torque acting on the inner ring from the rollers, that is frictional torque M acting on the bearing can be obtained by the following equation: M = Σ m' = m' d where is the number of rollers. M is expressed from equations a ~ d as follows: ro Q o D W dm B Q i r i Q r Fig. Geometry of tapered roller bearing xo F i Fo Fr xr mi mi mo xi D W Fig. 3 Forces and moments acting on a tapered roller e r i l mo Fr Fo F i r o e F a M = [ ( r i + ) m i + r i mo ] + Fr e ( r i + cos h) + Fr l sin b r i sinc Here, f sin b r i sin c= r i AB AB r i + r i + cosh= ro = 0 g h Then equation f can ultimately be abbreviated as follows: M = M + M j M = ( i r i mo ) ro m + M = ro efr where M is frictional torque generated in the raceway contacts and M is frictional torque generated in the rib contacts. 3. Theoretical Analysis of Viscous Rolling Resistance 3) 3. Viscous Rolling Resistance Viscous rolling resistance is the frictional moment caused by the eccentricity of pressure distribution in the EHL contact. As shown in Fig. 4, pressure distribution on a roller-raceway contact is not symmetrical due to the EHL effect. The center of pressure is dislocated from the center of Hertian contact. The coordinate x cp of the center of pressure is given as follows: xcp = w x L x L px dx The moment due to w acting on the coordinate x cp is equivalent to the moment due to the EHL pressure distribution. The viscous rolling resistance at a raceway contact, m v, is given as follows: l 0 mv = xcpwdy = xcpw l Dimensionless coordinate of center of pressure, X cp is expressed from equation 0 as follows X L xcp Xcp = = PX dx b p X L Thus, dimensionless viscous rolling resistance, M v, is given as the following equation: mv Mv = = X cpw be' l R 3 When pressure distribution is obtained from EHL analysis, M v can be estimated from equations and 3. r i k l 0 54 KOYO Engineering Journal English Edition No.59E (00)

3 Dimensionless film thickness, H = hr/b Film thickness Center of pressure Mv X X cp Center of Hertian contact W Pressure Dimensionless coordinate, X = x/b Fig. 4 Pressure distribution in EHL contact 3. Conditions To obtain dimensionless viscous rolling resistance M v, EHL analysis has been carried out using line contact model shown in Fig. 5. Roller R H Elastic half space min Fig. 5 Line contact model H = hr/b Convergence was solved by numerical analysis using a combination of the Reynolds equation, the oil film thickness equation, the force equilibrium equation, the Roelands equation for relationship between pressure and viscosity, and the Dowson-Higginson equation for relationship between pressure and density. The details are given in an existing paper 3) and have been omitted here. In this numerical analysis, the dimensionless speed parameter U, dimensionless material parameter G, and dimensionless load parameter W determined to cover the operating conditions of tapered roller bearings used in automotive driving systems such as differential gear units or transmissions; U = 0 3 to 0 9, G = 500 to 9 000, W = 0 5 to 0 3. When the bearing lubricated with 85W-90 gear oil, these dimensionless parameters are evaluated as follows: bearing temperature is 0 to 50 C, rotational speed is 0 to min, and maximum Hertian pressure is 0.3 to 3.0 GPa. X Dimensionless pressure P = p/p h 3. 3 Results Figure 6 shows the effects of dimensionless parameters U, G and W on dimensionless viscous rolling resistance M v under full flooded conditions. M v obtained by numerical analysis is proportional to U to the 0.70 to 0.77 power, G to the 0.04 to 0.07 power, and W to the 0.4 to 0.47 power respectively. As a result, a simplified formula for dimensionless viscous rolling resistance M v is obtained as follows. Mv = 8.89U 0.75 G 0.04 W Figure 7 gives a typical example of the relationship between pressure distribution and dimensionless parameters. Pressure distribution is close to Hertian pressure when U = 0 3, but as U increases, the center of pressure transfers significantly towards the inlet, thus causing M v to increase. The reason G has little effect on M v is that increase of G has almost no effect on offset of center of pressure. The reason M v decreases as W increase is that the larger W is, the less center of pressure is offset. From equations 3 and 4, viscous rolling resistance m v under full flooded conditions is given by the following equation: mv = 4. E' l R U 0.75 G 0.04 W The exponent of W in equation 5 is the exponent for W in equation 4 plus 0.5. This is because half width of Hertian contact b is proportional to W to the 0.5 power. Equation 5 suggests that G and W have little effect on viscous rolling resistance and that the effect of U is large. Frictional torque at the raceway contacts can be predicted by the following equation, which is obtained by substituting m v of equation 5 for m i and m o of equation k. M = ( ro 4.E' l R i U i 0.75 G i 0.04 Wi r i 4.E' l R o Uo 0.75 Go 0.04 Wo 0.08 ) 4. Experimental Analysis of Frictional Torque in Raceway Contacts 4. Method Frictional torque of tapered roller bearings is measured with test apparatus to be compared with the simplified formula derived from theoretical analysis. Figure 8 provides an overview of the test apparatus. The rib is separated from the inner ring in order to obtain only frictional torque generated at the rib contacts in the test apparatus. Frictional torque due to rib contacts M is given by the following equation from moment m r acting on the shaft which the rib is attached to, obtained by strain gauges. romr M = e 7 r i + l sinb e cosh In order to obtain total bearing torque M, the housing of test bearing is supported by air static bearing. M is given by load cell. Because friction between rollers and cage and churning resistance of the lubricant can be neglected, frictional torque M due to the raceway contacts can be estimated by the following equation: M = M M 8 6 KOYO Engineering Journal English Edition No.59E (00) 55

4 Mv Dimensionless viscous rolling resistance, G = 3 300, =.5 0 W 5 G = 3 300, W = G = 4 500, W = 0 4 G = 4 500, W = Dimensionless speed parameter, U Dimensionless viscous rolling resistance, Mv U = 7.7 0, W = U =.5 0 0, W = U =.3 0 0, W = Dimensionless material parameter, G Fig. 6 Effects of dimensionless parameters on dimensionless viscous rolling resistance Dimensionless viscous rolling resistance, Mv U =.5 0 0, G = U = 7.7 0, G = U =.3 0 0, G = U = 5 0 0, G = Dimensionless load parameter, W Dimensionless pressure, P Dimensionless coordinate, X.6.6 W = W = 0 4 U = 0 9 G = U = 0 U = 0 3 P Dimensionless pressure, G = G = Dimensionless coordinate, X Fig. 7 Effects of dimensionless parameters on pressure distribution P Dimensionless pressure,..0 W = Dimensionless coordinate, X Inner ring Rib Strain gauges Slip ring Oil level Roller Outer ring Lubricant Housing Air static bearing Shaft Motor Table Test conditions Axial load ~ 3 kn Hertian maximum pressure 0.3 ~.3 GPa Rotational speed 00 ~ 500 min Lubricating oil Four types of paraffin-based mineral oils, pure paraffin oil, and traction oil Lubrication method Oil bath Oil temperature 6 ± 3 C Viscosity 3 ~ 0 mpa s Pressure-viscosity coefficient ~ 48 GPa Load cell Fig. 8 Schematic diagram of test apparatus 4. Conditions Test conditions are shown in Table. Tapered roller bearings with inner diameter of 45 mm, outer diameter of 9.5 mm, and width of 35 mm were used. Because the number of rollers was reduced to increase rolling element load for some tests, Table gives axial load converted to the bearing with standard number of rollers. The frictional torque of bearings measured with reduced number of rollers is converted to those with standard number of rollers. Four types of non-additive paraffin-based mineral oils (ISO VG0, VG3, VG46, VG68), pure paraffin oil and traction fluid (both equivalent to VG3) were used for the tests. Frictional torque is largely influenced by the oil level. The tests described herein were conducted using oil bath lubrication (oil level at the center of the lowest roller). A range of rotational speed whereby Goksem-Hargreaves thermal reduction factor 9) (explained later) was at least 0.95 was selected in order to minimie the effect of heat generation. Variation in oil temperature and bearing temperature was kept within C during the test. The conditions correspond to U = 0 to 0 0, G = 700 to 000, and W = 0 5 to ~ KOYO Engineering Journal English Edition No.59E (00)

5 4. 3 Results Figure 9 shows the relationship between frictional torque M at the raceway contacts obtained by the test and dimensionless parameters U, G and W. Figure 0 gives an example of comparison of the calculated values for frictional torque and those obtained in the tests. Because dimensionless parameters differ for the raceways of the inner and outer rings, U, G and W given below are representative for dimensionless parameters for raceway contacts. U = U i / ( + cm ) = U o / ( cm ) = p dm Ngo /( 60 E' ) G = Gi = G o = ao E' W = ( cm ) Wi = ( + cm ) = Fa /( E' l sina) In Fig. 9, the symbols represent the values obtained in the test and the solid lines represent regression lines. Figure 0 also shows the values calculated by the previously proposed equations for viscous rolling resistance 6), 7), 9), ). Figure 9 and Fig.0 show that the values calculated from equation 6 for friction torque M generally agree with those obtained in the test. Concerning the relationship between M and W, however, the slopes of the regression lines of the test data are larger than those of the theoretical estimate lines, and difference in the effect of W is observed. Concerning the relationship between M and U, the slopes of the regression Wo 9 0 lines of the tests are smaller than those of the theoretical estimate lines. M obtained in the tests is proportional to U to the 0.5 to 0.69 power, G to the 0.03 to 0.05 power, and W to the 0.8 to 0.5 power respectively. The effect of G on M is found to be slight, just as was determined by theoretical analysis. The values obtained from the formulae proposed in the past largely differ from those obtained in the test, with all equations providing values smaller than those obtained in the test. The effect of G on M, furthermore, cannot be predicted accurately. It is thought that the equations are not suitable for estimating frictional torque of the bearings lubricated in an oil bath Discussion The reason that the exponent of W obtained in the test is larger than that of the equation 5 for estimating viscous rolling resistance could possibly be that the range of W for the test was not as wide as that of the theoretical analysis. As shown in Fig., the exponent of W determined in the range of W = 0 5 to 0 4 is larger than that of equation 4. For the range of W = 0 5 to 0 4, the exponent of W for equation 5 is 0.0, which is close to the exponent obtained in the test. This could be caused by the fact that the correlation of M and W indicted in the two logarithmic coordinate planes is not linear. There could be a problem with arranging in the form of M W c. The reason the exponent for U obtained in the test is slightly smaller than that of equation 5 could possibly be due to the thermal effect of inset shear heating. The effect of Frictional torque in raceway M conta Frictional torque in raceway contacts M, N m 0 G = 3 300, W = U =.0 0, W = G = 3 800, W = U = 3.4 0, W = G = 4 000, W = U = 4.8 0, W = G = 4 00, W = U = 6.8 0, W = Frictional torque in raceway contacts M, N m U =.0 0, G = U = 3.4 0, G = U = 5.6 0, G = U = 8.5 0, G = Dimensionless speed parameter, U Dimensionless material parameter, G Dimensionless load parameter, W Fig. 9 Effects of dimensionless parameters on frictional torque in raceway contacts Test data Equation (6) Goksem-Hargreaves Hamrock-Jacobson Aihara Zhou-Hoeprich Frictional torque in raceway contacts M, N m 0 G = 3 800, W = Test data Equation (6) Goksem-Hargreaves Hamrock-Jacobson Aihara Zhou-Hoeprich Frictional torque in raceway contacts M, N m Test data Equation (6) Goksem-Hargreaves Hamrock-Jacobson Aihara Zhou-Hoeprich U = 3.4 0, W = U = 3.4 0, G = Dimensionless speed parameter, U Dimensionless material parameter, G Dimensionless load parameter, W Frictional torque in raceway contacts M, N m 0 Fig. 0 Comparisons between calculated frictional torque in raceway contacts and test results KOYO Engineering Journal English Edition No.59E (00) 57

6 Dimensionless viscous rolling resistance, Mv U =.5 0 0, G = Regression in range of W= 0 5 to 0 4 Mv W 0.30 Regression in range of W = 0 5 to 0 3 Mv W Dimensionless load parameter, W Fig. Effect of regression range on exponents of W difference of regression range observed in the exponent of W is not observed in the exponent of U. Due to taking the effect of apparatus capacity and heat generation into account, the range of U, G and W experimentally analyed could not be covered by a wide range of theoretically analyed U, G and W. The fact that the exponent of the simplified formula for calculating oil film thickness analyed together with viscous rolling resistance more or less agrees with the exponents of equations proposed by numerous researchers 4) suggests that the theoretical analysis described in this paper is adequate. Experimental verification within the same range as used for theoretical analysis will be a theme in the future. 5. A New Formula for Frictional Torque 5. Frictional Torque in Roller-raceway Contact As was previously mentioned, it was found that the exponents for U i, U 0, W i and W 0 of equation 6 obtained from EHL analysis are different from those confirmed by testing. Thus, by correcting equation 6, a new frictional torque formula for the test conditions has been derived in this paper. r i, r 0, R i and R 0 in equation 6 can be expressed using D w, d m and c m, so if these are substituted for equations 9 ~ as well as for equation 6, the expressions are altered as follows: 0.75 M = dm E' l ( c m ) 4. U G W [ ( + cm ) 0.75 ( cm ) ( cm ) 0.75 ( + cm ) 0.9 ] 0.57 M = 0.8 g ut U G 0.04 W 0. 3 where, g is the parameter determined only by bearing dimensions, and is given by the following equation: g = dm E' l ( cm ) 8 [uci ( +cm) 0.75 ( cm ) uco ( cm ) 0.75 ( + cm ) 0.9 ] 4 u T is the thermal reduction factor to consider the thermal effect of inlet shear heating in the EHL contact. The following equation from the result of Goksem-Hargreaves analysis is used 9) exp (.06 0 L ln D ) ut = 5 l L u Ci and u C0 in equation 4 are factors for easily correcting for the effect of torque reduction due to crowning of the inner and outer raceways respectively. u Ci and u C0 obtained by testing are arranged by ratios R Ci /R i and R C0 /R 0, where R C is crowning radius and R is equivalent radius. 5. Frictional Torque in Rib-roller End Contact When lis coefficient of friction at rib-roller end contact and Q r is normal load acting on rib, F r in equation l is expressed as follows. Fr = lq r 6 And then Qr = Fa sin c/ ( sina) ro sin c/ ( sina) = Equation l can be written as follows. M = le Fa cosc It was experimentally confirmed that ldecreased with increase of rotational speed and lwas nearly constant after a certain rotational speed. lis generally arranged by oil film parameter, K, because lwas changed by lubricating conditions between rib and roller end. The correlation of land K, such as shown in Fig., is approximately expressed as follows: c 3 l= c exp ( c K ) + c 4 0 where K is the ratio of central oil film thickness to composite surface roughness (Ra), and c, c, c 3 and c 4 are constants determined from testing. lwas nearly constant at K in a VG3 oil bath, and the mean value was In the case of well run-in tapered roller bearings, lcan be regarded as small and constant at practical speed conditions. Coefficient of friction, l Test data Approximate curve Oil film parameter, K Fig. Coefficient of friction of rib-roller end contact versus oil film parameter 5. 3 Calculation Examples Frictional torque of bearing is obtained as the sum of one at the raceway contacts estimated by equation 3 and one at the rib contacts estimated by equation M = 0.8 g ut U G 0.04 W 0. +lefacosc Examples of comparisons between measured values and calculated values for M are shown in Fig. 3 and Fig. 4. In these figures, the symbols represent the measured values and KOYO Engineering Journal English Edition No.59E (00)

7 the solid lines represent the calculated values. Figure 3 shows a comparison of results when crowning radius of outer raceway is changed. Figure 4 shows a comparison of results when the rotational speed is increased up to min with bearings of different sies and internal dimensions. These figures show that frictional torque of tapered roller bearings lubricated in an oil bath of non-additive mineral oil can be predicted by equation. Frictional torque of bearing, N m Rco/Ro = u45 u VG3 oil bath, Fa = 4kN Rco/Ro = 40 Rco/Ro = Rotational speed, min Fig. 3 Comparison of calculated total frictional torque and test results Frictional torque of bearing, N m Bearing Inner dia. Outer dia. Width A B C VG3 oil bath, Fa = 4kN Bearing C Bearing A Bearing B Rotational speed, min Fig. 4 Comparison between calculated total frictional torque and test results 5. 4 Prediction of Frictional Torque under Practical Conditions However, there is a problem in using equation to predict frictional torque of tapered roller bearings lubricated with gear oil used for automotive driving systems. Because equation 3 for calculating the frictional torque in raceway contacts has been derived based on test data using an oil bath of non-additive mineral oil, the effects of oil amount, base oil types and additives cannot be taken into account. The frictional torque lubricated by gear oil can be estimated from the following equation, which is derived correcting equation 3 based on test data under the same conditions: k M = k ut F a N k 3 k go 4 k ao 5 +le Fa cosc G G = f (, d m, D w, E', l, a, cm, uci, uco ) 3 where k, k, k 3, k 4 and k 5, which are constants determined from testing, are influenced on lubricating conditions and G is the parameter determined only by bearing dimensions. An example of comparisons between measured values and calculated values for frictional torque of tapered roller bearings lubricated in a gear oil are shown in Fig. 5. Frictional torque of bearing, N m Bearing Inner dia. Outer dia. Width D E Gear oil 85W-90 Oil supply 800 cm 3 /min Fa = 6kN, g0 = 95mPa s Bearing E Bearing D Rotational speed, min Fig. 5 Comparison between calculated total frictional torque and test results 6. Conclusions As the first step in the study of frictional torque for tapered roller bearings, theoretical and experimental analyses of frictional torque in raceway contacts were conducted. The results are as follows: ) A simplified formula for viscous rolling resistance under full flooded conditions was obtained by conducting full EHL analysis in the range of U = 0 3 to 0 9, G = 500 to and W = 0 5 to 0 3. m v = 4. E' l R U 0.75 G 0.04 W 0.08 ) As a results of experimental analysis in the range of U = 0 to 0 0, G = 700 to 000 and W = 0 5 to 0 4, it was confirmed that frictional torque in the raceway contacts obtained by testing generally agreed with one predicted from EHL theory. 3) The frictional torque obtained by experiment was proportional to U to the 0.5 to 0.69 power, G to the 0.03 to 0.05 power and W to the 0.8 to 0.5 power respectively, and difference between experiments and theoretical analyses was observed concerning effects of U and W on frictional torque. 4) By correcting the simplified formula for viscous rolling resistance, the following formula for calculating frictional torque of the bearings lubricated in an oil bath has been proposed. M = 0.8 g ut U 0.57G 0.04 W 0. + le Fa cosc KOYO Engineering Journal English Edition No.59E (00) 59

8 As indicated in this paper, it is currently difficult to estimate frictional torque under all operating conditions with a single calculation formula; different equations must be used according to each individual operating condition to accurately estimate frictional torque. However, the formulae proposed in this paper are sufficient for studying effect of internal design specifications on frictional torque and would be extremely effective for low-torque bearing design in the future. <Symbols> b = half width of Hertian contact, m, b = R (8W/p) / d m = pitch circle diameter of roller, m e = contact height of rib and roller-end, m h = oil film thickness, m k = thermal conductivity of oil, W/m/ C l = effective roller length, m m V = viscous rolling resistance, N m m i = viscous rolling resistance of roller-inner raceway contact, N m m 0 = viscous rolling resistance of roller-outer raceway contact, N m p = pressure, Pa p h = maximum Hertian pressure, Pa p h = E' (W/p) / r i = mean inner raceway diameter, m r i = d m ( c m )/ r o = mean outer raceway diameter, m r o = d m (+ c m )/ u = mean surface velocity, m/s, u = (u +u )/ w = load per unit length, N m, w = Q/l x = coordinate of rolling direction, m x cp = x coordinate of center of pressure, m y = coordinate in longitudinal direction, m = number of rollers D = dynamic loading parameter D = (9p 3 /) / GUW 3/ D w = mean roller diameter, m E, = Young's modulus, Pa E' = equivalent Young's modulus, Pa E' = [( m )/E + ( m )/E ] F a = Axial load, N F r = frictional force of rib-roller end contact, N G = dimensionless material parameter, G =a 0 E' H = dimensionless oil film thickness, H = hr/b L = thermal loading parameter, L = g 0 b 0 u /k M = frictional torque of bearing, N m M = frictional torque due to raceway contacts, N m M = frictional torque due to rib contacts, N m M V = dimensionless viscous rolling resistance M V = m V /(be'lr) N = rotational speed, min P = dimensionless pressure, P = p/p h Q = normal load, N Q i, o = load on rollers from inner and outer raceways, N Q i Q o = F a /( sina) Q r = load on roller-end from rib, N R = equivalent radius, m, R = (/R + /R ) <Symbols> R, = radius of curvature, m R Ci, Co = crowning radius of inner and outer raceways, m R i = equivalent radius of roller-inner raceway contact, m R i = D W ( c m )/ R o = equivalent radius of roller-outer raceway contact, m R o = D W ( +c m )/ U = dimensionless speed parameter, U = u g 0 /(E'R) W = dimensionless load parameter, W = w/(e'r) X = dimensionless coordinate, X = x/b X cp = dimensionless x coordinate of center of pressure X cp = x cp /b a = outer raceway half angle, rad a 0 = pressure-viscosity coefficient of lubricant, Pa b = inner raceway half angle, rad b 0 = temperature-viscosity coefficient of lubricant, C c = roller taper half angle, rad c m = D w cosh/d m g 0 = viscosity of lubricant at atmospheric pressure, Pa s h = angle formed by center of roller and center of bearing rad, h=a c=b+c l = coefficient of friction at rib-roller end contact m, = Poisson's ratio r, = surface roughness of rib and roller-end, lmra x i, o, r = angular velocity of inner ring, outer ring and rollers, rad/s K = film parameter, K= (r +r ) / subscript letters i and o represent roller-inner raceway contact and roller-outer raceway contact respectively. References ) M. Takeuchi: Koyo Engineering Journal, 7 (985) 5. ) Y. Asai, H. Ohshima: Koyo Engineering Journal, 43 (993) 3. 3) H. Ohshima: Koyo Engineering Journal, 47 (995) 37. 4) A. Palmgren: VDI-Berichte Bd., 0 (957) 7. 5) D. C. Witte: ASLE Trans., 6, (973) 6. 6) S. Aihara: Trans. ASME, Jour. of Tribology, 09, July (987) 47. 7) R. S. Zhou and M. R. Hoeprich: Trans. ASME, Jour. of Tribology, 3, July (99) ) C. L. Karna: ASLE Trans., 7 (974) 4. 9) P. G. Goksem and R. A. Hargreaves: Trans. ASME, Jour. of Lubrication Technology, 00, July (978) ) P. G. Goksem and R. A. Hargreaves: Trans. ASME, Jour. of Lubrication Technology, 00, July (978) 353. ) P. G. Goksem and R. A. Hargreaves: Trans. ASME, Jour. of Lubrication Technology, 00, Oct. (978) 47. ) B. J. Hamrock and B. O. Jacobson: ASLE Trans., 7, 4 (984) 75. 3) H. Matsuyama, S. Kamamoto and K. Asano: SAE Technical paper, 9809 (998). 4) H. Matsuyama, S. Kamamoto, K. Asano: Synopses, Tribology Conference (Nagoya 998-) KOYO Engineering Journal English Edition No.59E (00)