Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 54, 2, pp. 345-352, Warsaw 2016 DOI: 10.15632/jtam-pl.54.2.345 A CONCENTRIC HYDRODYNAMIC JOURNAL BEARING WITH THE BOUNDARY SLIPPAGE Yongbin Zhang College of Mechanical Engineering, Changzhou University, Changzhou, Jiangsu Province, China e-mail: engmech1@sina.com Thepaperproposes a concentrichydrodynamic journal bearing constructedby theboundary slippage, which is opposed by conventional lubrication theory. Analysis for the carried load and friction coefficient of this bearing is presented.The optimumcondition for themaximum load-carrying capacity of this bearing is examined. It is shown that the whole circumference of the bearing should be taken as the lubricated area,while onmost of the stationary surface of the bearing there should be a hydrophobic coating covered so that the boundary slippage would occur on this surface, In this condition, the load-carrying capacity of the bearing is the highest but its friction coefficient is the lowest. Keywords: hydrodynamics, boundary slippage, load, friction, bearing 1. Introduction Conventional lubrication theory says that no hydrodynamic lubrication effect can be generated between two sliding parallel smooth plane surfaces (Pinkus, 1961). It also denies a concentric hydrodynamic journal bearing,where the lubricatingfilmthickness is circumferentially constant. However, in practice, a concentric hydrodynamic journal bearing is very useful because of its high supporting precision, high lubricating film thickness, low viscous friction and low energy consumption. The boundary slippage has been found to be capable of improving the performance of a hydrodynamic lubrication (Salant and Fortier, 2004; Zhang, 2008, 2010, 2013, 2014, 2015b; Li et al, 2014). It was found that hydrodynamic lubrication can be generated between two sliding parallel smooth plane surfaces because of the boundary slippage (Zhang, 2008). While, in conventional bearing configurations, the artificial introduction of the boundary slippage can increase the load-carrying capacity of the bearing but reduce its friction coefficient Zhang (2010, 2013, 2014, 2015b). In Zhang (2015b), the performance of a hydrodynamic journal bearing with an eccentricity was found to be able to be significantly improved by the boundary slippage. However, in that paper, a concentric hydrodynamic journal bearing was not addressed. This paper proposes a concentric hydrodynamic journal bearing which is formed dependent on the boundary slippage. Analysis of this bearing is presented. It has been found that a signi- ficant load-carrying capacity can be generated depending on the design method applied. The optimum condition for the maximum load-carrying capacity of the bearing is also analyzed. In this optimum condition, the bearing also works with the lowest friction coefficient. The study shows a potential application value of such a bearing in practice. 2. The bearing configuration Figure 1 shows the configuration of the studied bearing. The bearing is formed by a rotating shaft (with circumferential speed u) and a stationary sleeve. The two elements are concentric 346 Y. Zhang and the lubricating film thickness in the bearing is constant and equal to the bearing clearance c(= R−r).The clearance of thebearing is filledwith afluid.The lubricated area of thebearing is divided into two subzones, i.e. “I” and “II” subzoneswhich are, respectively, the inlet and outlet zones of the bearing. On the stationary (sleeve) surface in “I” subzone there is a hydrophobic coating covered to yield a low fluid film-contact surface interfacial shear strength (τsa) at this surface so that the boundary slippage would occur at this coated surface. The envelope angles of “I” and “II” subzones are respectively φI and φII. On the other surfaces of the bearing, the fluid film-contact surface interfacial shear strength is relatively high so that the boundary slippage is absent on these surfaces. The radii of the shaft and sleeve are r and R, respectively. The carried load per unit contact width and attitude angle of the bearing are respectively w and γ. The coordinate system used in the analysis is also shown in Fig. 1. Fig. 1. Configuration of the proposed bearing 3. Analysis Theanalysis carried out byZhang (2015b) is also applicable to the present bearing.The analysis is based on the following assumptions: (a) The lubricant film thickness is high enough so that the lubricant is continuum across the film thickness; (b) The lubricant film is Newtonian within the film; (c) The lubricant is isoviscous and imcompressible; (d) Contact surface deformations are negligible; (e) The side leakage in the bearing is negligible and the lubricant is in laminar flow; (f) The operating condition is isothermal. Accordingly to Zhang (2015b), the following dimensionless parameters are defined: W = wc2 uηr2 P = pc2 uηr Qv = qv uc F̄x = Fxc 2 uηr2 F̄y = Fyc 2 uηr2 A concentric hydrodynamic journal bearing with the boundary slippage 347 F̄f,h = Ff,hc 2 uηr2 F̄f,s = Ff,sc 2 uηr2 τ̄ = τc2 uηr kτ = τsac uη DU = ∆u u Here, η is fluid viscosity, p is film pressure, qv is volume flow rate in the bearing per unit contact width, Fx and Fy are respectively components of the carried load in the x and y coordinate directions, Ff,h and Ff,s are respectively friction forces per unit contact width acting on the sleeve and shaft surfaces, τ is shear stress, and∆u is fluid film interfacial slipping velocity. The pressure boundary conditions in the bearing are: P|φ=0 =0 P|φ=φI+φII =0 (3.1) When the eccentricity ratio ε is zero, a lot of the analytical results obtained by Zhang (2015b) are applicable to the present bearing. The following Sections demonstrate those results. 3.1. “I” subzone The dimensionless Reynolds equation in “I” subzone is: dPslip dφ =3−3Qv,slip − 3kτ 2 (3.2) Using the boundary condition in Eq. (3.1), integrating Eq. (3.2) gives dimensionless pressure in “I” subzone: Pslip = ( 3−3Qv,slip − 3kτ 2 ) φ for 0¬ φ ¬ φI (3.3) The dimensionless pressure on the boundary between “I” and “II” subzones is: Pslip,max = ( 3−3Qv,slip − 3kτ 2 ) φI (3.4) 3.2. “II” subzone The dimensionless Reynolds equation in “II” subzone is: dPslip dφ =6−12Qv,slip (3.5) Using the boundary condition in Eq. (3.1), integrating Eq. (3.5) gives dimensionless pressure in “II” subzone: Pslip =(6−12Qv,slip)(φ−φI −φII) for φI ¬ φ ¬ φI +φII (3.6) According to Eq. (3.6), the dimensionless pressure on the boundary between “I” and “II” sub- zones is: Pslip,max =(12Qv,slip −6)φII (3.7) Equations (3.3) and (3.6) show that the pressure is respectively linearly distributed in “I” and “II” subzones in the present bearing. Figure 2 schematically shows the pressure distribution in the present bearing. 348 Y. Zhang Fig. 2. Illustration of the pressure distribution in the proposed bearing 3.3. Volume flow rate and condition for the bearing Define ψφ = φII/φI, solving coupled equations (3.4) and (3.7) gives: Qv,slip = 1+2ψφ − 1 2 kτ 1+4ψφ (3.8) and Pslip,max = P̄slip,max(φI +φII) (3.9) where P̄slip,max =6ψφ(1−kτ)/[(1+ψφ)(1+4ψφ)]. From Qv,slip > 0, it is obtained that kτ < 2+4ψφ. From Pslip,max > 0, it is obtained that kτ < 1. Therefore, kτ < 1 is the condition for the present bearing. It is noted fromEq. (3.9) that for given values of kτ and φI +φII, when ψφ =1/2, Pslip,max reaches the maximum, and its maximum value is 2(1−kτ)(φI +φII)/3. 3.4. Carried load and attitude angle of the bearing The dimensionless hydrodynamic force component in the x axis direction acting on the shaft per unit contact width is: F̄x,slip =− φI+φII ∫ 0 Pslipcosφ dφ = P̄slip,maxf1(ψφ,φtot) (3.10) where φtot =φI +φII and (also Zhang (2015a)) f1(ψφ,φtot)= ( 1+ 1 ψφ ) [ ψφφtot 1+ψφ sin ( φtot 1+ψφ ) +cosφtot − cos ( φtot 1+ψφ ) ] − (1+ψφ) [ φtot sin ( φtot 1+ψφ ) 1+ψφ +cos ( φtot 1+ψφ ) −1 ] (3.11) The dimensionless hydrodynamic force component in the y axis direction acting on the shaft per unit contact width is: F̄y,slip = φI+φII ∫ 0 Pslip sinφ dφ = P̄slip,maxf2(ψφ,φtot) (3.12) A concentric hydrodynamic journal bearing with the boundary slippage 349 where (also Zhang (2015a)) f2(ψφ,φtot)= ( 1+ 1 ψφ ) [ ψφφtot 1+ψφ cos ( φtot 1+ψφ ) − sinφtot +φtot cosφtot +sin ( φtot 1+ψφ ) ] +(1+ψφ) [ sin ( φtot 1+ψφ ) − φtot 1+ψφ cos ( φtot 1+ψφ ) ] (3.13) The dimensionless load per unit contact width carried by the bearing is: Wslip = √ F̄2 x,slip + F̄2 y,slip = P̄slip,maxfw(ψφ,φtot) (3.14) where fw(ψφ,φtot)= √ f21(ψφ,φtot)+f 2 2(ψφ,φtot) (Zhang, 2015a). Figure 3 plots the values of fw against φtot for the given values of ψφ. It is shown that for a given ψφ, the value of fw reaches the maximum when φtot = 2π. This means that for the maximum load-carrying capacity of the bearing, φtot should be taken as 2π. Fig. 3. Plots of fw against φtot for given ψφ values (Zhang, 2015a) When φtot =2π, the dimensionless load is: Wslip =(1−kτ)G(ψφ) (3.15) where G(ψφ)= 6ψφfw(ψφ,2π)/[(1+ψφ)(1+4ψφ)]. The attitude angle of the bearing is: γ =arctan [ f2(ψφ,φtot) f1(ψφ,φtot) ] (3.16) 3.5. Friction coefficient and interfacial slipping velocity The dimensionless shear stress on the shaft surface is: τ̄s,slip =        c r ( 3− kτ 2 −3Qv,slip ) for 0¬ φ ¬ φI c r (4−6Qv,slip) for φI < φ ¬ φI +φII (3.17) 350 Y. Zhang The dimensionless shear stress on the sleeve surface is: τ̄h,slip =      kτ c r for 0¬ φ ¬ φI c r (6Qv,slip −2) for φI < φ ¬ φI +φII (3.18) The dimensionless friction force on the shaft surface per unit contact width is: F̄f,s,slip = φI+φII ∫ 0 τ̄s,slip dφ = φI ∫ 0 τ̄s,slip dφ+ φI+φII ∫ φI τ̄s,slip dφ = c r ( 3− kτ 2 −3Qv,slip ) φI + c r (4−6Qv,slip)φII (3.19) The dimensionless friction force on the sleeve surface per unit contact width is: F̄f,h,slip = φI+φII ∫ 0 τ̄h,slip dφ = φI ∫ 0 τ̄h,slip dφ+ φI+φII ∫ φI τ̄h,slip dφ = kτφI c r + c r (6Qv,slip −2)φII (3.20) The friction coefficients on the sleeve and shaft surfaces are respectively: fh,slip = F̄f,h,slip Wslip fs,slip = F̄f,s,slip Wslip (3.21) The dimensionless slipping velocity of the fluid film at the sleeve surface is: DU =      3Qv,slip 2 − 1 2 − kτ 4 for 0¬ φ ¬ φI 0 for φI < φ ¬ φI +φII (3.22) where DU should be positive for 0¬ φ ¬ φI. 4. Results and discussion Figure 4a plots the values of G against ψφ when φtot = 2π. It is shown that G significantly increases with the reduction of ψφ when ψφ ­ 0.1. While, for ψφ < 0.01, G is weakly influenced by ψφ. According to Eq. (3.15), it means that for a given kτ the load-carrying capacity of the bearing increases with the reduction of ψφ, especially when ψφ ­ 0.1, while too low values of ψφ have no benefits in increasing the load-carrying capacity. As the optimum value of ψφ for the maximum value of Pslip,max is 0.5, in the engineering design, the value of ψφ may be recommended to be chosen between 0.1 and 0.5. Figure 4b plots values of γ against ψφ when φtot = 2π. The minimum value of γ is about 57o, and it occurs when ψφ is around 1.0. For ψφ < 0.1 or ψφ > 20, γ approaches 90 o. Figures 5a and 5b plot respectively values of fs,slipr/c and fh,slipr/c against ψφ for different kτ when φtot =2π. It is shown that for given values of kτ and c/r, the friction coefficients fs,slip and fh,slip both are linearly reduced with the reduction of ψφ. This indicates that a relatively low value of ψφ has also benefit of giving a low friction coefficient to the bearing. The reduction of kτ is shown to significantly reduce the friction coefficient, especially when ψφ is high. A concentric hydrodynamic journal bearing with the boundary slippage 351 Fig. 4. Plots of (a) G, (b) γ against ψφ when φtot =2π Fig. 5. Plots of fs,slipr/c and fh,slipr/c against ψφ for different kτ when φtot =2π 5. Conclusions This paper proposes a concentric hydrodynamic journal bearing which is formed dependent on the boundary slippage. The configuration of the bearing is presented. The lubricated area of the bearing is divided into two subzones, which may respectively be the inlet and outlet zones. In the inlet zone, on the stationary surface a hydrophobic coating is covered to yield a low fluid film-contact surface interfacial shear strength so that the boundary slippage could occur on this surface. On the other bearing surfaces, the boundary slippage is absent because of relatively high interfacial shear strengths on these surfaces. Analysis for the load-carrying capacity and friction coefficient of the bearing is presented. Typical calculations have been carried out. It has been found that the optimumvalue of the ratio of the circumferential length of the outlet zone to that of the inlet zone, i.e. the optimum value of ψφ is 0.5 for the maximum hydrodynamic pressure building-up. However for this value ψφ, the load-carrying capacity of the bearing is still not the maximum. The whole circumference of the bearing should be taken as the lubricated area for achieving a high load-carrying capacity. 352 Y. Zhang In this condition, the carried load of the bearing is found to be increased with the reduction of ψφ, especially when ψφ ­ 0.1. Nevertheless, for ψφ ¬ 0.01, the load-carrying capacity of the bearing is weakly influenced by variation of ψφ. It is recommended that in engineering design the value of ψφ should be chosen between 0.1 and 0.5. A low value of ψφ also has the benefit of giving a low friction coefficient to the bearing. References 1. Pinkus O., Sternlicht B., 1961,Theory of hydrodynamic lubrication, McGraw-Hill, NewYork 2. Salant R.F., Fortier A.E., 2004, Numerical analysis of a slider bearing with a heterogeneous slip/no-slip surface,Tribology Transactions, 47, 328-334 3. ZhangY.B., 2008,Boundaryslippage forgeneratinghydrodynamic load-carryingcapacity,Applied Mathematics and Mechanics, 29, 1155-1164 4. Zhang Y.B., 2010, Boundary slippage for improving the load and friction performance of a step bearing,Transactions of the Canadian Society for Mechanical Engineering, 34, 373-387 5. Zhang Y.B., 2013, A tilted pad thrust slider bearing improved by boundary slippage,Meccanica, 48, 769-781 6. ZhangY.B., 2014,Hydrodynamic lubrication in line contacts improvedby the boundary slippage, Meccanica, 49, 503-519 7. ZhangY.B., 2015a,A concentricmicro/nano journal bearing constructed by physical adsorption, Journal of the Balkan Tribological Association, 21, 937-951 8. Zhang Y.B., 2015b, An improved hydrodynamic journal bearing with the boundary slippage, Meccanica, 50, 25-38 9. Li G., Zhang Y.B., Jiang X.D., 2014, A study on the performance of s hydrodynamic step bearing by controlling its boundary slippage in the outlet zone, Journal of Changzhou University, Natural Sciences Edition, 26, 33 Manuscript received April 24, 2015; accepted for print July 29, 2015