# Mixed and Fluid Film Lubrication Characteristics of Worn Journal Bearings

Authors: Toshiharu Kazama; Yukihito Narita
Journal: Advances in Tribology
 (2012)

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## Abstract

The mixed and fluid film lubrication characteristics of plain journal bearings with shape
changed by wear are numerically examined. A mixed lubrication model that employs both of
the asperity-contact mechanism proposed by Greenwood and Williamson and the average
flow model proposed by Patir and Cheng  includes the effects of adsorbed film and
elastic deformation is applied. Considering roughness interaction, the effects of the dent depth
and operating conditions on the loci of the journal center, the asperity-contact and
hydrodynamic fluid pressures, friction, and leakage are discussed. The following conclusions
are drawn. In the mixed lubrication regime, the dent of the bearing noticeably influences the
contact and fluid pressures. For smaller dents, the contact pressure and frictional coefficient
reduce. In mixed and fluid film lubrication regimes, the pressure and coefficient increase for
larger dents. Furthermore, as the dent increases and the Sommerfeld number decreases, the
flow rate continuously increases.


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## Body

## 1. Introduction

Hydrodynamic journal bearings are widely used as main shaft bearings of various machinery and equipment such as gas turbines, turbo generators, pumps, and machine tools [1–4]. Punctual and appropriate maintenance of the bearings that ensures they are lubricated by a fluid film enables them to operate for a long time.However, the sliding surfaces partially contact at the startup, shutdown, and under low-speed conditions, namely, the bearings are imposed to operate in the mixed lubrication regime. The materials of the sliding parts of the bearing bush are often composed of relatively soft metals such as white metal and copper-aluminum alloys. Thus, when the bearings are operated in the mixed lubrication regime for a specific period, the sliding surfaces are plastically deformed and worn [5], forming dents and resulting in clearance shape changes from its dimensions at design. This shape change affects the bearing performance.The static and dynamic characteristics of worn journal bearings have been studied by Hashimoto and Wada [6, 7], Kumar and Mishra [8, 9], and Tanaka and Suzuki [10, 11]. In parallel, their thermohydrodynamic characteristics have also been studied by Fillon and Bouyer [12]. The mixed lubrication characteristics of journal bearings have been studied by many researchers, for example, S. A. Mckee and T. R. Mckee [13], Burwell et al. [14], Kreisle [15], Kazama and Fujiwara [16], Nonogaki and Nakahara [17], and Wang et al. [18–20]. However, a few theoretical studies of dented journal bearings that include interactions between roughness asperities and surface contacts have been published.This study numerically analyzes plain hydrodynamic journal bearings with a wear dent model in the fluid film and mixed lubrication regimes. Including roughness interaction, the effects of shape changed by wear on friction and leakage as well as contact pressure and fluid pressure are examined. Furthermore, the bearing characteristics are discussed under the isothermal and steady-state condition.


## 2. Theory

### 2.1. Theoretical Model and Coordinates

Figure1 shows the coordinates of the hydrodynamic journal bearing model with a wear dent. The interference and contact of the mating surface roughness are considered, whereas misalignment of the journal, the effects of heat generation, and the condition of dynamic loading are neglected. Laminar flow of the lubricant in the bearing clearance is assumed.Figure 1
Schematic and coordinates of a plain hydrodynamic journal bearing with a wear dent.A mixed lubrication model [21] that employs the asperity-contact mechanism proposed by Greenwood and Williamson (hereafter, the GW model) [22] and the average flow model proposed by Patir and Cheng (hereafter, the PC model) [23, 24] is used for this analysis. The details of this lubrication model are described in [25, 26].


### 2.2. Nondimensional Basic Equations

The load-carrying capacity of asperity contact is estimated from the capacities in elastic and plastic contact. The asperity-contact pressuresp̅ae and p̅ap averaged over a small apparent area is given by [22](1)p̅ae=23E̅′ηβ′σ*(σ*β′)1/2[F3/2(de*)-F3/2(de*+wp*)],p̅ap=πH̅aηβ′σ*F1(de*+wp*),wherede* is the separation, E̅′ is the equivalent modulus of elasticity = E′/[6μω(R/C)2],  H̅a is the hardness of the softer material =Ha/[6μω(R/C)2],wp*=(β′/σ*)(2Ha/E′)2, β′ is the mean radius of asperities, η is the density of asperity, and σ* is the standard deviation of the asperity summit height = 0.7σ. The relationship de*=hs(hs≥1.8) and de* = 1.5 hs-0.9(hs<1.8) holds for these mean planes [27]. The subscripts e and p represent the elastic and plastic deformation of asperities, respectively. Moreover, the function Fm is expressed as follows:(2)Fm(hs)=∫hs∞(s-hs)mψ(s)ds,whereψ(s) is the standard probability density function = (2π)−1/2 exp(-s2/2). The standard plane of rough surfaces is either the summit mean plane or the surface mean plane, where the former is used in the GW model and the latter in the PC model [23, 24].Following the PC model, the average Reynolds equation is described as follows:(3)∂∂θ(ϕθh3̅∂p̅f∂θ)+(RL)2∂∂y̅(ϕyh3̅∂p̅f∂y̅)=∂h̅T*∂θ+σC∂ϕs∂θ,whereϕθ and ϕy are the pressure flow factors along the θ and y axes, respectively, and ϕs is the shear flow factor. The pressure boundary conditions are as follows:p̅f = p¯e (= atmospheric pressure) at the side-end of the bearing and p̅f=p̅g (= fluid pressure in the groove) at the edge of the supply ports.The equilibrium of loadW̅ due to contacting asperity and fluid can be computed from the following integrals:(4)∫02π∫01(p̅a+p̅f)cos⁡θdy̅dθ=0,∫02π∫01(p̅a+p̅f)sin⁡θdy̅dθ+RLW̅=0.The side leakage flow rateQ̅s is given by(5)Q̅s=12(RL)2∫02πϕyh3̅(∂p̅f∂y̅|y̅=-1/2-∂p̅f∂y̅|y̅=1/2)dθ.In addition, the frictionF̅a due to the asperities in contact and the friction F̅f due to fluid acting on the sliding surface are formulated below [25]: (6)F̅a=LR∬A̅0ξτ̅p(τ̅eτ̅pdA̅e+dA̅p)+(1-ξ)τ̅addA̅r,F̅f=CL6R2∫02π∫011h̅{(1-2σ12σ2)[ϕf+(1-2σ12σ2)ϕfs]-3ϕfph̅∂p̅f∂θ2σ12σ2}dy̅dθ,
where A̅0(=A0/LR) is the land area, dA̅r(=dA̅e+dA̅p) is the real contacting area, dA̅ is the apparent area, W′(=1/(E′β2)ΔWa/[ηF0(de*)ΔA0]) is the load parameter,τ̅(=τ/[6μω(R/C)2]) is the shearing strength, τ̅adis the shearing strength of the adsorbed film.  ϕf, ϕfp, and ϕfs are the factors after the PC model [23, 24]. The parameter ξ(=ϕ′exp[(had/ar)m/(ΔAr/ΔA0)]) is given by the area ratio of solid-contact asperities [25], where ar(=[β′σ*F1(de*)/F0(de*)]1/2) is the mean radius of the asperity contact, had is the elastohydrodynamic (EHL) film thickness of the point contact, and ΔAr/ΔA0(=π(ηβ′σ*)F1(de*)) is the real contact area ratio under dry friction.Finally, the friction coefficientf is defined by(7)f=F̅a+F̅fW̅.The clearanceh̅of the worn bearing is given by(8)h̅=1+εsin⁡(θ-ϕ)+δ̅,whereδ̅ represents the wear dent defined as δ̅=δ̅0-1-cos(θ′-π/2)inθs≤θ′≤θe and δ̅0 is the maximum of wear depth [5].


### 2.3. Calculation Procedure and Parameters Used

The Reynolds equations are discretized by the finite difference method, and the solutions are obtained by an iterative scheme [16]. The operating conditions are specified by the following parameters: clearance ratio C/R= 0.001, equivalent elastic modulus E′ = 83.6 GPa, hardness of softer materials Ha = 250 MPa, bearing-width ratio L/(2R) = 0.5, groove width Lg/L = 0.8, lubricant supply pressure pg = 0.1 MPa, bearing radius R= 50 mm, bearing load W= 10 kN, equivalent radius of the asperity summit β′ = 0.05 mm, ηβ′σ*=0.05 (η′: asperity density, σ*: standard deviation of the asperity summit height), viscosity μ = 27.2 mPa·s, circumferential extent angle of groove θg= π/9 rad, fluid density ρ = 850 kg/m3, equivalent surface roughness σ = 1 μm, shear strength of the adsorbed film τad = 15 MPa, and shear strength at the plastic contact τp= 70 MPa. The roughness is assumed to be isotropic, so that the roughness parameter γ in the PC model is unity.


## 2.1. Theoretical Model and Coordinates

Figure1 shows the coordinates of the hydrodynamic journal bearing model with a wear dent. The interference and contact of the mating surface roughness are considered, whereas misalignment of the journal, the effects of heat generation, and the condition of dynamic loading are neglected. Laminar flow of the lubricant in the bearing clearance is assumed.Figure 1
Schematic and coordinates of a plain hydrodynamic journal bearing with a wear dent.A mixed lubrication model [21] that employs the asperity-contact mechanism proposed by Greenwood and Williamson (hereafter, the GW model) [22] and the average flow model proposed by Patir and Cheng (hereafter, the PC model) [23, 24] is used for this analysis. The details of this lubrication model are described in [25, 26].


## 2.2. Nondimensional Basic Equations

The load-carrying capacity of asperity contact is estimated from the capacities in elastic and plastic contact. The asperity-contact pressuresp̅ae and p̅ap averaged over a small apparent area is given by [22](1)p̅ae=23E̅′ηβ′σ*(σ*β′)1/2[F3/2(de*)-F3/2(de*+wp*)],p̅ap=πH̅aηβ′σ*F1(de*+wp*),wherede* is the separation, E̅′ is the equivalent modulus of elasticity = E′/[6μω(R/C)2],  H̅a is the hardness of the softer material =Ha/[6μω(R/C)2],wp*=(β′/σ*)(2Ha/E′)2, β′ is the mean radius of asperities, η is the density of asperity, and σ* is the standard deviation of the asperity summit height = 0.7σ. The relationship de*=hs(hs≥1.8) and de* = 1.5 hs-0.9(hs<1.8) holds for these mean planes [27]. The subscripts e and p represent the elastic and plastic deformation of asperities, respectively. Moreover, the function Fm is expressed as follows:(2)Fm(hs)=∫hs∞(s-hs)mψ(s)ds,whereψ(s) is the standard probability density function = (2π)−1/2 exp(-s2/2). The standard plane of rough surfaces is either the summit mean plane or the surface mean plane, where the former is used in the GW model and the latter in the PC model [23, 24].Following the PC model, the average Reynolds equation is described as follows:(3)∂∂θ(ϕθh3̅∂p̅f∂θ)+(RL)2∂∂y̅(ϕyh3̅∂p̅f∂y̅)=∂h̅T*∂θ+σC∂ϕs∂θ,whereϕθ and ϕy are the pressure flow factors along the θ and y axes, respectively, and ϕs is the shear flow factor. The pressure boundary conditions are as follows:p̅f = p¯e (= atmospheric pressure) at the side-end of the bearing and p̅f=p̅g (= fluid pressure in the groove) at the edge of the supply ports.The equilibrium of loadW̅ due to contacting asperity and fluid can be computed from the following integrals:(4)∫02π∫01(p̅a+p̅f)cos⁡θdy̅dθ=0,∫02π∫01(p̅a+p̅f)sin⁡θdy̅dθ+RLW̅=0.The side leakage flow rateQ̅s is given by(5)Q̅s=12(RL)2∫02πϕyh3̅(∂p̅f∂y̅|y̅=-1/2-∂p̅f∂y̅|y̅=1/2)dθ.In addition, the frictionF̅a due to the asperities in contact and the friction F̅f due to fluid acting on the sliding surface are formulated below [25]: (6)F̅a=LR∬A̅0ξτ̅p(τ̅eτ̅pdA̅e+dA̅p)+(1-ξ)τ̅addA̅r,F̅f=CL6R2∫02π∫011h̅{(1-2σ12σ2)[ϕf+(1-2σ12σ2)ϕfs]-3ϕfph̅∂p̅f∂θ2σ12σ2}dy̅dθ,
where A̅0(=A0/LR) is the land area, dA̅r(=dA̅e+dA̅p) is the real contacting area, dA̅ is the apparent area, W′(=1/(E′β2)ΔWa/[ηF0(de*)ΔA0]) is the load parameter,τ̅(=τ/[6μω(R/C)2]) is the shearing strength, τ̅adis the shearing strength of the adsorbed film.  ϕf, ϕfp, and ϕfs are the factors after the PC model [23, 24]. The parameter ξ(=ϕ′exp[(had/ar)m/(ΔAr/ΔA0)]) is given by the area ratio of solid-contact asperities [25], where ar(=[β′σ*F1(de*)/F0(de*)]1/2) is the mean radius of the asperity contact, had is the elastohydrodynamic (EHL) film thickness of the point contact, and ΔAr/ΔA0(=π(ηβ′σ*)F1(de*)) is the real contact area ratio under dry friction.Finally, the friction coefficientf is defined by(7)f=F̅a+F̅fW̅.The clearanceh̅of the worn bearing is given by(8)h̅=1+εsin⁡(θ-ϕ)+δ̅,whereδ̅ represents the wear dent defined as δ̅=δ̅0-1-cos(θ′-π/2)inθs≤θ′≤θe and δ̅0 is the maximum of wear depth [5].


## 2.3. Calculation Procedure and Parameters Used

The Reynolds equations are discretized by the finite difference method, and the solutions are obtained by an iterative scheme [16]. The operating conditions are specified by the following parameters: clearance ratio C/R= 0.001, equivalent elastic modulus E′ = 83.6 GPa, hardness of softer materials Ha = 250 MPa, bearing-width ratio L/(2R) = 0.5, groove width Lg/L = 0.8, lubricant supply pressure pg = 0.1 MPa, bearing radius R= 50 mm, bearing load W= 10 kN, equivalent radius of the asperity summit β′ = 0.05 mm, ηβ′σ*=0.05 (η′: asperity density, σ*: standard deviation of the asperity summit height), viscosity μ = 27.2 mPa·s, circumferential extent angle of groove θg= π/9 rad, fluid density ρ = 850 kg/m3, equivalent surface roughness σ = 1 μm, shear strength of the adsorbed film τad = 15 MPa, and shear strength at the plastic contact τp= 70 MPa. The roughness is assumed to be isotropic, so that the roughness parameter γ in the PC model is unity.


## 3. Results and Discussion

Figure2 shows the loci of the journal with respect to the wear dentδ̅. When the bearing bush is not worn, that is, the maximum wear depthδ̅o = 0, the locus is formed as a half circle. As δ̅o increases, the loci move in the direction of ϕ = 0, that is, vertically. The eccentricity ratio ε becomes greater than unity because the bush is worn (δ̅0> 0) and the local clearance is larger than the radial clearance C.Figure 2
Loci of journal center.Figures3(a)–3(d) illustrate the fluid pressure p̅f and the asperity-contact pressure p̅a under the operating conditions of So= 100–10−3. Under the conditions of So = 100 and 10−1, the asperity-contact pressure p̅a remains zero for the entire circumference of the bearing clearance, which corresponds to fluid film lubrication. Under the conditions of So = 10−2 and 10−3, p̅a is partially positive, which corresponds to mixed lubrication.(a) Distributions of fluid pressuresp̅f (So = 100) (b) distributions of fluid pressures p̅f (So = 10−1); (c) distributions of asperity-contact pressures p̅a and fluid pressures p̅f (So = 10−2); (d) distributions of asperity-contact pressure p̅a and fluid pressure p̅f (So = 10−3).
(a)(b)(c)(d)In fluid film lubrication (Figures3(a) and 3(b)), two peaks of p̅f or a peak and a hump seemingly appear for larger δ̅o. This is because two wedge films form in the bearing clearance owing to the wear dent. On the other hand, in mixed lubrication (Figures 3(c) and 3(d)), the contact pressurep̅a peaks at around θ = π rad and the fluid pressure p̅fpeaks somewhat upstream at the circumferential location θ of the p̅a peak. This is because of the surface contact at the minimum clearance and an effective wedge film located just before the minimum clearance. When δ̅o= 0, the peaks of p̅a and p̅f are sharp. As δ̅o slightly increases, the peaks become smooth, and as δ̅o increases further, bothp̅a and p̅f     increase.Figures4 and 5 depict the effects of the wear dent δ̅0 on the friction coefficient fR/C and the side leakage flow rate Q̅s, respectively. In the mixed lubrication regime, the coefficient fR/C is largest for δ̅0 = 0 and smallest for δ̅0= 0.1; for δ̅0> 0, as δ̅0 increases, fR/C increases. In the fluid film lubrication regime, the effects of dent on fR/Care relatively low. In the mixed lubrication regime, the leakage Q̅s increases almost linearly as δ̅0     increases and increases continuously as the Sommerfeld number So decreases.Figure 4
Effect of dentδ̅o on frictional coefficient fR/C.Figure 5
Effect of dentδ̅o on side leakage flow rate Q̅s.Figures6 and 7 show the maximum fluid pressure p̅fmax and the maximum asperity-contact pressure p̅amax in terms of the Sommerfeld number So, respectively. The maximum pressure p̅amax of the asperity contact increases as So decreases below So = 10−2. It is largest atδ̅0  = 0 and smallest at δ̅0= 0.1 but increases as δ̅0 increases. In contrast, the maximum pressure p̅fmax of the fluid pressure varies over So = 100–10−3, where it maximizes around So = 2 to 5 × 10−3 and minimizes in So = 10−2–10−1, except for the case of δ̅0= 0.1. In particular, p̅fmax is largest in the mixed lubrication regime for bearings with no dents and large wear dents, but it is largest in the fluid film lubrication regime for bearings with slight wear dents. Therefore, a slightly worn bearing bush (δ̅0 = 0.1) enables reducing the maxima of the fluid pressurep̅fmax and the asperity-contact pressurep̅amax.Figure 6
Effect of dentδ̅o on maximum asperity-contact pressure p̅amax.Figure 7
Effect of dentδ̅o on maximum fluid pressure p̅fmax.Figure8 illustrates the effect of the bearing-width ratio L/D (D = 2R) on the bearing performance. For bearings with and without dents, as the ratio L/(2R) decreases, the friction coefficient fR/C and the side leakage Q̅s increase over the entire range of So = 10−3–100 (δ̅ = 0.5), where there is larger change in the coefficient fR/C in the mixed lubrication regime.Figure 8
Effect of bearing-width ratioL/(2R) on frictional coefficient fR/C and side leakage flow rate Q̅s in terms of Sommerfeld number So  (no dent and δ̅o= 0.5).Figure9 shows the effects of the surface roughness σ/C on the friction coefficient fR/C and the side leakage Q̅s in terms of the Sommerfeld number So. So for the transition from fluid film to mixed lubrication is larger for worn bearings. In contrast, the effect of σ/C on the leakage Q̅s is relatively low.Figure 9
Effect of surface roughnessσ/C on frictional coefficient fR/C and side leakage flow rate Q̅s in terms of Sommerfeld number So (no dent and δ̅o= 0.5).


## 4. Conclusions

The tribological characteristics of plain journal bearings with wear dents in mixed and fluid film lubrication are examined numerically under the steady-state condition. The effects of both the wear dent depth and the surface roughness interaction on the journal center loci, asperity pressure distributions, fluid pressure distributions, friction coefficient, and leakage flow rate are discussed in a wide range of operating conditions. The salient conclusions are as follows.In mixed lubrication, for bearings with no dents, pressure peaks of the asperity-contact and fluid pressures are sharp; as the dent slightly increases, the peaks become small, and as the dent increases further, both the pressures increase. The friction coefficient is largest for bearings with no dents and smallest for those with shallow dents; for bearings with dents, this coefficient increases with the dent depth. In fluid film lubrication, the fluid pressure distribution shows two peaks or a peak and a hump for large dents. In addition, the friction coefficient is less influenced by the dent, and the leakage increases with the dent depth.Similar tendencies are observed for smaller width bearings and rough surface bearings. A bearing with a slightly worn bush could reduce the maxima of the fluid and asperity-contact pressures.



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*Source: 296464-2012-03-13.xml*