Mixed and Fluid Film Lubrication Characteristics of Worn Journal Bearings

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.


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][2][3][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.
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.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.

Theoretical Model and Coordinates
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].

Nondimensional Basic Equations.
The load-carrying capacity of asperity contact is estimated from the capacities in elastic and plastic contact.The asperity-contact pressures p ae and p ap averaged over a small apparent area is given by [22] where d * e is the separation, E is the equivalent modulus of elasticity = E /[6μω(R/C) 2 ], H a is the hardness of the softer material = H a /[6 μω(R/C) 2 ], w * p = (β /σ * )(2H a /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 d * e = h s (h s ≥ 1.8) and d * e = 1.5 h s − 0.9 (h s < 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 F m is expressed as follows: where ψ(s) is the standard probability density function = (2π) −1/2 exp(−s 2 /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: where φ θ and φ y are the pressure flow factors along the θ and y axes, respectively, and φ s is the shear flow factor.The pressure boundary conditions are as follows: p f = p e (= atmospheric pressure) at the side-end of the bearing and p f = p g (= fluid pressure in the groove) at the edge of the supply ports.The equilibrium of load W due to contacting asperity and fluid can be computed from the following integrals: The side leakage flow rate Q s is given by ( Advances in Tribology In addition, the friction F a due to the asperities in contact and the friction F f due to fluid acting on the sliding surface are formulated below [25]: where A 0 (= A 0 /LR) is the land area, dA r (= dA e + dA p ) is the real contacting area, dA is the apparent area, ) is the shearing strength, τ ad is the shearing strength of the adsorbed film.φ f , φ f p , and φ f s are the factors after the PC model [23,24].The parameter ξ (= φ exp [(h ad /a r ) m /(ΔA r /ΔA 0 )] ) is given by the area ratio of solid-contact asperities [25], where a r (= [β σ * F 1 (d * e )/F 0 (d * e )] 1/2 ) is the mean radius of the asperity contact, h ad is the elastohydrodynamic (EHL) film thickness of the point contact, and ΔA r /ΔA 0 (= π(ηβ σ * )F 1 (d * e )) is the real contact area ratio under dry friction.
Finally, the friction coefficient f is defined by The clearance h of the worn bearing is given by where δ represents the wear dent defined as δ = δ 0 − 1 − cos(θ − π/2) in θ s ≤ θ ≤ θ e and δ 0 is the maximum of wear depth [5].

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 = 83.6GPa, hardness of softer materials H a = 250 MPa, bearing-width ratio L/(2R) = 0.5, groove width L g /L = 0.8, lubricant supply pressure p g = 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 θ g = π/9 rad, fluid density ρ = 850 kg/m 3 , equivalent surface roughness σ = 1 μm, shear strength of the adsorbed film τ ad = 15 MPa, and shear strength at the plastic contact τ p = 70 MPa.The roughness is assumed to be isotropic, so that the roughness parameter γ in the PC model is unity.

Results and Discussion
Figure 2 shows the loci of the journal with respect to the wear dent δ.When the bearing bush is not worn, that is, the maximum wear depth δ o = 0, the locus is formed as a half circle.As δ o 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 (δ 0 > 0) and the local clearance is larger than the radial clearance C. Figures 3(a)-3(d) illustrate the fluid pressure p f and the asperity-contact pressure p a under the operating conditions of S o = 10 0 -10 −3 .Under the conditions of S o = 10 0 and 10 −1 , the asperity-contact pressure p a remains zero for the entire circumference of the bearing clearance, which corresponds to fluid film lubrication.Under the conditions of S o = 10 −2 and 10 −3 , p a is partially positive, which corresponds to mixed lubrication.
In fluid film lubrication (Figures 3(a) and 3(b)), two peaks of p f or a peak and a hump seemingly appear for larger δ o .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 pressure p a peaks at around θ = π rad and the fluid pressure p f peaks somewhat upstream at the circumferential location θ of the p a peak.This is because of the surface contact at the minimum clearance and an effective wedge film located just before the minimum clearance.When δ o = 0, the peaks of p a and p f are sharp.As δ o slightly increases, the peaks become smooth, and as δ o increases further, both p a and p f increase.
Figures 4 and 5 depict the effects of the wear dent δ 0 on the friction coefficient fR/C and the side leakage flow rate Q s , respectively.In the mixed lubrication regime, the coefficient fR/C is largest for δ 0 = 0 and smallest for δ 0 = 0.1; for δ 0 > 0, as δ 0 increases, fR/C increases.In the fluid film lubrication Figures 6 and 7 show the maximum fluid pressure p f max and the maximum asperity-contact pressure p a max in terms of the Sommerfeld number S o , respectively.The maximum pressure p a max of the asperity contact increases as S o decreases below S o = 10 −2 .It is largest at δ 0 = 0 and smallest at δ 0 = 0.1 but increases as δ 0 increases.In contrast, the maximum pressure p f max of the fluid pressure varies over S o = 10 0 -10 −3 , where it maximizes around S o = 2 to 5 × 10 −3 and minimizes in S o = 10 −2 -10 −1 , except for the case of δ 0 = 0.1.In particular, p f max 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 (δ 0 = 0.1) enables reducing the maxima of the fluid pressure p f max and the asperity-contact pressure p amax .
Figure 8 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 Q s increase over the entire range of S o = 10 −3 -10 0 (δ = 0.5), where there is larger change in the coefficient fR/C in the mixed lubrication regime.Figure 9 shows the effects of the surface roughness σ/C on the friction coefficient fR/C and the side leakage Q s in terms of the Sommerfeld number S o .S o for the transition from fluid film to mixed lubrication is larger for worn bearings.In contrast, the effect of σ/C on the leakage Q s is relatively low.

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.

Figure 1 :
Figure 1: Schematic and coordinates of a plain hydrodynamic journal bearing with a wear dent.
Figure 3: (a) Distributions of fluid pressures p f (S o = 10 0 ) (b) distributions of fluid pressures p f (S o = 10 −1 ); (c) distributions of asperitycontact pressures p a and fluid pressures p f (S o = 10 −2 ); (d) distributions of asperity-contact pressure p a and fluid pressure p f (S o = 10 −3 ).

Figure 6 :
Figure 6: Effect of dent δ o on maximum asperity-contact pressure p amax .

Figure 7 :
Figure 7: Effect of dent δ o on maximum fluid pressure p f max .

5 Figure 8 :Figure 9 :
Figure 8: Effect of bearing-width ratio L/(2R) on frictional coefficient f R/C and side leakage flow rate Q s in terms of Sommerfeld number S o (no dent and δ o = 0.5).