Performance of Elastohydrodynamic Inclined Fixed Pad Thrust Bearing with Ultra Low Clearance

This study investigates the performance of elasto-hydrodynamic inclined fixed pad thrust bearing with ultra-low clearance. The analysis was conducted using a multiscale approach by considering the surface elastic deformation, the physical adsorption layer and the fluid piezo-viscous effect. Numerical calculations reveal that in this bearing the magnitude of the surface elastic deformation is comparable to both the surface separation and the adsorption layer thickness. Compared to the case of rigid bearing surfaces, the surface elastic deformation results in a significant reduction of the maximum hydrodynamic pressure in the bearing. For a given speed and load, when the fluid-bearing surface interaction is stronger, the influence of the surface elastic deformation on the bearing pressure is more significant. For different surface materials with quite different Young’s modulus of elasticity, when the load and speed are given, the distributions of the film pressure and the surface separation in the bearing are obviously different. For a given load, the surface elastic deformation results in the reduction of the minimum surface separation. For a given minimum surface separation, the increase of the load with the sliding speed is more rapid for rigid bearing surfaces than for elastic bearing surfaces.


Introduction
Hydrodynamic thrust bearings are one of the fundamental mechanical components for supporting the rotating shafts with the axial loads with the purpose of low friction and negligible surface wear.Classical hydrodynamic lubrication theories [1] have been developed well for these bearings based on the assumptions of the Newtonian fluid, the smooth bearing surfaces and the isothermal condition.In severe operating conditions with heavy loads and high sliding speeds, the effects of the fluid non-Newtonian shear thinning [2,3], the surface roughness [4,5], the fluid film viscous heating [6,7] and the surface elastic deformation [8,9] were also incorporated, since in such bearings the surface separations are comparable to both the surface roughness and the magnitudes of the surface elastic deformations and the film shear strain rates are high.Yan et al. [10] showed that the wall slippage effect can result in the greatly reduced surface clearance and even the film breakdown in a hydrodynamic step bearing for a given load.In the big-size hydrodynamic thrust bearings with heavy loads and high sliding speeds such as applied in hydrogenerators, the minimum surface clearances are very low and even vanishing owing to the thermoelastic deformation of the bearing surfaces [11][12][13].On the other hand, the surface clearances in micro thrust bearings are intrinsically as low as on the 1nm or 10nm scales [14][15][16].In these macro or micro thrust bearings with ultra-low clearances, the hydrodynamic status is actually more complicated than ever expected, and it is in the mixed lubrication where different lubrication regimes (physically adsorbed layer boundary lubrication and sandwich film lubrication) respectively occur in different irregular areas (owing to different surface separations) if the surface roughness is incorporated.The elastic deformations of the bearing surfaces surely significantly change the areas where the different films are distributed, as they are at least on the same scale with or even much greater than the bearing clearances.Conventional modeling of mixed lubrication in the thrust bearing omitted these complicated phenomena in the condition of much higher bearing clearances [17][18][19][20].
With the increase of the load or with the reduction of the sliding speed, the bearing clearance will be ultra-low and only on the 1nm scale.In this case, locally or even in the whole bearing the continuum fluid film can disappear and only the adsorbed layers on the bearing surfaces exist.This type of bearing may be in the new mode of mixed lubrication where different flow regimes occur respectively in different areas i.e. the pure adsorbed layer and the sandwich film simultaneously occur respectively in different lubricated areas.Surely, in such a bearing the effect of the surface elastic deformation should be significant as the magnitude of the surface elastic deformation is much comparable to or even much greater than the surface separation.Currently, the study is absent on this type of bearing.
The present paper aims to analytically investigate the performance of the hydrodynamic inclined fixed pad thrust bearing with ultra-low clearance by incorporating both the effects of the adsorbed layer and the surface elastic deformation.In the pure adsorbed layer area, the flow is described by the molecular-scale non-continuum flow equation; In the other area, the sandwich film flow is described by Zhang's multiscale flow equations [21].The present study is more progressive than the research before in two regards i.e. by exploring the much smaller bearing clearance with the intermediate continuum fluid film locally vanishing and by considering the elastic deformations of the bearing surfaces.The mathematical equations have been derived respectively for calculating the hydrodynamic pressures and the carried load of the studied bearing.The numerical calculations were made for solving these equations.The obtained results reveal the important influences of both the adsorbed layer and the surface elastic deformation on the performance of this new mode of the bearing.

Elastohydrodynamic thrust bearing with ultra-low clearance
Fig. 1 shows the studied hydrodynamic inclined fixed pad thrust bearing with ultra-low clearance involving the effect of the surface elastic deformation.The bearing clearance is so low that in part of the lubricated area (in the outlet zone) the intermediate continuum fluid film is squeezed out and only the adsorbed layer exists, while in the other lubricated area the surface separations are big enough so that the continuum fluid film is contained between the two adsorbed layers.For a given operating condition, the surface elastic deformation results in the denting of the surface and the increase of the surface separation so that the area where the pure adsorbed layer exits is reduced and the sandwich film area is increased.The films in this bearing appear quite mixed with different rheologies.The previous analysis thus fails and the new analytical tool should be developed for this bearing.
In Fig. 1, u is the sliding speed, hbf is the thickness of the physically adsorbed layer, h is the thickness of the intermediate continuum fluid film, htot is the surface separation, htot,o is the surface separation on the exit of the bearing,  is the tilting angle, and l is the width of the bearing.Fig. 1 The studied hydrodynamic inclined fixed pad thrust bearing with ultra-low clearance when the effect of the surface elastic deformation is involved

Mathematical analysis
In the pure adsorbed layer area, there are only several molecule layers adhering to the bearing surfaces, the rheological properties of which are largely influenced by the interaction between the fluid and the bearing surface.The flow in this area is essentially molecular-scale noncontinuum and the rheological evolution such as the en-hancement of both the effective viscosity and the average density of the adsorbed layer should be considered.Also the discontinuity and inhomogeneity across the adsorbed layer thickness should be incorporated.In these circumstances, the molecular-scale non-continuum flow equation is used to describe the flow in the pure adsorbed layer area.In the sandwich film area, the flow regimes of the adsorbed layer and the intermediate continuum fluid are radically different, and also there are the coupled effect between the adsorbed layer and the intermediate continuum fluid [21].Zhang's multiscale flow equations are used to describe the flow in the sandwich film area.
For simplifying the problem and being focused on the issues, the present analysis is based on the assumptions of two identical adsorbed layers, negligible side leakage and film viscous heating, smooth bearing surfaces, and no interfacial slippage.

Core equations
According to Zhang's multiscale flow equations [21], in the sandwich film area the total mass flow rate per unit contact length through the bearing is:  − is the separation between the neighboring fluid mole- cules across the adsorbed layer thickness just on the boundary between the adsorbed layer and the continuum fluid film, and the formulations of F1, F2, and  have been shown in [21].
In the pure adsorbed layer area, the total mass flow rate per unit contact length through the bearing is [21] where ,2 eff bf  are respectively the average density and the effective viscosity of the adsorbed layer across the whole surface separation, and S is the parameter accounting for the discontinuity and inhomogeneity effect of the adsorbed layer.

Numerical calculation for film pressure and load
The incorporation of the surface elastic deformation causes the high non-linearity of the present problem.The numerical calculation is thus mandatory for finding the film pressure and load of the bearing.In the numerical analysis, there are evenly distributed (N+1) discretized points in the whole lubricated area in the present bearing.
By forward difference, the pressure gradient on the i-th discretized point is expressed as: where the formulations of, a, b, c, d, A and B have been shown in [21], the subscript "i" denotes the values on the i-th discretized point, and By considering the surface elastic deformation, the surface separation on the i-th discretized point is expressed as: here hoo is constant, Ev is the compound Young's modulus of elasticity of two bearing surfaces, and s is an integral variable.
The thickness of the intermediate continuum fluid film on the i-th discretized point is: Based on the boundary condition p0 = 0, the pressure on each discretized point was calculated from the following equation: ( ) where (pk − pk-1) is calculated from Eq. ( 3).The load per unit contact length carried by the bearing is: 3.2.2.Numerical integration for the surface elastic deformation and numerical solution procedure The equation for the surface elastic deformation shows the following integral term: The exact calculation of Eq. ( 8) is impossible because of the unknown film pressure distribution.The numerical integration of Eq. ( 8) has been shown in [22].The detailed numerical solution procedure is also shown in [22].

Formulation for fundamental parameters
The bulk density and bulk viscosity of the fluid on the i-th discretized point are respectively formulated as: is the critical thickness for characterizing the rheological properties of the adsorbed layers in the pure adsorbed layer area.The general form (Cq(Hbf)) of Cq1 and Cq1 have been shown in [21].The general form (Cy(Hbf)) of Cy1 and Cy1 as well as the function S(Hbf,2) have also been shown in [21].
In all the calculations, the following parameter values were chosen: Most of the characteristic parameter values respectively for the weak, medium and strong fluid-bearing surface interactions have been shown in [21].Some of them are shown in Table 1.Different bearing surface materials were used.The corresponding values of Ev for these materials are shown in Table 2.It is assumed that , , 1 0 / line j line j q   + = , where  is constant and , -1 line j  is the local viscosity between the j-th and (j-1)-th fluid molecules across the adsorbed layer thickness.

Film pressure distributions
Fig. 2, a-c shows the influences of the surface elastic deformation on the film pressure distributions when the fluid-bearing surface interactions are respectively weak, medium and strong under different loads and the bearing surface is made of steel.Why the different loads were used for different fluid-bearing surface interactions is to obtain the hydrodynamic status shown in Fig. 1 where the sandwich film area and the pure adsorbed layer area coexist.It is shown that the surface elastic deformation significantly reduces the maximum film pressure and modifies the film pressure profile especially when the load is heavier and thus the magnitude of the surface elastic deformation is greater.This is particularly the case for the strong fluidbearing surface interaction.Fig. 3 shows the influences of different bearing surface materials on the film pressure distribution in the bearing for the given load and sliding speed when the effect of the surface elastic deformation is considered.For the same operating conditions, these four different materials give different maximum film pressures and different film pressure profiles owing to their different Young's modulus of elasticity, which results in different magnitudes of the surface elastic deformation.Greater the Young's modulus of elasticity of the bearing surface material, steeper the film pressure distribution.This effect should be more significant when the load is heavier.Fig. 4, a-c shows the corresponding surface separation profiles involving the surface elastic deformation when the operating conditions are respectively the same as shown in Fig. 2, a-c.The surface elastic deformation significantly modifies the surface separation profile, reduces the surface separation in the inlet zone, but increases the surface separation in most of the outlet zone (except on the exit of the bearing), especially when the load is heavier.Although the pressures in the bearing are normal and no more than 10 MPa as shown in Fig. 2, a-c, the changes of the surface separations by the surface elastic deformation are on the same scale with the original bearing clearance, and their effects on the film pressures and load-carrying capacity of the bearing should undoubtedly be pronounced.Fig. 5 shows the corresponding surface separation profiles for different bearing surface materials when the effect of the surface elastic deformation is considered and the film pressures are shown as in Fig. 3.For a given operating condition, the surface material has a strong influence on the surface separation profile in the bearing.For silicon carbonized, which has the highest Young's modulus of elasticity ( 4 8E11 Pa v E. = ), the bearing surfaces are nearly rigid under the load; In this case, the calculation can ignore the effect of the surface elastic deformation.However, for non crystalline silica, which has the lowest Young's modulus of elasticity ( 7 5E10 Pa Strong interaction,elastic surfaces,u=18μm/s Strong interaction,rigid surfaces,u=18μm/s Medium interaction,elastic surfaces,u=40μm/s Medium interaction,rigid surfaces,u=40μm/s Weak interaction,elastic surfaces,u=50μm/s Weak interaction,rigid surfaces,u=50μm/s Fig. 6 Variations of the carried load of the bearing with the minimum surface separation htot,min respectively for elastic and rigid bearing surfaces for different fluidbearing surface interactions when the bearing surface is made of steel Fig. 6 compares the load (w) versus minimum surface separation (htot,min) curves respectively for elastic and rigid surfaces when the sliding speed is given, the fluidbearing surface interaction is respectively weak, medium and strong, and the bearing surface is made of steel.htot,min is located on the exit of the bearing, and the slope of the curve is the film stiffness in the bearing.It is shown that the surface elastic deformation results in the reduced film stiffness.Fig. 7 shows that for a given operating condition, when the bearing surface material is different, the film stiffness is different; The surface material with a higher Young's modulus of elasticity gives a higher film stiffness.
For the same value of htot,min the carried load of the bearing is greater for the less elastic surface material, or for the same load the value of htot,min is greater.This shows that the surface material influences the load-carrying capacity of the bearing owing to the surface elastic deformation.Strong interaction,elastic surfaces Strong interaction,rigid surfaces Medium interaction,elastic surfaces Medium interaction,rigid surfaces Weak interaction,elastic surfaces Weak interaction,rigid surfaces Fig. 8 Variations of the load of the bearing with the sliding speed respectively for elastic and rigid surfaces for different fluid-bearing surface interactions when htot,min = 7 nm and the bearing surface is made of steel Fig. 8 shows that whenever the bearing surface is rigid or in the elastic deformation, the carried load by the bearing is linearly increased with the increase of the sliding speed.However, the surface elastic deformation results in the reduced proportionality of the load with the sliding speed especially for the strong fluid-bearing surface interaction, which yields the highest film pressures and thus the greatest surface elastic deformation when the sliding speed is given.For the weak and medium fluid-bearing surface interactions, the two curves are nearly overlaid owing to the low film pressures and the resulting negligible effect of the surface elastic deformation.

Conclusions
The effect of the surface elastic deformation is analytically investigated in the hydrodynamic inclined fixed pad thrust bearing with ultra-low clearance.The hydrodynamic status in this bearing is mixed and it consists of the molecular-scale non-continuum hydrodynamics of the pure adsorbed layer and the hydrodynamics of the sandwich film which contains both the adsorbed layer and the intermediate continuum fluid film across the surface separation.The analysis should solve these two different flow regimes.Here, the molecular-scale non-continuum flow equation is used to describe the pure adsorbed layer flow, and Zhang's closed-form explicit multiscale flow equations are used to describe the sandwich film flow.Owing to the high non-linearity of the problem caused by the surface elastic deformation, the numerical calculations were carried out for finding the film pressures and carried load of the bearing.Different fluid-bearing surface interactions and different bearing surface materials were considered.
Based on the calculation results, the conclusions are drawn as follows: The surface elastic deformation has a significant influence on the performance the bearing.For a given load, it reduces the maximum film pressure and the minimum surface separation, and modifies both the film pressure profile and the surface separation profile.
The effect of the surface elastic deformation strongly depends on the fluid-bearing surface interaction.For a given operating condition, stronger the fluid-bearing surface interaction, more significant the effect of the surface elastic deformation.
The surface elastic deformation results in the reduced film stiffness in the bearing.The bearing surface material significantly influences the performance of the bearing.The less elastic surface material gives the higher film stiffness and the greater load-carrying capacity of the bearing.
Whenever the bearing surface is rigid or in the elastic deformation, the carried load of the bearing is linearly increased with the sliding speed.However, the surface elastic deformation results in the reduced proportionality of the bearing load with the sliding speed.
The effect of the surface elastic deformation heavily depends on the carried load of the bearing, which determines the magnitude of the surface elastic deformation.Whenever the film pressures are normal or very high in the bearing with ultra-low clearance, the effect of the surface elastic deformation should be involved when analyzing the bearing performance.This study investigates the performance of elastohydrodynamic inclined fixed pad thrust bearing with ultralow clearance.The analysis was conducted using a multiscale approach by considering the surface elastic deformation, the physical adsorption layer and the fluid piezoviscous effect.Numerical calculations reveal that in this bearing the magnitude of the surface elastic deformation is comparable to both the surface separation and the adsorption layer thickness.Compared to the case of rigid bearing surfaces, the surface elastic deformation results in a significant reduction of the maximum hydrodynamic pressure in the bearing.For a given speed and load, when the fluidbearing surface interaction is stronger, the influence of the surface elastic deformation on the bearing pressure is more significant.For different surface materials with quite different Young's modulus of elasticity, when the load and speed are given, the distributions of the film pressure and the surface separation in the bearing are obviously different.For a given load, the surface elastic deformation results in the reduction of the minimum surface separation.For a given minimum surface separation, the increase of the load with the sliding speed is more rapid for rigid bearing surfaces than for elastic bearing surfaces.


p is the film pressure, x is the coordinate shown in Fig.1,  and  are respectively the bulk density and the bulk viscosity of the fluid, are respectively the average density and the effective viscosity of the adsorbed layer, D is the fluid molecule diameter, x is the separation between the neighboring fluid molecules in the x coordinate direction in the adsorbed layer, q0 is the average value of 1 is the sepa- ration between the (j+1)-th and j-th fluid molecules across the adsorbed layer thickness), n is the equivalent number of the fluid molecules across the adsorbed layer thickness, 2 n

Fig. 2
Fig. 2 Film pressure distributions in the bearing for different fluid-bearing surface interactions respectively for elastic and rigid surfaces when u = 2.0E−5 m/s and the bearing surface is made of steel: afor the weak interaction, w = 220 N/m; bfor the medium interaction, w = 306 N/m; cfor the strong interaction, w = 562 N/m

Fig. 3 5 . 2 .
Fig. 3 Film pressure distributions in the bearing with different surface materials involving the surface elastic deformation for the medium fluid-bearing surface interaction when u = 2.0E−5 m/s and w = 306 N/m 5.2.Surface separation profile

Fig. 4 Fig. 5 5 . 3 .
Fig. 4 Surface separation in the bearing for different fluidbearing surface interactions considering elastic deformation when u = 2.0E−5 m/s and the bearing surface is made of steel: afor the weak interaction, w = 220 N/m; bfor the medium interaction, w = 306 N/m; cfor the strong interaction, w = 562 N/m

Fig. 7
Fig. 7 Variations of the carried load of the bearing with the minimum surface separation htot,min for different bearing surface materials for u = 4.0E−5 m/s when the surface elastic deformation is involved 5.4.Variation of the load of the bearing with the sliding speed

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. Wang, Y. Zhang PERFORMANCE OF ELASTOHYDRODYNAMIC IN-CLINED FIXED PAD THRUST BEARING WITH UL-TRA LOW CLEARANCE S u m m a r y