Analysis of the effect of oil inlet size on the static performance characteristics of non-Newtonian lubricated hole-entry hybrid journal bearings

In this study, the influence of inlet pocket size on the static performance of non-Newtonian lubricated hole-entry hybrid journal bearings is theoretically analyzed. The oil film of the bearing is discretized into a nonuniform mesh containing the geometric characteristics of the oil inlet pocket, and the inlet pocket is treated as a micro-oil recess. The Reynolds equation is solved by the finite element method based on Galerkin’s techniques, and a new solution strategy to solve the recess/pocket pressure is proposed. The power-law model is used to introduce the non-Newtonian effect. The results show that the static performance characteristics of this type of bearing are greatly affected by the pocket size at both zero speed and high speed.


INTRODUCTION
Compared with rolling bearings and hydrodynamic bearings, liquid hydrostatic bearings are characterized by high load carrying capacity, high stiffness, high precision, less heat generation, less thermal expansion and controllability. They are widely used as critical components in a large number of high-speed, high-precision and heavy-duty CNC machine tools, such as automatic CNC crankshaft/camshaft grinding machines, large gantry milling machines or ultra-precision lathes.
The hydrodynamic effect starts to appear when the spindle of the hydrostatic bearing is rotated, and the bearing becomes a hybrid bearing rather than a simple hydrostatic bearing. To maximize the hydrodynamic effect, the design of the oil recess is avoided. The design avoids the need for oil cavities in order to maximize the hydrodynamic effect, thus allowing much attention to be paid to slot-entry and hole-entry structures. Rowe [1,2] found that the hole-entry hybrid bearing has a very distinct advantage over the recessed hydrostatic bearing due to the reduction of hydrostatic area and the increase in the hydrodynamic area. Later, Koshal and Rowe [3] experimentally found that hole-entry hybrid bearings have many advantages over recessed hydrostatic bearings under most operating conditions, and are particularly suitable for heavy-duty applications with high dynamic loads.
The treatment of the oil inlet hole/pocket is a critical operation for hole-entry hybrid bearings. Rowe et al. [4] conducted a comprehensive study on the hole-entry hybrid bearing using the finite difference method, in which the oil inlet hole/pocket is represented as a single node, for the solution of flow out from the entry port, eight radial flow elements are used to solve the pressure gradient term and two rectangular velocity units are used to solve the entrainment flow. To some extent, this treatment can reflect the influence of the oil inlet hole/pocket size, but the solution of the oil flow rate is heavily dependent on the grid density, resulting in the final pressure distribution showing a distinct spike in the oil inlet hole/pocket region. Yoshimoto et al. [5] pointed out small pockets are usually manufactured in the outlet area of restrictors, and the smaller the pockets, the larger the hydrodynamic load surface and the smaller the oil leakage between the high to low pressure areas. The ratio of inlet pocket and bearing diameter is defined in the paper, the node pressure inside the pocket is equalized and the flow rate of the pocket is solved in a way similar to Rowe et al. [4] The result shows that the load capacity of this type of bearing is greatly affected by the pocket size at zero speed and high speed. In [6], a form of annular orifice restriction is also proposed. When the area of the annular feed πd r h is smaller than the throat area πd 2 r /4, the annular orifice restriction form will play a major role, and the bearing stiffness is considerably reduced. This phenomenon particularly appears in aerostatic bearings, which therefore often have shallow small air chambers in the outlet area [7].
Due to the complexity of the actual operating conditions of hydrostatic bearings, the Reynolds equation and boundary conditions need to be corrected by a combination of influencing factors. To obtain both accuracy and efficiency, self-programmed programs to calculate the bearing performance are often used.
The finite difference and finite element methods with uniform grids are often used, and the oil/air inlet hole is simply treated as a single node, ignoring the influence of the inlet hole/pocket size. Compared with the finite difference method, the main advantages of the finite element method are highly adaptable, less restricted by geometry, dealing with various definite solution conditions, arbitrarily selected unit size and node number, and higher computational accuracy.
For finite element calculations of hydrostatic bearings with unconventional or complex-shaped oil recesses, a large amount of work lies in the meshing of the oil film structures. Prabhu and Ganesan [8] developed a finite element method to study the hydrostatic thrust bearings, linear triangle elements were used and the circular recess was approximated by sector or octagon shapes. With the development of commercial FEM software, meshing has become more convenient. Sharma et al. [9] used the eight-node two-dimensional isoparametric elements to model the circular thrust pad hydrostatic bearing with different recess geometric shapes, and the FEM meshes were generated using the commercial software ANSYS. Yadav and Sharma [10] numerically analyzed the influence of the tilt and recess shape on the static and dynamic performance characteristics of the hydrostatic thrust pad bearing system with a Rabinowitsch lubricant, and the tilt had an obvious influence on the bearing performance. Similarly, Kumar and Sharma [11] theoretically investigated the combined influence of couple stress lubricant, recess geometry and method of compensation on the performance of hydrostatic circular thrust pad bearing. Also, the FEA mesh of four-node quadrilateral elements was discretized by ANSYS.
Non-Newtonian lubrication is a fascinating area of interest in the study of hydrodynamic and hydrostatic bearings. The lubricants used in bearings are mineral in nature, and additives are mixed with them to enhance the lubricating performance. To predict behaviors of these lubricants, various non-Newtonian models such as power law [12], couple stress [13], micropolar [14] and cubic law [15,16] have been employed by many researchers. Singh et al. [17] investigated the hydrodynamic thrust pad bearing with a power-law lubricant by using Galerkin's technique to solve the modified Reynolds equation. Lin [13] theoretically studied the combined effects of couple stresses, fluid inertia and recess volume fluid compressibility on the steadystate performance of hydrostatic circular step thrust bearings. Sharma and Yadav [18] investigated the influence of misalignment on the performance of a micropolar lubricated hybrid journal bearing system, and pointed out that the bearing performance can be improved significantly by using the micropolar fluid. Khatri and Sharma [12] studied the performance of capillary compensated hybrid bearings with a non-Newtonian lubricant, and found that surface texture can improve the steady-state threshold speed.
To solve the problem of neglecting the size of the oil inlet hole/pocket in the usual self-programming finite element program, this paper uses the mature commercial software platform to generate a quadrilateral isotropic unit mesh containing the structure of the oil inlet hole/pocket. A new iterative approach to solving the inlet hole/pocket pressure is then proposed, and the new calculation method allows for the unification of solution methods of hole-entry and recessed hydrostatic bearings. At the same time, the power-law non-Newtonian lubrication is considered, and the static characteristics under different eccentricities are fully discussed. Therefore, the purpose of this paper is to evaluate the influence of the oil inlet hole/pocket size on the performance of hole-entry bearings with non-Newtonian lubrication. The results of this paper will be of great help to bearing designers.

ANALYSIS
The geometric details and coordinate system of the capillarycompensated hole-entry hybrid journal bearing are shown in Fig. 1. The following nondimensional Reynold's equation governs the flow of lubricant in the bearing clearance space:

Fluid film thickness
The nondimensional fluid film thickness expression of bearing land is [12,[19][20][21][22][23] To avoid the hydrodynamic effects caused by the shallow depth of the oil recess/pocket, the film thickness of this area is taken as a finite large number, which is 100 in this paper.

Finite element equation
The Reynolds equation is solved by the finite element method to obtain the oil film pressure distribution. Based on the AN-SYS Workbench, the fluid film was divided into high-quality structured quadrilateral isoparametric element mesh. The block topology and mesh generation are shown in Fig. 2a. The mesh data contain the node array and the coordinates of all nodes. The mapping relationship between the arbitrary quadrilateral element and the four-node rectangular element is shown in Fig. 2b, and the Langrangian interpolation functions are given by [24,25] ⎧ The pressure at any interpolation point (ξ , η) in the element can be approximately expressed as [12,22,26] According to Galerkin's technique, the node pressure can be used to solve the minimization of residue. By orthogonalizing the residue with interpolation function, the following global system equation can be obtained [12,[19][20][21][22]: The integral transformation relation of function G from global coordinates (α, β) to local coordinates (ξ , η) is as follows [25]: where |J| is the rank of the Jacobian matrix J. The local coordinate integral form of Eq. (6) is as follows: where superscript e indicates that Eq. (8) is an equation for the eth element, l 1 and l 2 are the direction cosines and i, j = 1, 2, . . ., n e l . e is the boundary and e is the domain of the eth element.

Power-law model
The behavior of most non-Newtonian oils can be expressed using the power-law model [12,18,27]. The relation between the shear strain rate and shear stress rate of the power-law lubricant is described asτ where n = 1 indicates a Newtonian lubricant, n < 1 indicates a pseudoplastic lubricant showing shear thinning and n > 1 indicates a dilatant lubricant showing shear thickening. The shear strain rateγ is expressed as [28] where ∂p/∂α = (∂N j /∂β ), N j is the Lagrangian interpolation function and n e l is the number of nodes per element [29]. The apparent viscosityμ a is used to describe the viscosity of a power-law lubricant as follows:

Restrictor flow equation
For capillary-compensated hybrid journal bearings, the nondimensional restrictor flow equation can be expressed as follows [4,9,12,19,30]:Q Because the dimensionless coefficientsC S2 of the flow equations for orifice and capillary restrictors are different, it is difficult to make a comparative analysis. In this paper, the restriction form of the annular orifice is not considered. For the case that the size of the oil inlet hole/pocket d r is equal to the diameter of the capillary d c , it is assumed that the restriction form of the simple capillary is still satisfied.

Performance characteristics
This paper compares the static performance characteristics of the hole-entry bearings under both the zero-speed ( = 0) and high-speed ( = 1) conditions, and the static performance characteristics discussed here include bearing load supportW , maximum fluid film pressurep max , lubricant flowQ and frictional torqueT fric . The static performance characteristics are solved with reference to the citation [12,21,24], which is not repeated here.

Boundary conditions
1. The external boundary node pressure is zero. 2. The flow rate after restrictor is equal to the input flow of the bearing. 3. When the depth of the oil recess or inlet pocket is large enough, the nodal pressures in the oil inlet pocket and oil recess area can be equal. 4. The Reynolds boundary condition is applied to deal with the cavitation problem [21,24], and at the trailing edge of the positive region,p

SOLU TION SCHE ME
In this paper, the numerical calculation procedure is shown in Fig. 3. A new grid generation strategy is used, as shown in Fig. 2a. First, the oil film geometry is blocked in the CFD ICEM platform, and high-quality structured quadrilateral isoparametric element mesh generation can be achieved through the reasonable setting of block topology. Based on the Workbench platform, the mesh data are passed to APDL, and then the element and node data are achieved in APDL by macro command. The Reynolds equation is solved in the MATLAB platform. The convergence study with the number of elements is illustrated in Fig. 4. With the increase of the mesh density, the new method has good convergence in bearing load supportW and bearing flow rateQ . The mesh data for d r /D = 0 used in this study include 1536 elements and 1625 nodes, as shown in Fig. 2b. When d r /D = 0, a uniform mesh of 49 × 21 quadrilateral isoparametric elements is used, and the oil inlet hole/pocket is represented as a single node. After reading the grid data and bearing parameters, the coefficient matrices in Eq. (6) can be generated. The coefficient matrices of each element are obtained by Gauss quadrature numerical integration [30], which requires the calculation ofF 0 ,F 1 andF 2 at each Gaussian point in the element. The iteration procedure (Fig. 3) is briefly described as follows: 1. In this paper, a new method is used to solve the pressure distribution. First, the fixed eccentricity is given, the film thickness distribution is calculated by Eq. (4) and the dimensionless film thickness of the oil inlet hole/pocket region is set to a finite large number (100 for this study). Then, the initial pressure distribution and values ofF 0 ,F 1 andF 2 are given. The pressure in the oil inlet hole/pocket area is taken as a constant value, and the Dirichlet boundary condition [31] is used to deal with it. The Newton-Raphson method [12,[19][20][21][22] is used to solve Eq. (6), and then the pressure distribution and the flow rate at each supply hole can be obtained. Then, utilizing the restrictor flow equation [Eq. (12)], the new pressure of each oil supply hole is obtained, and the initial value of oil supply hole pressure in the next pressure iteration is adjusted by a low relaxation factor. The above cycle is iterated repeatedly until the total difference of successive iteration nodal pressures of supply holes is <0.001. 2. The pressure distribution in the Newtonian case is taken as the initial value of the non-Newtonian cycle. ∂p/∂α and ∂p/∂β of all Gauss points in every element are calculated, and thenγ is obtained by Eq. (10). Finally,F 0 , F 1 andF 2 of each Gauss point are obtained by using the Newton-Raphson method, in which the six-point Gauss-Legendre integral is used for numerical integration of film thickness direction. The coefficient matrices in Eq. (6) are updated, and then the iteration of step 1 is repeated to obtain the new pressure distribution. Taking the new pressure distribution as the initial value of the next iteration, the above cycle is repeated until the total difference of successive iteration nodal pressures is <0.001. Then, the final  non-Newtonian pressure distribution for the bearing with fixed eccentricities is obtained. 3. Finally, the static performance characteristics are evaluated based on the obtained pressure distribution.

RESULTS AND DISCUSSION
The bearing geometric and operating parameters are shown in Table 1. A series of inlet pocket diameters are used for analysis. At the same time, single-node oil supply, and four-recess and sixrecess bearings are used for comparative analysis. The bearing geometric and operating parameters are carefully selected from the published literature [4,5,12,19], and the concentric design pressure ratio β* is 0.5 for both hole-entry hybrid bearing and Table 1 Bearing operating and geometric parameters for the holeentry hybrid bearing.

Parameters Value/range
Bearing aspect ratio (λ) 1 Land width ratio (ā b ) 0 . 2  recessed hydrostatic bearing. Considering the manufacturing error of bearings, the maximum safe eccentricity is generally set at 0.5 [6], so the eccentricity range in this paper is 0-0.5. To validate the computational methods and self-programming procedures in this paper, the present results are compared with those of [5,12]. As shown in Fig. 5a and b, the circumferential pressure distribution in the plane of the entry ports and the zero-speed load capacity against eccentricity ratio are compared; the results in this paper are in good agreement with those calculated by the finite difference method in [5]. In Fig. 5b-d, the single-node oil supply and the finite element method are used. Compared with the results of the finite element method in [12], Fig. 5c shows the pressure distribution in the axial middle plane of the hole-entry bearing with a Newtonian lubricant, and Fig. 5d shows the minimum film thicknessh min in the non-Newtonian case. The calculation results in the above figures are in good agreement, which proves the correctness of the calculation results in this paper. For the non-Newtonian lubricated hole-entry hybrid journal bearings with inlet pocket, the pressure distribution of the fluid film under different operating parameters is first compared, and then the static performance characteristics (bearing load sup-portW , maximum fluid film pressurep max , lubricant flowQ and frictional torqueT fric ) are discussed in detail. Figure 6 compares the oil film pressure distribution of the Newtonian lubricated hole-entry hybrid journal bearing under the condition of zero speed and high speed with different pocket/bearing diameter ratios d r /D. Figure 6a and c shows the single-node oil supply situation, and the pressure distribution has an obvious peak near the oil inlet hole; especially in the zerospeed condition, the pressure mutation is particularly obvious. The pocket/bearing diameter ratio d r /D in Fig. 6b and d is 0.1, and the pressure distribution near the oil inlet hole is gentler and there is no peak. Figure 7 compares the circumferential pressure distribution in the plane of the entry ports and axial mid-plane of the holeentry hybrid journal bearing with different speed parameters , pocket/bearing diameter ratio d r /D and power-law index n. In the plane of the entry ports (i.e. step state in Fig. 7a and c), the "dispersion effect" is significant [6], which is due to the wide separation between the holes; the flow from the holes has to flow sideways to fill the gap, so as to produce pressure drop between the restrictors.

Influence on pressure distribution under fixed eccentricity ratio
For the circumferential pressure distribution in the plane of the entry ports in Fig. 7a and c, when the pocket/bearing diameter ratio d r /D is 0, there is no pocket at the outlet of the capillary restrictor, and the pressure drop between adjacent oil supply holes is very obvious. For the bearing with a large inlet pocket (d r /D = 0.15), the pressure drop between adjacent oil inlet holes is the smallest, and the bearing capacity is the largest. At the same time, compared with the pressure distribution under different non-Newtonian factors, the pressure difference is obvious when = 0, and the order of pressure is n = 0.7 > n = 1.0 > n = 1.3.
For the circumferential pressure distribution in the axial mid-plane in Fig. 7b and d, the effect of the non-Newtonian factor on the mid-plane pressure distribution is more pronounced in the case of = 0. When = 1, the hydrodynamic effect is dominant. As the pocket/bearing diameter ratio d r /D increases, the peak pressure distribution shows a decreasing trend. As the non-Newtonian factor increases, the peak of the pressure distribution shows an increasing trend. Figure 8 shows the variation trend of static performance characteristics (bearing load supportW , maximum fluid film pressurē p max , lubricant flowQ and frictional torqueT fric ) under a fixed eccentricity ratio. In Fig. 8a, after considering the size of the oil inlet hole/pocket, the bearing load supportW under zero-speed ( = 0) condition is the minimum at d r /D = 0.005. With the  size, which is due to the decrease of bearing land surface area, resulting in the weakening of the hydrodynamic effect. With the increase of non-Newtonian factor n, the bearing load support under zero speed decreases gradually, while that under high speed increases gradually. A pseudoplastic lubricant (n < 1) enhances the bearing load capacity under zero-speed condition, and weakens the bearing load capacity under high-speed condition; however, the influence trend of a dilatant lubricant (n > 1) is the opposite. In Fig. 8b, considering the size of the oil inlet hole/pocket, the maximum film pressurep max changes slightly under zerospeed condition ( = 0). The maximum oil film pressure reaches the maximum value when d r /D = 0.005. When d r /D = 0.005, the maximum oil film pressure is lower than that of the recessed bearings. When n = 1, the maximum oil film pressure of d r /D = 0.005 decreases by 17.86% compared with that of the six-recess bearing. Considering the non-Newtonian effect, the maximum film pressure decreases with the increase of non-Newtonian factor n. Under high-speed ( = 1) condition, the maximum oil film pressure increases first and then decreases with the increase of d r /D, and reaches the minimum value when d r /D = 0.05. With the increase of non-Newtonian factor n, the maximum oil film pressure presents an increasing trend.

Influence on static characteristics under fixed eccentricity ratio
In Fig. 8c, after considering the size of the oil inlet hole/pocket, the flow rateQ of the bearing shows a similar trend under both zero-speed ( = 0) and high-speed ( = 1) conditions: both change significantly. When d r /D = 0.005, the flow rate decreases to the minimum value compared with single-node oil supply d r /D = 0, and then increases significantly with the increase of d r /D, reaching the minimum value when d r /D = 0.15. When d r /D = 0.15, the flow rate is higher than that of the six-recess bearing. When d r /D = 0.15 and = 0, the flow rate increases by 30.00% compared with that of d r /D = 0.005, and increases by 12.66% compared with that of the six-recess bearing. After considering the non-Newtonian effect, the flow shows an obvious growth trend with the increase of non-Newtonian factor n under zero-speed conditions ( = 0). However, under highspeed ( = 1) condition, the flow rate decreases slightly with the increase of non-Newtonian factor n.
The variation of the frictional torqueT fric with the size of the oil inlet hole/pocket is shown in Fig. 8d. From Eq. (13), it is found that the frictional torqueT fric is velocity dependent, so only the high-speed ( = 1) condition is discussed. Considering the size of the oil inlet hole/pocket, the frictional torque shows an obvious downward trend. The frictional torque reaches the minimum when d r /D = 0.15. When d r /D = 0.15, the frictional torque is higher than that of the six-recess bearing. When d r /D = 0.15, the frictional torque decreases by 12.27% compared with the case of d r /D = 0, and increases by 31.70% compared with that of the four-recess bearing. Considering the non-Newtonian effect,T fric increases slightly with the increase of non-Newtonian factor n.

CONCLUSIONS
The present study numerically investigates the influence of inlet pocket size on the static performance characteristics of non-Newtonian lubricated hole-entry hybrid journal bearings. Based on the discussion in this paper, several conclusions can be drawn: 1. The oil film of the bearing is discretized into a nonuniform mesh containing the geometric characteristics of the oil inlet pocket, and a new solution strategy to solve the recess/pocket pressure is proposed. With the increase of the mesh density, the new method has good convergence in bearing load supportW and bearing flow rateQ . 2. With an appropriate pocket/bearing diameter ratio d r /D, the pressure distribution near the oil inlet hole is gentler and there is no peak. Under zero-speed ( = 0) condition, the peak valley difference of the axial mid-plane pressure distribution decreases with the increase of d r /D. 3. For the influence on static characteristics of hybrid bearings with fixed eccentricity ratio, under zero-speed ( = 0) condition, with the increase of d r /D, the bearing load supportW and lubricant flowQ increase greatly and the maximum fluid film pressurep max decreases slightly. The frictional torqueT fric is greatly reduced under the highspeed ( = 1) condition. 4. A pseudoplastic lubricant (n < 1) enhances the bearing load capacity under zero-speed condition, and weakens the bearing load capacity under high-speed condition; however, the influence trend of a dilatant lubricant (n > 1) is the opposite. 5. To get optimum use of the non-Newtonian lubricated hole-entry hybrid journal bearing, sufficient numerical calculation and discussion about the size of the oil inlet hole/pocket are essential.

NOMENCL ATURE
a b = axial land width c = radial clearance D = journal diameter d c = capillary diameter d r = oil inlet hole/pocket diameter e = journal eccentricity F x , F z = X and Z components of fluid film reactions g = gravitational acceleration h = fluid film thickness K = number of holes per row L = bearing length l c = length of capillary M c = critical mass N = number of rows of holes n = power-law index n e l = number of nodes per element p = pressure p s = supply pressure Q = lubricant flow R J = journal radius T fric = frictional torquē W 0 = external load X, Y, Z = Cartesian coordinate system X J , Z J = journal center coordinates x = circumferential Cartesian coordinate y = axial Cartesian coordinate z = coordinate across the fluid film thickness α = circumferential cylindrical coordinate β = axial cylindrical coordinate τ , τ θ = shear stress in a lubricant film and shear stress in the circumferential direction λ = aspect ratio μ = lubricant viscosity μ 0 = reference viscositẏ γ = shear strain rate ϕ = attitude angle ω J = journal rotational speed