Abstract

The present paper deals with a theoretical analysis of 4-pocket capillary compensated conical hybrid journal bearing system operating with micropolar lubricant. The modified Reynolds equation for a conical journal bearing system operating with micropolar lubricant has been derived. In the present study, Eringen's micropolar theory has been used to model micropolar lubrication in cylindrical coordinate system. The study has been carried out for the different values of micropolar parameters, that is, characteristic length () and coupling number (). The conical bearing configurations having semicone angle 10^∘, 20^∘, 30^∘, and 40^∘ have been studied. The study suggests that micropolar lubricant offers better performance vis- á-vis Newtonian lubricant for bearing configurations having values of semicone angle 10^∘and 20^∘.

1. Introduction

The noncircular journal bearings such as lobed bearing, pressure dam bearing, and conical journal bearing provide better performance than that of the circular journal bearings for different considerations. Conical journal bearing system can support both radial and axial load within less space vis- á-vis different combinations of circular journal and rotary thrust bearing. These bearings are also more economical for the machining point of view [1, 2]. In recent times, many researchers have focused their studies [2–15] concerning the analysis and design of conical bearings.

Murthy et al. [3] studied a new type of conical multilobe special precision bearing. It was reported that axial preload and lobe depth affect the working clearance and the fluid film stiffness of the bearing significantly. Prabhu and Ganesan [4, 6, 7] analyzed the dynamic stiffness characteristics of capillary compensated annular recess conical hydrostatic thrust bearings. In their study, they discussed the conditions of tilt, eccentricity, and rotation. Srinivasan and Prabhu [5] studied the steady state performance of externally pressurized gas-lubricant conical bearings under rotation. The influence of bearing operating parameters on the radial and axial load capacity, attitude angle, friction loss, and stiffness was discussed for the semicone angle 10°–40°. Khalil et al. [8] investigated the influence of turbulent lubrication on the performance of an externally pressurized circular and conical thrust bearings. Both the inertia forces and thermal effects were neglected. Their study indicates that the circular thrust bearing has slightly higher dimensionless pressure, load, and torque than the conical thrust bearing.

The mono and multigrade lubricants used in present day industrial applications are blended with additives which are polymeric in nature. This blending results into deviation of lubricant rheology from Newtonian to non-Newtonian (nonlinear). Yousif and Nacy [9] presented experimental and theoretical studies dealing with the effect of solid additives in lubricating oils on the steady state performance of conical journal bearings using the mass transfer theory. Yoshimoto et al. [10, 11] investigated water-lubricated hydrostatic conical bearings with spiral grooves for high-speed spindles. It was found that the compliant surface bearing had a larger load capacity in a relatively large bearing clearance than the rigid surface bearing and lower bearing power consumption in a small bearing clearance although the load capacity is reduced. Abdel-Rahman [12] used an average clearance method to solve the equation of stationary pressure distribution in flow between the conical surfaces. Li et al. [13] studied Taylor vortex flow between two conical cylinders, with the inner one rotating and the outer one stationary by the numerical method. It was found that the basic flow becomes unstable with an increase in Reynolds number (Re) above a certain critical value (Re = 117) and with the further increase of Re (to about Re = 300), the first stable vortex is formed near the top of the flow system. These were confirmed by experimental observations. Sharma et al. [14, 15] carried out a theoretical study of a 4-pocket hydrostatic conical journal bearing system. The bearing static and dynamic performance characteristics were presented for the values of external load ranging = 0.1–1.0and for the values of semicone angles (, 20°, 30°, and 40°). The influence of wear in the analysis of a four- pocket orifice compensated hybrid conical journal bearing was considered [15]. The studies [14, 15] were limited for Newtonian lubricants only. Very recently Zhang et al. [16] studied the end plate effect on Taylor vortices between rotating conical cylinders. It was reported that the number of vortices increases with decreasing of the gap size and the end plates play an important role in the parity of the number of the vortices.

The lubricant plays a vital role on the overall performance of bearing. A lot of research has been carried out with non Newtonian lubricants. Polymer additives are mixed to the Newtonian lubricant in order to improve their performance characteristics. These polymer additives have the different flow characteristics than the parent fluid. There are many theories available to describe the flow characteristics of non-Newtonian lubricants such as electrorheological lubricant, coupled stress lubricant, and micropolar lubricant. The theory of micropolar fluid gives the best descriptions of the non-Newtonian fluids containing additives substructures. There are many physical examples of micropolar fluids as ferrofluids, blood flows, bubbly liquids, liquid crystals [17]. This class of fluids exhibit special properties to support couple stress and body couple due to their microstructure and microrotational inertia at microscopic level. The theory of micropolar fluids was formulated by Eringen [18]. The first application of micropolar lubricant in fluid film bearings was carried out by Allen and Kline [19]. Prakash and Sinha [20] discussed the lubrication theory for micropolar fluid. They assumed a two dimensional flow field and gave argument for order of magnitude analysis. Later on, Singh and Sinha [21] derived 3-dimensional Reynolds equation for micropolar lubricated cylindrical journal bearing. Huang et al. [22] and Khonsari and Brewe [23] obtained steady state characteristics of finite-width journal bearings with micropolar lubricants. Huang and Weng [24] presented the study of the dynamic characteristics of finite-width journal bearings with micropolar lubricants. They observed that it is easier for the oil whirl to occur in a micropolar fluid under low Sommerfield number condition, although the micropolar fluid increases the effective viscosity and load capacity. Wang and Zhu [25] studied the lubricating effectiveness of micropolar lubricants in a dynamically loaded journal bearing. Das et al. [26] presented the dynamic characteristics of hydrodynamic journal bearings lubricated with micropolar lubricants. Their results showed that micropolar fluid exhibits better stability in comparison with Newtonian fluid. A numerical study of the non-Newtonian behavior for a finite journal bearing lubricated with micropolar fluids was carried out by Wang and Zhu [27]. They studied the influence of material characteristic length and the coupling number on the thermohydrodynamic performance of a journal bearing.

Nair et al. [28] considered the effect of deformation of the bearing liner on the static and dynamic performance characteristics of an elliptical journal bearing operating with micropolar lubricant. Rahmatabadi et al. [29] considered the noncircular bearing configurations for two-, three- and four-lobe bearings lubricated with micropolar lubricants.

A thorough scan of the existing literature reveals that the available studies in the area of conical hydrostatic and hybrid journal bearing system have been mainly restricted to Newtonian lubricant. Now a days the lubricants are additized in order to improve their lubricating performance. As a consequence of which, they exhibit nonlinear behavior and the analysis becomes quite involved. Therefore, the present work is planned to bridge the gap in literature and to analyze the conical hybrid journal bearing system operating with micropolar lubricant as shown in Figure 1(a). The numerically simulated results in present study are expected to be beneficial to bearing designers as well as to academic community.

(a)

(b)

(a)
(b)

Figure 1 

(a) Schematic of conical journal bearing system. (b) Developed surfaces of conical journal bearing system in cylindrical coordinates.

2. Analysis

The governing equations describing the behavior of micropolar lubricants as derived by Eringen [18] are expressed as

In a conical journal bearing system, clearance between the journal and bearing is very small and the ratio of the fluid film thickness to the mean radius of conical journal approaches to 10⁻³ to 10⁻⁴. Therefore a conical journal bearing system can be modeled as flow of lubricant between two curvilinear plates as shown in Figure 1(b). Thus in cylindrical coordinate system, the velocity, microrotation velocity, and pressure for a conical hybrid journal bearing system may be expressed as

In cylindrical coordinate system, for a steady incompressible flow (), (1) can be written as follows:

2.1. Assumptions

In general, the following assumptions are made in the analysis of a journal bearing system using micropolar theory.(i)Body force and body couples are negligible.(ii)The inertia forces and inertia couples are very small, and therefore neglected.(iii)All the characteristics coefficients are independent of .(iv)For steady incompressible fluids, .(v)The pressure variation in direction is negligible.(vi)Flow of lubricant in the clearance space between the journal and bearing is two dimensional.(vii)The velocity and microrotation velocity fields are independent of direction.

The velocity, microrotation velocity, and pressure, for the conical journal bearing system as shown in Figure 1 are given as follows:

Incorporating the above assumptions and order of the magnitude analysis, ((4)–(9)) reduces to the following forms:

To determine the expressions of velocity field for micropolar lubricant, the following boundary conditions are incorporated in (11):

2.2. The Lubricant Flow

The local lubricant flow per unit width in and direction can be obtained by integrating the respective velocity components across the local fluid film thickness, expressed as where is known as micropolar function. Consider the following: where is a dimensionless quantity, named as coupling number, and has the unit of length, named as characteristic length. These and are two main characteristics parameters of micropolar lubricant.

2.3. Modified Reynolds Equations

The continuity equation in cylindrical coordinate system for 2-dimensional flow is given as follows [30]:

Integrating (15) with respect to across the fluid film thickness, that is, , and using (13), yields

Substituting the value of and in (16), we have

Substituting the values of the parameters as , [, (see Figure 2(a)), we have

(a)

(b)

(a)
(b)

Figure 2 

(a) Development of the surface of conical journal bearing. (b) Load carrying capacity distribution of a conical journal bearing.

Using the following nondimensional parameter in (18), we get

Nondimensionalization of (18) yields where .

Equation (20) is the modified Reynolds equation for a conical journal bearing system operating with micropolar lubricant. For the values of micropolar parameters, , (20) changes into the form applicable for Newtonian lubricant as given in [14, 15].

2.4. Finite Element Formulation

Applying the finite element method using the orthogonality condition of Galerkin’s technique in (20), the following elemental equation (21) in matrix form is obtained: where where and are the direction cosines and , (number of nodes per element). represents the area domain and is the boundary domain of the th element.

The Equation (21) represents the finite element formulation of the bearing system. The Equation (21) has been derived by orthogonalizing the residue (From (20)) using Galerkin’s technique. The fluid film domain has been discretized using 4-noded isoparametric 2D element.

The fluid film pressure is approximated over an element as follows:

2.5. Fluid Film Thickness

For a conical journal bearing system, the fluid film thickness in nondimensional form is expressed as [14]

2.6. Restrictor Flow Equation

The continuity of flow between restrictor and bearing is required to be maintained for a compensated hydrostatic/hybrid journal bearing system. Therefore, the flow through the restrictor should be taken as the boundary constraint to solve the Reynolds equation. The flow rate of the lubricant is given as follows [14, 31]:

2.7. Boundary Conditions

The boundary conditions used for the analysis of lubricant flow field are described as follows.(i)The nodes lying on external boundary of bearing have zero relative pressure with respect to ambient pressure, that is, .(ii)Nodes lying on a recess have equal pressure.(iii)Flow of lubricant through the restrictor is equal to the bearing input flow.(iv)At the trailing edge of positive region, the Reynolds boundary condition is applied, that is, .

2.8. Journal Centre Equilibrium Position

Assuming the steady state condition , (21) is solved for pressure field for a specific value of journal center coordinate and the iterative procedure is continued until the equilibrium position of journal centre is achieved. For a specified vertical external radial load, the following iterative scheme is used to establish the journal centre equilibrium positions:

The fluid film reaction and are expanded using Taylor series for th journal centre position and correction in the journal centre displacements are obtained as [14, 31] where is the determinant of the matrix .

The new journal center position coordinates are expressed as where are the coordinates of th journal center position. Iterations are continued until the following convergence criterion (30) is satisfied [31, 32] as

2.9. Load Carrying Capacity

Figure 2(b) depicts the distribution of load carrying capacity of a conical journal bearing.

The load carrying capacity in radial direction is expressed as follows: The radial load carrying capacity is further resolved into and directions as follows:

2.10. Fluid Film Stiffness and Damping Coefficients

The fluid-film stiffness coefficients are defined as the negative derivative of the load capacity with respect to the journal center displacement. Mathematically, fluid film stiffness coefficients are defined as [14] where : direction of force, : components of the journal centre displacement. Here, represents the velocity component of journal center .

2.11. Stability Parameters

The stability of a journal bearing system is expressed in terms of threshold speed (), critical mass . The stability parameters are defined using the Routh’s stability criterion.

The nondimensional critical mass of the journal for lateral motion is expressed as where ; ; .

Threshold speed, that is, the speed of journal at the threshold of instability, can be obtained using the relation as follows: where is resultant fluid film force or reaction at .

3. Solution Procedure

The solution for the fluid flow field of four pockets conical hybrid journal bearing system operating with both Newtonian and micropolar fluid is obtained using an iterative scheme for vertical external load as explained in Section 2.8. The pressure field equation (21) and restrictor flow equation (26) are solved together with the boundary conditions given in Section 2.7. A flow chart to describe the overall solution scheme is shown in Figure 3.

Figure 3 

A flow chart to describe the overall solution scheme.

The unit INDAT reads the input data and the 2D mesh of the bearing system. Then unit FTHIK calculates the fluid film thickness at nodal points for the tentative values of the journal centre position. In unit FEQ, the fluidity matrix is generated and globalized as per the connectivity of the elements. The unit BCNS applies all the boundary conditions. In unit SLEQ, the modified Reynolds equation is solved for the fluid film nodal pressure using Gaussian elimination technique. The load carrying capacities are also computed. Utilizing the value of and , the correction in journal centre position is computed, hence the journal centre equilibrium position is computed in unit PREQ. This iterative process is continued until the convergence criterion described in (30) is satisfied. The other performance characteristics parameters are computed in unit POPRO.

4. Results and Discussion

The performance characteristics of four-pocket hybrid journal bearing system having different semicone angles for the values of micropolar parameters (coupling number and characteristic length) have been discussed. A source program has been developed in Fortran 77 using the analysis as described in earlier sections. In order to validate the methodology followed, the results of the present study have been compared with the results of Stout and Rowe [2]. Figure 4(a) depicts a favorable validation of the results of the present study with the already published results [2] with an error of 1–4%. As there are no results available for micropolar lubricated conical hydrostatic/hybrid journal bearing system, for the validation of the micropolar analysis, the results of micropolar effect in case of a circular hydrodynamic journal bearing have been compared with the already published results of Khonsari and Brewe [23] and Wang and Zhu [27]. It may be noticed from the Figures 4(b) and 4(c) the present results validate well with already published results [21, 27].

(a)

(b)

(c)

(a)
(b)
(c)

Figure 4 

(a) Radial load carrying capacity () versus eccentricity ratio . (b) Load carrying capacity () versus eccentricity ratio . (c) Load carrying capacity () versus eccentricity ratio .

The values of bearing geometric and operating parameters chosen for conical journal bearing system are shown in Table 1. The influence of micropolar lubricant on the bearing performance characteristics has been presented with respect to the external radial load as shown through Figures 5(a)–12(a). To depict the influence of different micropolar parameters on the performance of conical hybrid journal bearing, the results of performance characteristics parameters are also plotted with respect to characteristic length for different value of coupling number at the constant value of external radial load through Figures 5(b)–12(b).

(a)

(b)

(a)
(b)

Figure 5 

(a) Variation of attitude angle () versus radial load (). (b) Variation of attitude angle () versus characteristic length .

(a)

(b)

(a)
(b)

Figure 6 

(a) Variation of maximum fluid film pressure () versus radial load (). (b) Variation of maximum fluid film pressure () versus characteristic length ().

(a)

(b)

(a)
(b)

Figure 7 

(a) Variation of minimum fluid film thickness () versus radial load (). (b) Variation of minimum fluid film thickness () versus characteristic length ().

(a)

(b)

(a)
(b)

Figure 8 

(a) Variation of versus radial load . (b) Variation of versus characteristic length .

(a)

(b)

(c)

(d)

(a)
(b)
(c)
(d)

Figure 9 

(a) Variation of   versus radial load . (b) Variation of   versus characteristic length . (c) Variation of   versus radial load . (d) Variation of   characteristic length .

(a)

(b)

(c)

(d)

(a)
(b)
(c)
(d)

Figure 10 

(a) Variation of versus radial load . (b) Variation of characteristic length . (c) Variation of versus radial load . (d) Variation of characteristic length .

(a)

(b)

(a)
(b)

Figure 11 

(a) Variation of   versus radial load . (b) Variation of   versus characteristic length .

(a)

(b)

(a)
(b)

Figure 12 

(a) Variation of   versus radial load . (b) Variation of   characteristic length .

The performance characteristics parameters of hybrid conical journal bearing for Newtonian and micropolar lubricant at the constant value of external radial load , computed in the present study are tabulated in Table 2. In the present study, performance characteristics parameters of a conical journal bearing system configuration having semicone angles 10°, 20°, 30°, and 40° have been presented for Newtonian lubricant and micropolar lubricant in the following sections.

4.1. Influence on Attitude Angle

For a conical hybrid journal bearing system, the value of attitude angle increases with an increase in the value of external radial load for both Newtonian and micropolar lubricant as depicted in Figure 5(a); however, the use of micropolar lubricant results in higher value of the attitude angle than Newtonian lubricant. Figure 5(b) depicts the variation of attitude angle with respect to characteristic length at the radial load ) for different value of micropolar parameters. For the value of characteristic length , the bearing system attains the larger value of attitude angle for each bearing configuration. Moreover, the value of attitude angle decreases rapidly up to , beyond that attitude angle is almost constant. Further, it may be observed that an increase in the value of characteristic length results in a decrease in the value of the attitude angle . Another notable observation from Figures 5(a) and 5(b) is that higher semicone angle results in higher value of attitude angle . The bearing configuration having semicone angle () results in the higher value of attitude angle than other bearing configurations. Figure 5(b) also depicts that for a constant values of characteristic length, (), and radial load , an increase in the value of coupling number parameter () results in an increase in the value of attitude angle .

4.2. Influence on Maximum Fluid Film Pressure ()

It may be noticed from Figure 6(a) that an increase in the value of external radial load results in an increase in the value of maximum fluid film pressure for either Newtonian or micropolar lubricant. The notable observation is that the values of are higher for micropolar lubricant vis- á-vis Newtonian lubricant for a specified value of radial load. Further, it may also be noticed that for bearing configurations having semicone angle (), the value of increases more rapidly with an increase in the value of radial load than other bearing configurations for either of the Newtonian and micropolar lubricants. However, for a specified radial load, the value of maximum fluid film pressure is higher for the bearing configuration having semicone angle () than other bearing configurations for either of the lubricants. For bearing configuration having semicone angle (), at a specified value of radial load, , the micropolar lubricant with characteristics and results an increase of 14.769% in the value of vis- á-vis Newtonian lubricant. The variation of maximum fluid film pressure with respect to characteristic length, () of micropolar lubricant has been shown in Figure 6(b). It may be observed that for different values of coupling parameters for various configurations of semicone angle, the value of decreases rapidly up to the value of characteristic length , beyond that the value of is less sensitive. It may also be noticed that for a constant values of characteristic length, (), and radial load , an increase in the value of coupling number parameter () results in an increase in the value of . For the value of characteristic length, (), and a coupling parameter , the value of is higher by a factor of 21.64%, 21.023%, 19.19%, and 16.055% vis- á-vis Newtonian lubricant for semicone angles 10°, 20°, 30°, and 40° sequentially.

4.3. Influence on Minimum Fluid Film Thickness ()

It may be seen in Figure 7(a), as the value of external radial load increases, the value of minimum fluid film thickness reduces for either Newtonian or micropolar lubricant. It may be noticed that for bearing configuration having semicone angle (), the value of decreases more rapidly with an increase in the value of radial load than other bearing configurations for either of Newtonian or micropolar lubricants; however, for a specified radial load, the value of    is lower for the bearing configuration having semicone angle ( = 40°) than other bearing configurations for either of Newtonian or micropolar lubricants. The value of decreases with an increase in the value of radial load due to increase in the value of maximum fluid film pressure. For a specified value of radial load, , for semicone angle = 10°, the value of of a Newtonian lubricated bearing is higher by a factor of 1.025, 1.09 and 1.209 for semicone angle 20°, 30°, and 40° sequentially. The variation of with respect to characteristic length, () at a specified value of radial load, , has been shown in Figure 7(b). The maximum increase in the value of reported at the value of characteristic length, () for different values of coupling number parameter for different semicone angle vis- á-vis Newtonian lubricant.