performance analysis of flexible multirecess hydrostatic journal bearing operating with micropolar...

LUBRICATION SCIENCELubrication Science (2012)Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/ls.1181

Performance analysis of flexible multirecess hydrostatic journalbearing operating with micropolar lubricant

Suresh Verma1,*,†, Vijay Kumar2 and K. D. Gupta1

1Department of Mechanical Engineering, D.C.R. University of Science and Technology, Murthal 131039, India2Gyan Jyoti Group of Institutions, Shambu Kalan, Punjab, India

ABSTRACT

This paper presents the analytical study of the effect of the bearing shell flexibility on the performance ofmultirecess hydrostatic journal bearing system operating with micropolar lubricant. The modified Reynoldsequation for the flow of micropolar lubricant through constant flow valve-compensated hydrostatic journalbearing has been solved by finite element technique based on Galerkins method, and the resulting elasticdeformation in the bearing shell due to fluid-film pressure has been determined iteratively, in which thedeformation coefficient accounts for the bearing shell flexibility. The computed results suggest that theinfluence of the micropolar effect on bearing performance characteristics is significantly affected by thebearing shell flexibility. Copyright © 2012 John Wiley & Sons, Ltd.

Received 26 May 2011; Revised 19 January 2012; Accepted 10 February 2012

KEYWORDS: constant flow valve; elastic deformation; finite element method; flexible hydrostatic journal bearing;micropolar lubricant; multirecess bearing; restrictor

INTRODUCTION

The multirecess hydrostatic journal bearings compensated through constant flow valve are used inseveral machines for the reason that they provide a smooth relative motion, low friction and wear evenat low speed, large fluid-film thickness, high stiffness and high damping. It has been realised that thehydrostatic journal bearing system undergoes elastic deformation when operating under heavy loads.The bearing deformations are generally of the order of the magnitude of fluid-film thickness; thus,the fluid-film profile is modified, and the performance of a bearing system is changed. Therefore,the studies carried out with rigid bush assumptions may not be appropriate for an accurate predictionof the performance of the bearing system.For Newtonian lubricant, Sinhasan et al. analytically studied the effect of bearing shell elasticity in

hydrostatic journal bearing using capillary,1 constant flow valve 2 and orifice flow restrictors.3 Theypresented that with Newtonian lubricant for heavily loaded journal bearings, the deformation due to

*Correspondence to: Suresh Verma, Department of Mechanical Engineering, D.C.R. University of Science andTechnology, Murthal 131039, India.†E-mail: [email protected]

Copyright © 2012 John Wiley & Sons, Ltd.

S. VERMA, V. KUMAR AND K. D. GUPTA

elasticity of bearing shell is comparable in magnitude with fluid-film thickness. This deformation altersthe lubricant-film profile and consequently the behaviour of a journal bearing system. They concludedthat the static and dynamic characteristics of bearing for a given load decrease as bearing flexibilityincrease. Their study suggested that to establish an optimum design of compensated hydrostaticjournal bearing to support a particular external load, a judicious selection of restrictor design parameter,bearing shell deformation coefficient and bearing geometry is essential. In their extended work,Sharma et al.4 have studied the performance characteristics of flexible hydrostatic/hybrid multirecessjournal bearing system operating with Newtonian lubricant and using membrane type variable-flowrestrictor as compensating element. It has been concluded that a careful selection of flexibility ofbearing shell is required to obtain the improved stability margin of the hydrostatic journal bearingsystem.Furthermore, while designing the hydrostatic journal bearing, generally the assumption is that the

lubricant behaves as a Newtonian fluid. The development in the field of lubrication in the recent yearshas increased the quest of the lubricating effectiveness of non-Newtonian fluids. Nowadays, most of themodern lubricants in practice use polymeric additives to enhance their performance. The behaviour ofpolymer-added lubricant is no longer Newtonian. In view of the inadequacies of classical Newtoniantheory, lubrication theory for micropolar fluids is applied to solve the lubrication problems of suchfluids. Micropolar fluids are fluids with microstructure. They represent fluids consisting of rigid,randomly oriented particles suspended in a viscous medium, where the deformation of fluid particlesis ignored.As reported by Khader and Vachon,5 in most of the practical usages, the lubricants, which are

mainly the polymer-thickened oils or lubricants blended with additives and are mostly contaminatedwith suspended metal particles or dirt, exhibit a non-Newtonian behaviour. As these contaminated sub-structures can translate, rotate or even deform independently, the classical Newtonian theory becomeslimited to predict the accurate flow behaviour; thus, the microcontinuum theories have been developed.The experimental results given by Scott and Suntiwattana6 support the achievement of better lubricat-ing effectiveness on blending a small amount of long-chained additives with the Newtonian lubricants.For the study of such long-chained polymer solution like polyisobutylene, the microcontinuum theoryis better suited.A number of studies7–14 on micropolar lubrication have been reported. The practical applications of

micropolar model can be found in the cases of contamination with metal particles or dirt, as given byAllen and Kline,7 and in the design of journal bearings in the area of a nuclear power plant where theheat transfer agent sodium is used as a lubricant, as suggested by Balaram.8 The study of the flow be-haviour using the theory of micropolar lubrication was initiated with the problem of a two-dimensionalslider bearing by Allen and Kline.7 For such micropolar lubrication, the nondimensional materiallength was found to have considerable influence on the lubricating properties. The steady-state analysisusing micropolar lubricants for infinitely long journal bearing studied by Prakash and Sinha 9 revealedthat such fluids increase the effective viscosity, especially in thin films, which supported the experi-mental evidence also. The squeeze film flow effects in micropolar lubrication were also studied byPrakash and Sinha10 for the journal bearings under a cyclic sinusoidal load with no journal rotation.Later on, Singh et al.11 presented the three-dimensional Reynolds equation using micropolar lubrica-tion theory.The enhancement in the bearing performance under micropolar lubrication was observed by Isa

et al.12 in the analysis of the characteristics of squeeze film two-dimensional porous journal bearings.The similar enhancement in the performance in terms of higher load capacity, higher frictional moment

Copyright © 2012 John Wiley & Sons, Ltd. Lubrication Science (2012)DOI: 10.1002/ls

FLEXIBLE MULTIRECESS HSJB WITH MICROPOLAR LUBRICANT

and lower frictional coefficient was observed in micropolar lubrication compared with those observed inthe Newtonian lubrication with the same viscosity for a short bearing by Tipei.13 The similar improve-ments were also observed for finite journal bearings and attributed to the characteristic length andcoupling number of micropolar fluid by Hung et al.14,15 and Khonsari et al.16 Further, Das et al.17 havepresented the dynamic characteristics of hydrodynamic journal bearings lubricated with micropolarfluids. They obtained the dynamic characteristics in terms of the components of stiffness and dampingcoefficients, critical mass parameter and whirl with respect to the micropolar property for varyingeccentricity ratios. They concluded that the micropolar fluid exhibits better stability in comparison withNewtonian fluid. In their extended work, Das et al.18 presented the performance of misaligned hydro-dynamic journal bearings lubricated with micropolar fluids. Wang et al.19 have studied the lubricatingeffectiveness of micropolar fluids in a dynamically loaded hydrodynamic journal bearing. A numericalstudy of the non-Newtonian behaviour for a finite journal bearing lubricated with micropolar fluids hasbeen undertaken by Wang et al.20 considering both thermal and cavitation effects. They derived themodified Reynolds equation and energy equation on the basis of Eringen’s micropolar fluid theoryand investigated the effects of the size of material characteristic length and the coupling number onthe thermohydrodynamic performance of a journal bearing.The micropolar theory of lubrication applied to hydrodynamic bearings has been categorically

reviewed by authors in Verma et al.21 In this, the influence of micropolar parameters, i.e. characteris-tics length and the coupling number, on the performance of four-pocket hydrostatic journal bearingcompensated through constant flow valve has been presented. But in that analysis, the bearing shellflexibility has not been taken into account. A study of hole-entry hybrid journal bearing systemcapillary compensated and operating with micropolar lubricant is presented in Verma et al.,22 and ithas been concluded that there exists an optimum value of restrictor design parameter correspondingto micropolar lubricant at which the stiffness coefficient and stability parameters are maximum.Rahmatabadi et al.23 investigated the noncircular bearing configurations with three-lobe and four-lobebearings lubricated with micropolar fluids and showed that micropolar lubricants can produce signifi-cant enhancement in the static performance characteristics. Further, the trend of effects also depends onthe bearing configurations. Nicodemus et al.24 studied the influence of wear on the performance ofcapillary-compensated four-pocket hydrostatic journal bearing operating with micropolar lubricantfor two different loading arrangements. The study revealed that the influence of wear on bearing staticperformance characteristics is significantly higher for a bearing lubricated with micropolar as com-pared with bearing lubricated with Newtonian lubricant. It was also reported that the influence of wearon bearing dynamic performance characteristics for a bearing operating with micropolar lubricantsignificantly depends on type of loading arrangement. Recently, Nicodemus et al.25 presented theanalytical study of four-pocket orifice-compensated hydrostatic/hybrid journal bearing system ofvarious geometric shapes of recess operating with micropolar lubricant. They concluded that theinfluence of micropolar effect of lubricant on bearing performance is predominantly affected by thegeometric shape of recess and restrictor design parameter. Very recently, Verma et al.26 have theoret-ically studied the performance of capillary-compensated multirecessed hydrostatic journal bearingsoperating with micropolar lubricant, and the performance has been compared with Newtonian lubricant.To the best knowledge of authors so far, no investigation is yet available in literature that considers theinfluence of bearing shell flexibility on the performance of hydrostatic journal bearing operating withmicropolar lubricant; therefore, this issue has been addressed in this paper. The present work dealswith the simultaneous solution of the modified Reynolds equation and elasticity equations to studythe performance of four-pocket flexible hydrostatic journal bearing compensated with constant flow



valve restrictor and operating with micropolar lubricant. The results presented in the study areexpected to be quite useful to the bearing designers.

ANALYSIS

Figure 1 shows geometric configuration of a four-pocket hydrostatic journal bearing compensatedthrough constant flow valve. In a hydrostatic bearing design, it must be possible to support the fulloperating load at zero speed as well as at high speed. It is therefore necessary to present load data forthe worst condition, which is when the bearing is at zero speed.

Modified Reynolds Equation for Micropolar Lubricant Flow Domain

The flow of incompressible micropolar lubricant in the convergent area of the journal bearing isgoverned by the modified Reynolds equation. With the usual assumptions of the lubrication film,the modified Reynolds equation is given as11

@

@x

h3Φ12m

@p

@x

� �þ @

@y

h3Φ12m

@p

@y

� �¼ oJRJ

2@h

@xþ @h

@t(1)

Figure 1. (a) Four-pocket hydrostatic bearing coordinate system; (b) bearing geometry.



where

Φ ¼ 1þ 12l2

h2� 6Nl

hcoth

Nh

2l

� �; N ¼ k

2mþ k

� �1=2

; l ¼ g4m

� �1=2

Here, m is the viscosity coefficient of the Newtonian fluid, k is the spin viscosity, g is the materialcoefficient, h is the film thickness and p is the micropolar film pressure. N and l are two parametersdistinguishing a micropolar lubricant from Newtonian lubricant. N is a dimensionless parameter calledthe coupling number that couples the linear and angular momentum equations arising from the microrota-tional effects of the suspended particles in the lubricant. l represents the interaction between themicropolarlubricant and the film gap and is termed as the characteristic length of the micropolar lubricant.Equation (1) in its nondimensional form can be given as

@

@a

�h3Φ12�m

@�p

@a

( )þ @

@b

�h3Φ12�m

@�p

@b

( )¼ Ω

2@�h

@aþ @�h

@�t(2)

With the use of the finite element method based on Galerkins’ technique and Equation (2), thesystem equation for the discretised flow field is derived, and in matrix form, it is given as 21

�F½ � �pf g ¼ �Qf g þ Ω �RHf g þ �_X j �Rxj

� �þ �_Z j �Rzj

� �(3)

A dot over terms represents first derivative of the respective terms with respect to time. Each term ofrespective matrix/vector is computed using the following expressions:

�Feij ¼∬�A

e

�h3

12�mΦ@Ni

@a@Nj

@aþΦ

@Ni

@b@Nj

@b

� �( )da db (3a)

�Qe ¼ZΓe

�h3

12�mΦ@�p

@a

� ��

�Ω2�h

!l1 þ

�h3

12�mΦ@�p

@b

� �l2

( )Ni d�Γ

e (3b)

�ReHi ¼∬�Ae

�h

2@Ni

@adadb (3c)

�Rexji ¼∬�A

ecosaNi dadb (3d)

�Rezji ¼∬�A

esina Ni dadb (3e)

where l1 and l2 are direction cosines and i,j= 1,2,. . .. nel (number of nodes per element) are the localnode numbers. Γ is the boundary of the eth element.



Fluid-film Thickness

The journal bearing is required to maintain an appropriate minimum fluid-film thickness to minimisethe chances of metal to metal contact under the operating load. For a rigid journal bearing system,the fluid-film thickness expression is given as

�h ¼ �ho þ Δ�h

whereΔ�h is the perturbation due to dynamic condition on the fluid-film thickness and �ho is the fluid-filmthickness when the journal centre is at the static equilibrium position and is given as

�ho ¼ 1� �Xj cosa� �Zj sina (4)

where �Xj and �Zj are equilibrium coordinates of the journal centre.Now, for a flexible bearing, the fluid-film thickness becomes modified due to elastic deformation,

and the modified film thickness is given as

�h ¼ �ho þ Δ�hþ �dr (5)

where �dr represents nondimensional radial elastic deformation due to the fluid-film pressure.

Restrictor Flow Equation

For a compensated journal bearing system, the continuity of flow between restrictor and bearing isrequired to be maintained. The flow through the restrictor is therefore taken as constraint in the solu-tion domain. The constant flow valve restrictor should be able to supply a fixed quantity of lubricantthrough it; hence, the flow �QR of lubricant through it is expressed as

�QR ¼ constant ¼ �Qc (6)

Here, �QR and �Qc represents the restrictor flow and pocket flow, respectively.

Fluid-film stiffness and damping coefficients

The fluid-film stiffness coefficients are defined as

�Sij ¼ � @�Fi

@�qji ¼ x; zð Þ (7)

where i represents the direction of force; �qj is direction of journal centre displacement �qj ¼ �Xj; �Zj

� �.



Stiffness coefficient in matrix form will be

�Sxx �Sxz�Szx �Szz

� ¼ �

@�Fx

@�Xj

@�Fx

@�Zj

@�Fz

@�Xj

@�Fz

@�Zj

2664

3775 (8)

For the computation of stiffness coefficients �Sij i; j ¼ �Xj; �Zj

� � of a journal bearing system, the nodal

pressure derivatives at steady-state conditions are to be calculated by differentiating system equation(Equation (3)) with respect to journal displacement �Xj;�Zj

� . The elements of the RHS matrices in the

differentiation of system equation (Equation (3)) are computed, and the values of pressure derivatives@�p0=@�Xj; @�p0=@�Zj

� can be obtained. With the use of the values of pressure derivatives, the components

of the RHS matrix of Equation (8) can be computed.The fluid-film damping coefficients are defined as

�Cij ¼ � @�Fi

@�_qj

; i ¼ x; zð Þ (9)

where �_qj represents the velocity component of journal centre �_qj ¼ �_X j;�_Z j

� �.

Damping coefficients in matrix form is

�Cxx �Cxz�Czx �Czz

� ¼ �

@�Fx

@�_X j

@�Fx

@�_Z j

@�Fz

@�_X j

@�Fz

@�_Z j

26664

37775 (10)

For the computation of damping coefficients �Cij i; j ¼ �_X j;�_Z j

� �, the nodal pressure derivatives

@�p0=@�_X j; @�p0=@

�_Z j

�are required. These may be obtained by differentiating the global system equation

(Equation (3)) with respect to �_qj ¼ �_X j;�_Z j

�.

Elastic continuum

In general, bearing shell or bush is considered to be a cylindrical structure of finite length enclosed in arigid housing. With the use of the linear elasticity equation, virtual work principle and finite elementformulation, the system equation governing deformation in an elastic continuum is derived. At a pointin elastic continuum, the displacements in the circumferential (dx), axial (dy) and radial (dr) directionsare defined. The radial component at fluid-film and shell interference is needed for the computation offluid-film thickness. Generally, in practical conditions, the rigidity of the journal is more as comparedwith that of shell; hence, deformation in the journal due to fluid-film pressure has been neglected in thepresent study.



By using the nondimensional scheme given as

a ¼ x=RJð Þ; b ¼ y=RJð Þ; �r ¼ r=RJð Þ; �D½ � ¼ D½ �=Ebð Þ; �d� � ¼ df g=cð Þ;

the discretised elastic continuum system equation is as follows:1

�K½ � �d� � ¼ �Cd �FΓf g (11)

where �K½ � is the system stiffness matrix, �d� �

is the system nodal displacement vector, �FΓf g is the sys-tem traction force vector and �Cd is the elastic deformation coefficient (= (psth)/(Ebc))

.

The examples of some bearing material specimens and their elastic deformation coefficientvalues are as follows:Babbit (tin base): �Cd= 0.017025, Textolite: �Cd= 0.140650, Nylon (polyamide):�Cd = 0.398228, Teflon (polytetrafluoroethylene): �Cd = 2.156494.The global system equations from the governing Equations (3), (4), (6) and (11) are obtained by

employing Galerkins orthogonality criterion and then solved after applying appropriate boundaryconditions. The entire lubricant flow field is discretised using four-noded quadrilateral isoparametricelements. The two-dimensional grid is used for the solution of modified Reynolds’ equation alongthe two directions (i.e. circumferential and axial). The displacement field in elastic continuum isdiscretised using eight-noded isoparametric hexahedral elements.

BOUNDARY CONDITIONS

The relevant boundary conditions are

• Nodes situated on the external boundary of the bearing have zero pressure �p b¼�l�� = 0.0.

• All the nodes situated on a pocket have equal pressure.• Flow of lubricant through the restrictor (�QR) is equal to the bearing input flow.• At the trailing edge of the positive region �p ¼ @�p=@að Þ ¼ 0.• The displacement of the nodes on shell-housing interface is zero (�d ¼ 0).

SOLUTION SCHEME

The modified Reynolds equation governing the flow of micropolar lubricant in the clearance space of afour-pocket hydrostatic journal bearing system has been solved by using finite element method togetherwith required boundary conditions. The solution of a constant flow valve compensated hydrostaticjournal bearing system problem needs iterative solution scheme for solving Equation (3). Under

steady-state condition �_Xj;�_Z j ¼ 0

�, assuming the rigid bearing shell �Cd ¼ 0ð Þ, the lubricant flow

field system equation (Equation (3)) is solved for a specified journal centre position �Xj; �Zj

� , after

adjustment for flow through constant flow valve restrictor Equation (6), and modified for theboundary conditions. But if the solution is to be obtained for a specified vertical load, oneadditional iterative loop is needed to establish the equilibrium journal centre position using thefollowing equations:



�Fx ¼ 0 and �Fz � �W0 ¼ 0

Under a given bearing geometric parameters and for a given external vertical load, journal centreposition (�Xj; �Zj) is unique. For a given external load, tentative values of journal centre coordinatesare fed as input. The corrections (Δ�Xj;Δ�Zj) on the assumed journal centre coordinates (�Xj; �Zj) arecomputed using the following algorithm.The fluid-film reaction components �Fx; �Fz are expressed by Taylor’s series about ith journal centre pos-

ition. Assuming that the alteration in the journal centre position are quite small and retaining terms only upto first order in the Taylor’s series expansion, the corrections (Δ�Xj;Δ�Zj) on the coordinates are obtained as

Δ�Xjji¼ � 1

Dj½ @�Fz

@�Zjji� @�Fx

@�Zjji� �Fx ji

�Fz ji � �W0

( )(12a)

Δ�Zjji¼ � 1

Dj½� @�Fz

@�Xjji

@�Fx

@�Xjji� �Fx ji

�Fz ji� �W0

( )(12b)

where

Dj ¼ ½ @�Fx

@�Xjji

@�Fz

@�Zjji� @�Fx

@�Zjji

@�Fz

@�Xjji� (13)

The new journal centre position coordinates ½�Xjjiþ1;�Zjjiþ1� are expressed as

�Xj iþ1 ¼ �Xj i þ Δ�Xj ij�� (14a)

�Zj iþ1 ¼ �Zj i þ Δ�Zj ij�� (14b)

Iterations are continued until the following convergence criterion is satisfied:

Δ�Xij

�2þ Δ�Zi

j

�2� �1=2

�Xij

�2þ �Zi

j

�2� �1=2

��

��< 0:001 (15)

When the journal centre equilibrium position is established, the nodal displacements �d�

in the elasticdomain (bearing shell) are computed using the pressure developed in the fluid film, the system equation(Equation (11)) and boundary conditions. The fluid-film thickness �hð Þ is modified using Equation (5) andthe radial displacement component �dr

� of the nodes on the fluid film-elastic domain interface. With the

use of the modified film thickness, the flow field system equation (Equation (3)) is again solved for thesteady-state case, and new nodal pressures and flows are obtained. With the use of these nodal pressures,



nodal displacements �d�

in the elastic domain are again computed using the system equation (Equation(11)). Iterations are continued until the differences in the nodal pressures of successive iterations do notcome within the specified tolerance limit of 0.1%. The flow chart of the iteration scheme is shown inFigure 2. The matched steady-state solutions for nodal pressures obtained for the modified film geometryare used to determine the static and dynamic performance characteristics of the bearing system.

RESULTS AND DISCUSSION

The validity of the computer program developed is established by computing the load at different ec-centricity ratios for rigid hydrodynamic bearing operating with Newtonian and micropolar lubricants.The results obtained from present work have been compared with the available theoretical results ofWang et al.20 and found to be quite close as shown in Figure 3(a). The maximum deviations of about8% and 7% are noted for Newtonian and micropolar lubricant, respectively, at maximum eccentricityratio of 0.8. The difference in the analytical solutions may be attributed to the different computationalscheme used. Also, the deviation in the results can be attributed to the fact that the present study dealswith isothermal hydrodynamic condition, and the results of Wang et al.20 were presented consideringthermal effects. Further, Figure 3(b) shows the comparison of the present results for a constant flowvalve-compensated four-pocket flexible hydrostatic journal bearing system operating with Newtonianlubricant, for minimum fluid-film thickness (�hmin) with restrictor flow (�Qc) at different values of thedeformation coefficient (�Cd), with existing results of Sinhasan et al.2 They compare very well.Here, the results have been presented for constant flow valve-compensated four-pocket flexible

hydrostatic journal bearing system having aspect ratio l = 1, bearing shell thickness ratio �th = 0.1,Poison’s ratio n = 0.3, flow �Qc= 0.5, for various values of deformation coefficient �Cd and operating withmicropolar lubricant. Since for hydrostatic bearing design it must be possible to support the fulloperating load at zero speed, the results are therefore presented for zero speed only. The followingnondimensional values of bearing operating and geometric parameters are used:

Ω ¼ 0:0; l ¼ 1:0; �ab ¼ 0:14; �Wo ¼ 0:5; �Qc ¼ 0:5; θ ¼ 18�

The direction of the load is along z-axis as shown in Figure 1b. Two nondimensional parameters,i.e. N2 and lm, are used to distinguish a micropolar lubricant from a Newtonian lubricant. As N2

approaches zero and/or lm approaches infinity, the micropolar effect becomes insignificant and themicropolar lubricant behaves like Newtonian one. In other words, the micropolar effect of lubricantis said to increase with decrease in the value of parameter lm and/or increase in the value of parameterN2. The physical interpretation of this can be given as follows: higher value of N2 implies highmicrorotation, and small value of lm implies big suspended particle size in the micropolar lubricant.As per definition, deformation coefficient �Cd indicates the degree of bearing flexibility. The higherthe value of �Cd , the higher is the degree of bearing flexibility. The performance of journal bearingsystem changes with alteration in fluid-film pressure profile around the bearing. The variation of thefluid-film pressure in circumferential and longitudinal directions is presented in the following sectionfor the deformation coefficient �Cd = 0.0, 0.5 and various values of micropolar parameters (lm and N2).Figures 4 and 5 depict the contrasting difference as compared with rigid bearing (�Cd= 0.0) caused by

elastic effect ( �Cd = 0.5) on the fluid-film pressure distribution in all the four pockets along


Diagrams:

Problem index and other input data

dC =0.0

Bearing Lubrication Data

Compute bearing Characteristics

IECH=1

IT=IT+1

Compute nodal displacement ( )

Output results

1

YES

YES

IL= 0 Bearing Elastic data

IL=0

NO

dC = dC + incdC

INCV=0; IT=0

dC > limdC

IECH=0

Compute nodal pressure

Compute journal centre equilibrium position

INCV=1

NO

IL= 0

IT>IT max

p < pIL=1

STOP

Modify fluid film thickness ( h )

INCV=1

YES

NO

YES

NO

NOYES

STOP

NO

YES

YES

YES

NO

NO

NO

YES

1

IL = 0 NHS or MHS Problem= 1 EHS or MEHS Problem

IECH = 0 Journal centre equilibrium not achieved

= 1 Journal centre equilibrium achieved

INCV = 0 Convergence criteria notachieved

= 1 Convergence criteria achieved

incdC = Increment in dC

limdC = Limiting value of dC

tolp = Pre-assigned tolerance on

pressure

difp = 1

1

mi

mi

mi

p

pp

mip = Pressure at the thi node in

the fluid for thm iteration

dif tol

Figure 2. Iterative scheme.



(a)

(b)

0.675

0.7

0.725

0.75

0.775

0.8

0.825

0.85

0.875

0.9

0.925

0.2 0.3 0.4 0.5 0.6 0.7

resultsPresent

CFV:Restrictor

0.0dC

1.0dC

5.0dC

minh

cQ

From Sinhasan et al (1989)

0123456789

10

0 0.2 0.4 0.6 0.8 1

---o--- Ref.-20 ( N2 = 0.2,lm = 20 )------ Ref.-20 ( Newtonian )

Present ( Newtonian )Present ( N2 = 0.2,lm = 20 )

W (

kN)

Figure 3. (a) Variation of load-carrying capacity (W) with eccentricity ratio (e) for rigid hydrodynamicjournal bearing. (b) Comparison of present result of flexible four-pocket bearing with Newtonian lubricant

and CFV restrictor.


circumferential and longitudinal directions, respectively, for pure hydrostatic mode of operation(Ω= 0.0). As a usual and obvious trend, the maximum pressure is at pocket that is just below thejournal centreline (in this case, pocket number 4) in the direction of external load. It can be seen thatthe fluid-film pressure in all four pockets of bearing decreases with increase in bearing flexibility forboth Newtonian as well as the micropolar lubricants.As compared with rigid bearing (�Cd= 0.0), the decrease in the value of maximum pressure (�pmax) due

to elasticity effect (�Cd = 0.5) is about 19% in case of the bearing operating with Newtonian lubricant,and this decrease in �pmax is 27% due to elasticity effect (�Cd = 0.5) in the case of bearing operating withmicropolar lubricant (N2 = 0.5, lm= 10). It indicates that the decrease in �pmax caused by bearing elasticeffect is more in case of micropolar lubricant as compared with the Newtonian lubricant.The variations of bearing performance characteristics against different values of deformation coef-

ficient (�Cd) are shown in Figures 6–11. The variation in the maximum pressure (�pmax) with deformation


0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 30 60 90 120 150 180 210 240 270 300 330 360

p

Figure 4. Circumferential pressure profile of rigid and flexible hydrostatic journal bearing.

Figure 5. Longitudinal pressure profile of hydrostatic journal bearing.


coefficient (�Cd) at different values of micropolar parameters (lm and N2) is shown in Figure 6. It may beobserved that �pmax decreases with increase in �Cd for both Newtonian and micropolar lubricants. Withmicropolar lubricant, a maximum of about 55% increase in �pmax with flexible bearing having �Cd = 0.1and 34% increase in �pmax with flexible bearing (�Cd = 0.5) have been observed corresponding to lm= 10,N2 = 0.9 as compared with Newtonian lubricant. Hence, it can be concluded that the micropolar effectof lubricant on �pmax is less pronounced at higher values of deformation coefficient (�Cd).


Figure 6. Variation of �pmax with �Cd.

Figure 7. Variation of �hmin with �Cd.

Figure 8. Variation of �Sxx with �Cd.



Figure 9. Variation of �Szz with �Cd.

Figure 10. Variation of �Cxx with �Cd.

Figure 11. Variation of �Czz with �Cd.



TableI.

Perform

ance

characteristicsof

aconstant

flow

valvecompensated

four-pocketrigid/flexiblehydrostatic

journalb

earing

forNew

tonian

andmicropolarlubricant.

� Cdl m

N2

Perform

ance

characteristics

p max

h min

S xx

S zz

Cxx

Czz

MN

m�2

�10

�3m

�10

3MN

m�1

�10

3MN

m�1

MNsm

�1

MNsm

�1

New

tonian

5.816644

0.044545

1.689098

1.928850

3.572506

3.880401

200.5

6.866003

0.045702

2.225007

2.430471

4.403808

4.649336

0.0

200.9

7.426962

0.046220

2.540731

2.735892

4.842367

5.065191

100.5

7.883483

0.046438

2.702109

2.871992

5.198804

5.404294

New

tonian

4.670812

0.040921

1.116725

1.304760

2.611632

2.851365

200.5

5.200948

0.042083

1.359885

1.487195

3.019595

3.158339

0.5

200.9

5.455986

0.042560

1.487431

1.585831

3.211416

3.301769

100.5

5.688462

0.042779

1.558920

1.627909

3.388049

3.440053

100.9

6.289759

0.043818

1.886388

1.864590

3.839357

3.746328

a b/L=0.14;R

J=0.05

m;c

=0.0502

�10�

1m;L

=0.1m;t

h=5�10

�3m;l

=1.0;

n=0.3;

m=0.0345

N�s�m

�2at38

� C;W

o=11.2kN

;� W0=0.5;

� Qc

=0.5;

Qc=16.42745�1

0�1m

3s�

1




The plot of minimum fluid-film thickness �hmin with �Cd is shown in Figure 7. It indicates that �hmin

decreases with increase in deformation coefficient (�Cd) for Newtonian as well as the micropolar lubri-cants. Further, as compared with Newtonian lubricant, a maximum increase of 6.5% in �hmin for rigidbearing (�Cd = 0.0) and 7% for flexible bearing (�Cd = 0.5) has been noticed corresponding to micropolarlubricant with lm= 10, N2 = 0.9. Therefore, for a given load, the flexible bearing shall operate at highereccentricity compared with rigid bearing for both Newtonian as well as micropolar lubricants.However, for a given load and lubricant flow, a specified value of �hmin can be maintained by selectingthe appropriate lubricant viscosity or radial clearance and so on.Figures 8 and 9 show the variation in stiffness coefficients (�Sxx; �Szz) with deformation coefficient (�Cd).

Figure 8 indicates that the direct stiffness coefficient �Sxx decreases for both Newtonian and micropolarlubricant with increase in bearing flexibility ( �Cd ). However, at a fixed value of �Cd , the value of �Sxxincreases with increase in micropolar effect of lubricant. A maximum of 124% increase in �Sxx has beennoted for rigid bearing (�Cd = 0.0), and correspondingly, about 68% increase has been noted for flexiblebearing (�Cd = 0.5) with micropolar lubricant (lm= 10, N2 = 0.9) as compared with Newtonian lubricant.The direct stiffness coefficient in the direction of load, i.e.�Szz, also decreases with increase in bearing

flexibility as shown in Figure 9 for micropolar as well as the Newtonian lubricants. But it increaseswith increase in micropolar effect of lubricant irrespective of the value of deformation coefficient�Cd. As compared with Newtonian lubricant, a maximum increase of about 104% in �Szz for rigid bearing(�Cd = 0.0) and 42% for flexible bearing (�Cd = 0.5) has been observed for micropolar lubricant (lm = 10,N2 = 0.9). Hence, it is concluded that the increasing trend of �Szz with increase in micropolar effect oflubricant is more pronounced for rigid bearing as compared with the flexible bearing.Under dynamic conditions, from vibration point of view, it is important to have suitable values of the

damping coefficients �Cxx and �Czz to minimise the oscillations. The variations of direct damping coefficients(�Cxx,�Czz) with �Cd are plotted in Figures 10 and 11, respectively, which show the decreasing trend for bothNewtonian and micropolar lubricants. However, for a fixed values of �Cd, the direct damping coeffi-cients (�Cxx,�Czz) are found to increase with increase in micropolar effect of lubricant. A maximum of 84%increase in �Cxx for rigid bearing (�Cd =0.0) as compared with Newtonian lubricant has been observed forlm=10, N

2 = 0.9 and correspondingly a maximum of 47% increase for flexible bearing (�Cd =0.5).Similarly, a maximum increase of 73% in �Czz for rigid bearing (�Cd =0.0) and about 31% for flexible

bearing (�Cd =0.5) has been observed corresponding to lm=10, N2 = 0.9 as compared with Newtonian lu-

bricant. Hence, it can be concluded that the increasing trends of damping coefficients (�Cxx,�Czz) with increasein micropolar effect of lubricant, is less pronounced at higher values of deformation coefficient (�Cd).The dimensional values of performance characteristics of a constant flow valve-compensated

four-pocket flexible hydrostatic journal bearing for Newtonian and micropolar lubricant are alsopresented in Table I for various geometric and operating parameters of the bearing.

CONCLUSION

The study of four-pocket hydrostatic journal bearing system with constant flow valve restrictor has beencarried out considering the effects of micropolar parameters of the lubricant and the bearing flexibility.Micropolar lubricant has the potential in describing the effect of polymeric additives in lubricants. Thecomputed results indicate that the influence of the flexibility of bearing shell and micropolar parametersare substantial on the performance characteristics of recessed hydrostatic journal bearing system. It is



shown that the decrease in �pmax caused by bearing elastic effect is more in case of micropolar lubricantas compared with the Newtonian lubricant. In addition, for both Newtonian and micropolar lubricants,the fluid-film direct stiffness coefficients (�Sxx,�Szz) and the direct damping coefficients (�Cxx,�Czz) show thedecreasing trend with increase in the value of deformation coefficient (�Cd).

NOMENCLATURE

DIMENSIONAL PARAMETERS

ab

Copyrigh

land width (m)
c radial clearance (m) D journal diameter (m) Eb modulus of elasticity (N/m-2) F fluid-film reaction (N) h fluid-film thickness (m) l characteristic length (m) L bearing length (m) p pressure (N.m-2) ps supply pressure (N.m-2) Q bearing flow (m3.s-1) r radial coordinate RJ journal radius (m) t time (s) th shell thickness u, v,w x, y and z velocity components (m�s�1) W0 external load (N) XJ, ZJ journal centre coordinates x, y circumferential and axial coordinates (m) z coordinate across film thickness (m) m dynamic viscosity (Pa�s) n Poisson’s ratio oJ journal speed (rad s�1) θ angle of inter-recess land width (Figure 1b)
NONDIMENSIONAL PARAMETERS

āb
=ab/L; land width ratio Āe =area of eth element �Cd =(psth/Ebc); elastic deformation coefficient� �F =F 1=psR2
J ; fluid-film reaction
�h =h/c lm =c/l
t © 2012 John Wiley & Sons, Ltd. Lubrication Science (2012)DOI: 10.1002/ls


N

Copyrig

=(k/(2m+ k))1/2; coupling number
�p =p/ps �Q =(mr/c
3ps).Q�
�t =t c2ps=mrR
2J

�th
= thRJ
�u;�v
=(u, v)(mrRJ/c2ps)
�w
=w(mrRJ/c2ps)(RJ/c)�
�W0
= Wo=psR2J
�Xj; �Zj
=(Xj, Yj)/c (a, b) =(x, y)/RJ; circumferential and axial coordinates e =e/c; eccentricity ratio l =L/D; aspect ratio �m =m/mr� Ω =oJ mrR
2J=c

2ps ; speed parameter

SUBSCRIPTS

b
bearing c pocket J journal r reference value R restrictor s supply condition
MATRICES AND VECTORS

Ni,Nj

ht © 20

shape function matrices
�pf g pressure vector �Qf g flow vector �RxJf g; �RzJf g vectors due to journal velocity �RHf g column vector (hydrodynamic term)
ABBREVIATIONS

EHS
elastohydrostatic MEHS micropolar elastohydrostatic MHS micropolar hydrostatic NHS Newtonian hydrostatic RHS right-hand side
12 John Wiley & Sons, Ltd. Lubrication Science (2012)DOI: 10.1002/ls


REFERENCES

1. Sinhasan R, Sharma SC, Jain SC. Performance characteristics of an externally pressurized capillary-compensated flexiblejournal bearing. Tribology International 1989; 22(4): 283–293.

2. Sinhasan R, Sharma, SC, Jain SC. Performance characteristics of a constant flow valve compensated multirecess flexiblehydrostatic journal bearing. Wear 1989; 134: 335–356.

3. Sinhasan R, Sharma S C, Jain SC. Performance characteristics of externally pressurized orifice compensated flexible journalbearing. STLE Tribology Trans. 1991; 34(3): 465–471.

4. Sharma SC, Sinhasan R, Jain SC. Performance characteristics of multirecess hydrostatic/hybrid flexible journal bearing withmembrane type variable-flow restrictor as compensating elements. Wear 1992; 152: 279–300.

5. Khader MS, Vachon RI. Theoretical effects of solid particles in hydrostatic bearing lubricant. J. Lub. Technol. Trans. ASME1973; 95: 104–106.

6. Scott W, Suntiwattana P. Effect of oil additives on the performance of a wet friction clutch material. Wear 1995; 183:850–855.

7. Allen S, Kline K. Lubrication theory of micropolar fluids. J. Appl. Mech. Trans. ASME 1971; 38: 646–650.8. Balaram M. Micropolar squeeze films. J. Lub Technol. Trans. ASME 1975; 97: 635–637.9. Prakash J, Sinha P. Lubrication theory of micropolar fluids and its application to a journal bearing. Int. J. Engg. Science

1975; 13: 217–232.10. Prakash J, Sinha P. A study of squeezing flow in micropolar fluid lubricated journal bearings. Wear 1976; 38: 17–28.11. Singh C, Sinha P. The three-dimensional Reynolds equation for micropolar fluid lubricated bearings. Wear 1982; 76(2):

199–209.12. Isa M, Zaheeruddin Kh. Characteristics of squeeze film porous journal bearings with micropolar lubricant. Transactions of

the Canadian Society for Mechanical Engineering 1981; 6(3): 146–150.13. Tipei N. Lubrication with micropolar liquids and its application to short bearings. J. Lub Technol. Trans. ASME 1979; 101:

356–363.14. Hung TW, Weng C, Chen CK. Analysis of finite width journal bearings with micropolar fluids. Wear 1988; 123: 1–12.15. Hung TW,Weng C. Dynamic characteristics of finite width journal bearings with micropolar fluids.Wear 1990; 141: 23–33.16. Khonsari MM, Brewe DE. On the performance of finite journal bearings lubricated with micropolar fluids. STLE, Tribology

Trans. 1989; 32(2): 155–160.17. Das S, Guha SK, Chattopadhyay AK. On the conical whirl instability of hydrodynamic journal bearings lubricated with

micropolar fluids. Proc. IMechE, Part-J 2001; 215: 431–439.18. Das S, Guha SK, Chattopadhyay AK. On the steady-state performance of misaligned hydrodynamic journal bearings

lubricated with micropolar fluids. J. Tribology Int. 2002; 35: 201–210.19. Wang Xiao-Li, Ke-Qin Zhu. A study of the lubricating effectiveness of micropolar fluids in a dynamically loaded journal

bearing. Tribology International 2004; 37: 481–490.20. Wang Xiao-Li, Ke-Qin Zhu. Numerical analysis of journal bearings lubricated with micropolar fluids including thermal and

cavitating effects. Tribology International 2006; 39: 227–237.21. Verma S, Kumar V, Gupta KD. Analysis of multirecess hydrostatic journal bearing operating with micropolar lubricant.

Journal of Tribology, ASME 2009; 131 / 021103–1.22. Verma S, Gupta KD, Kumar V. Analysis of capillary compensated hole-entry hydrostatic/hybrid journal bearing operating

with micropolar lubricant. IUTAM Springer book series 2011; 25: 241–252.23. Rahmatabadi AD, Nekoeimehr M, Rashidi R. Micropolar lubricant effects on the performance of non-circular lobed

bearings. Tribology International 2010; 43: 404–413.24. Nicodemus ER, Sharma SC. Influence of wear on the performance of multirecess hydrostatic journal bearing operating with

micropolar lubricant. Journal of Tribology 2010; 132 / 021703–1.25. Nicodemus ER, Sharma SC. Orifice compensated multirecess hydrostatic/hybrid journal bearing system of various geometric

shapes of recess operating with micropolar lubricant. Tribology International 2011; 44: 284–296.26. Verma S, Kumar V, Gupta KD. Analysis of capillary compensated hydrostatic journal bearing operating with micropolar

lubricant. Industrial Lubrication and Tribology 2011; 63(3): 192–202.


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