Influence of surface roughness effects on the performance of non-recessed hybrid journal bearings

Tribology International 35 (2002) 467–487 www.elsevier.com/locate/triboint

Influence of surface roughness effects on the performance of nonrecessed hybrid journal bearings T. Nagaraju b, Satish C. Sharma a,∗, S.C. Jain a a

Department of Mechanical and Industrial Engineering, Indian Institute of Technology, Roorkee 247 667, India b PES College of Engineering, Mandya 571 401, India

Abstract The effect of journal and bearing surface roughness on the performance of a capillary compensated hole-entry hybrid journal bearing system has been theoretically studied. The analysis considers the average Reynold’s equation for the solution of lubricant flow field in the clearance space of a rough surface journal bearing system. The finite element method and Galarkin’s technique has been used to derive the system equation for the lubricant flow field. The non-dimensional parameters ⌳ (surface roughness parameter) and g (surface pattern parameter) have been defined to represent the magnitude of height distribution of surface irregularities and their orientation, respectively. The influence of surface roughness on the bearing performance has been studied for the transverse, isotropic and longitudinal surface patterns. The bearing performance characteristics have been computed for both symmetric and asymmetric capillary compensated hole-entry journal bearing configurations for the various values of surface roughness parameter (⌳), surface pattern parameter (g) and restrictor design parameter (C¯S2). The computed results indicate that the inclusion of surface roughness effects in the analysis affects the performance of a bearing quite significantly vis-a`-vis smooth surface bearing. The study indicates that for generation of accurate bearing characteristic data, the inclusion of surface roughness effects in the analysis is essential.  2002 Elsevier Science Ltd. All rights reserved. Keywords: journal bearing; non-recessed; hybrid; surface roughness

1. Introduction Non-recessed (hole-entry type) hybrid journal bearings were developed so as to utilize both hydrostatic and hybrid actions in a more efficient and optimal manner. The applications of hole-entry journal bearings are quite wide and varied due to their superior performance, i.e. improved load carrying capacity at zero and high speed with low energy consumption and relative simplicity in manufacturing as compared to conventional recessed or pocketed bearings [16]. In a hole-entry journal bearing configuration, the holes are disposed around the circumference of the bearing either symmetrically or asymmetrically as shown in Figs 1(a) and (b) and are usually compensated by capillary, orifice or constant flow valve restrictors. The last few decades have witnessed rapid technologi-

Corresponding author. Tel.: +91-01332-85603; fax: +91-0133273560. E-mail address: [email protected] (Satish C. Sharma). ∗

cal advancements and the operating conditions of machines are becoming very stringent, exact and more demanding. Thus, it becomes imperative that the design of bearings should be based on more realistic bearing characteristic data so as to meet these requirements. In recent times considerable research work has been carried out and reported in the area of non-recessed hybrid journal bearings [6,16,18–21,27]. These studies focus on various aspects related to hole-entry bearings, i.e. the effect of size of small pockets at the hole-entry position [27], the effect of bearing shell flexibility [18– 21], the effect of journal misalignment [6], the effect of non-Newtonian behavior of lubricant [20] and the effect of flow control device [21] on the performance of nonrecessed (hole-entry) journal bearings. The available studies in the literature concerning the hole-entry journal bearings [6, 16, 18–21, 27] assume that both journal and bearing surfaces are perfectly smooth. However, in actual practice, on any finished surface, imperfections are bound to be there and these take the form of a succession of hills and valleys which often results from the characteristics of the machining process

0301-679X/02/$ - see front matter  2002 Elsevier Science Ltd. All rights reserved. PII: S 0 3 0 1 - 6 7 9 X ( 0 2 ) 0 0 0 3 7 - 3

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Nomenclature ab c Cij D e F Fx,Fz Fo g h h¯ l hT lc L RJ,Rb rc p Q Sij t Vr Wo X,Y,Z XJ,ZJ z zJ,zb

Bearing land width, mm Radial clearance, mm Damping coefficients (i,j ⫽ x,z), N s mm⫺1 Journal diameter, mm Journal eccentricity, mm Fluid film reaction (∂h / ∂t ⫽ 0), N Components of fluid film reactions in X and Z direction (∂h / ∂t ⫽ 0), N Fluid film reaction (∂h / ∂t ⫽ 0), N Acceleration due to gravity, m s⫺2 Nominal fluid-film thickness, mm Local fluid-film thickness, mm Average fluid-film thickness, mm Length of capillary, mm Bearing length, mm Radius of journal, radius of bearing, mm Radius of capillary, mm Pressure, N mm⫺2 Bearing flow, mm3 s⫺1 Stiffness coefficients (i,j ⫽ x,z), N mm⫺1 Time, s Variance ratio External load, N Cartesian coordinates Journal center coordinates, mm Combined roughness height, (z ⫽ zJ ⫹ zb),mm Roughness height distribution in journal and bearing, mm

冕 x

erf

Error function, erf(x) ⫽ 2 / p exp(⫺y2)dy

m l0.5x,y wJ wth wI s sJ sb

Dynamic viscosity of lubricant, N s m⫺2 0.5 correlation lengths of the x and y profile, mm Journal rotational speed, rad s⫺1 Threshold speed, rad s⫺1 (g / c)1 / 2, rad s⫺1 RMS value of combined roughness, s ⫽ 冑s2J ⫹ s2b, mm RMS value of journal surface roughness, mm RMS value of bearing surface roughness, mm

0

Non-dimensional parameters a¯ b ⫽ ab / L , Land width ratio C¯ ij ⫽ Cij(c3 / mR4J ) 1 C¯ S2 ⫽ (3pr4c / 2c3 lc), Restrictor design parameter 12 (F¯ ,F¯ o) ⫽ (F,F¯ o) / psR2J p¯ ,p¯ s,p¯ c,p¯ max ⫽ (p,ps,pc,pmax) / ps ¯ ⫽ Q(m / c3ps) Q ¯Sij ⫽ Sij(c / psR2J ) (V¯ rj,V¯ rb) ⫽ ((sj,sb) / s)2 ¯ o ⫽ Wo / psR2J W (X¯ J,Z¯ J) ⫽ (XJ,ZJ) / c z¯,z¯ J,z¯ b ⫽ (z,zJ,zb) / c

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469

a,b ⫽ (X,Y) / RJ , Circumferential, axial co-ordinates e = e / c , Eccentricity ratio l0.5x , Surface pattern parameter g⫽ l0.5y l ⫽ L / D , Aspect ratio ⍀ ⫽ wJ(mR2J / c2ps) , Speed parameter ¯ th ⫽ wth / wI w fx,fy Pressure flow factors Shear flow factor fs Shear flow factor related to single surface ⌽s s¯ ⫽ s / c ⌳ ⫽ 1 / s¯ , Surface roughness parameter t ⫽ t(c2ps / mR2J ) Matrices [F¯ ] Fluidity matrix {p¯ } Nodal pressure vector ¯} {Q Nodal flow vector {R¯ H} Vector due to hydrodynamic terms {R¯ Xj},{R¯ Zj} Right hand side vectors due to journal center velocities Subscripts and superscripts b Bearing c Capillary J Journal R Restrictor s Supply min,max Minimum, maximum — Corresponding non-dimensional parameter . First derivative w.r.t time

and its accompanying defects. The surface irregularities in bearing and journal may also result due to wear, impulsive damage, foreign particles, rust, etc. Moreover, it is a well known fact that if the surfaces were smooth, seizure may occur due to the difficulty of maintaining the lubricating oil film. Good bearing properties in any part are obtained when the surface has a large number of hills and valleys, as the hills in an irregular surface reduce the metal-to-metal contact and the valleys help to retain the film of lubricating oil. Therefore, the analyses based upon the assumption of a smooth surface may not be realistic. In reality, the surface roughness heights are typically of the same order as that of the fluid-film thickness of the journal bearing and hence the performance of the journal bearing system gets altered. Therefore, for an accurate prediction of the journal bearing performance characteristics, the consideration of the surface roughness in the analysis is imperative. Thus, the effect of surface roughness in the area of journal bearing

problems has attracted many researchers and has become an important activity over the past few years. In recent years many studies related to the effects of surface roughness on the performance of hydrodynamic journal bearings have been reported in the literature. Patir and Cheng [11,12] developed an average Reynold’s equation for the rough surface in terms of pressure and shear flow factors. Ai et al [1], and Shi and Wang et al. [22,26] used the average flow model developed by Patir and Cheng [11] in their mixed lubrication analysis of hydrodynamic journal bearing systems. Using the average flow model, Majumdar and Ghosh [10] and Ramesh et al. [13–15] studied the stability of finite rough oil and submerged oil journal bearing systems, respectively. Many investigators used a stochastic model developed by Christensen [2] to study the effect of surface roughness. Christensen and Tonder [3] applied the stochastic model to show the effect of transverse and longitudinal roughness on the characteristics of finite width journal

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Fig. 1.

Hole-entry journal bearing system.

bearings. Guha [4] studied the effect of isotropic roughness on the steady state characteristics of a misaligned hydrodynamic journal bearing system. Turaga et al. [23– 25] studied the stability of a hydrodynamic journal bearing with rough surfaces and Zhang et al. [28,29] analyzed the effects of two-sided roughness with three types of roughness texture. The study by Kato and Obara [7] indicates that a considerable improvement in dynamic characteristics can be achieved by providing longitudinal microgrooves on the bearing surface. Kazama and Yamaguchi [8] considered the effect of surface roughness in their study of the mixed lubrication problem concerning the hydrostatic thrust bearings of hydraulic equipment. San Andres [17] studied the pocketed hybrid journal bearing by considering the effect of surface roughness along with the fluid inertia effect. The surface roughness

in his analysis has been modeled by an effective roughness depth varying from 0.1 to 10 percent of radial clearance and showed an improvement of 27% in dynamic performance of the bearing. However, the work reported by San Andres did not take into account the height distribution of the surface irregularities and the surface pattern which are the inherent properties of any finished surfaces. The available studies in the literature which include the surface roughness effects in the analysis of journal bearing systems are concerned mainly with the hydrodynamic journal bearing systems, except the studies by Kazama and Yamaguchi [8] and San Andres [17]. The available studies in the case of hydrodynamic [1–5,7,9– 15,22–26,28,29] and hydrostatic [8,17] bearings demonstrates that their performance is quite significantly affected due to the inclusion of surface roughness effects in the analysis. To the best of the authors’ knowledge no results have yet been reported in the literature for the case of the non-recessed hole-entry type journal bearings. Therefore, the objective of this paper is to analyze the effects of surface roughness on the performance of capillary compensated hole-entry hybrid journal bearing systems. The effect of surface roughness is accounted for by defining a non-dimensional surface roughness parameter ⌳(which is the reciprocal of the non-dimensional rms value of composite roughness heights) and a surface pattern parameter g. The finite element method has been used to obtain the solution of the required governing equations. The bearing static performance characteristics such as minimum fluid film thickness, maximum pressure and bearing flow and dynamic characteristics in terms of fluid-film stiffness and damping coefficients are presented for the typical representative values of roughness parameters (⌳) for both transverse, isotropic and longitudinal surface texture for both symmetric and asymmetric bearing configurations shown in Fig. 1(a) and (b). The results presented in this paper are expected to be quite useful to the bearing designers and the academic community.

2. Analysis 2.1. Average fluid-film thickness(h¯T) The geometry of the externally pressurized hole-entry journal bearing system along with rough surface and coordinate system is shown in Fig. 1. The local fluid-film thickness hl in non-dimensional form is expressed as h¯l ⫽ h¯ ⫹ z¯. Assuming Gaussian distribution of surface heights, the non-dimensional form of average fluid-film thickness h¯T, which is equal to the expected or mean value of local fluid-film thickness, is expressed as

T. Nagaraju et al. / Tribology International 35 (2002) 467–487

冕 ⬁

h¯ T ⫽ E[h¯ ⫹ z¯] ⫽

(h¯ ⫹ z¯)y(z¯)dz¯

(1a)

⫺⬁

where y(z¯ ) is the probability density function of combined roughness z¯ and is expressed as y(z¯) ⫽

1

e⫺z¯ 冑2ps¯

2/2s ¯2

冕

where ⌳( ⫽ 1 / s¯ ) in the present work is defined as the surface roughness parameter and h¯ is the nominal fluidfilm thickness. It is same as the fluid-film thickness obtained for a smooth surface journal bearing and is expressed as [18–21] h¯ ⫽ 1⫺X¯ J cosa⫺Z¯ J sina.

.

(1b)

For the fully lubricated region, since h¯ is constant for a given value of circumferential coordinate a, the expected value of h¯ becomes as ⬁

E[h¯ ] ⫽ h¯

471

y(z¯) dz¯ ⫽ h¯

(4)

The average fluid-film thickness h¯ T is a function of the nominal fluid-film thickness (h¯ ) and rms value of the combined roughness height (s¯ ) of the corresponding surface pattern. Therefore, using Eqn. 3 the average fluidfilm thickness h¯ T can be computed for transversely, isotropically and longitudinally oriented roughness patterns. 2.2. Flow factors(fx, fy, fs)

⫺⬁

as

冕 ⬁

y(z¯ ) dz¯ ⫽ 1.

⫺⬁

Further, since z¯ is a random variable with mean zero, the expected value of z¯ in the fully lubricated region becomes equal to zero (i.e. E[z¯] ⫽ 0). Then, the expression for the average fluid-film thickness (i.e. Eqn. 1(a)) in the fully lubricated region becomes as (2a) h¯ T ⫽ h¯ For the partial lubrication region, the lower limit of integration in Eqn. (1a) is ⫺h¯ , the minimum value that z¯ can achieve in the partial lubrication region. Therefore, the Eqn. (1a) is expressed as

冕 ⬁

h¯ T ⫽

(h¯ ⫹ z¯) y(z¯) dz¯.

(2b)

⫺h¯

Substituting Eqn. (1b) into Eqn. (2b) and integration yields the average fluid-film thickness for the laminar partial lubrication region and is expressed as

冉

冉 冊冊

h¯ ⌳h¯ h¯ T ⫽ 1 ⫹ erf 2 冑2

⫹

1

⌳冑2p

2

e⫺(⌳h¯ ) /2.

(2c)

In the present work, the partial and fully lubricated regions are characterized as the region where ⌳h¯ ⬍ 3 and ⌳h¯ ⱖ3, respectively [11,12]. Using these conditions, the expression for average fluid-film thickness, from Eqn. (2a) and Eqn. (2c), can now be expressed as

冦

冉

冉 冊冊

h¯ ⌳h¯ 1 ⫹ erf h¯ T ⫽ 2 冑2

⫹

1

e⫺(⌳h¯ ) ⌳冑2p

h¯ for ⌳h¯ ⱖ3

2/2

In the analysis of rough surface journal bearings the pressure flow factors fx,fy are used to compare the average pressure flows (in axial and circumferential directions) in a rough journal bearing to that of a smooth bearing. Similarly, the shear flow factor fs is used to represent the additional flow transport of lubricant in the circumferential direction due to the sliding of the rough surface journal. The pressure flow factors in non-dimensional form are expressed as [11] fx ⫽ 1⫺Ce⫺r⌳h¯ fx ⫽ 1 ⫹ C(⌳h¯ )

⫺r

for gⱕ1

(5a)

for g ⬎ 1

(5b)

fy(⌳h¯ ,g) ⫽ fx(⌳h¯ ,1 / g).

(5c)

Assuming both journal and bearing surfaces have the same surface pattern (i.e. gJ ⫽ gb), the shear flow factor in non-dimensional form is expressed as [12] fs ⫽ (V¯ rj⫺V¯ rb)⌽s

(6a)

⫽ (2V¯ rj⫺1)⌽s since V¯ rj ⫹ V¯ rb ⫽ 1 where ⌽sis the shear flow factor related to a single surface. It is a positive function of (⌳.h¯ ) and the surface pattern parameter gof a given surface and is given by ⌽s ⫽ A1(⌳h¯ )a1e⫺a2(⌳h¯ )+a3(⌳h¯ ) ⌽s ⫽ A2e⫺0.25(⌳h¯ )

2

for ⌳h¯ ⱕ5

(6b)

for ⌳h¯ ⬎ 5

(6c)

where C,r,A1,A2,a1,a2 and a3 appearing in Eqns (5) and (6) are constants and g is the surface pattern parameter which is defined as g ⫽ (l0.5x / l0.5y). Purely transverse, isotropic and longitudinal roughness texture are having its value as 0, 1 and ⬁, respectively. 2.3. The average Reynold’s equation

for ⌳h¯ ⬍ 3 (3)

The average Reynold’s equation governing the laminar flow of an incompressible, Newtonian lubricant in the clearance space between the rough surfaces of a jour-

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nal and bearing in non-dimensional form is written [11– 15] as

冉

冊 冉

冊

∂ ⍀∂h¯ T h¯ 3 ∂p¯ h¯ 3 ∂p¯ ⍀ ∂fs ∂ fx ⫹ fy ⫽ ⫹ ∂a 12∂a ∂b 12∂b 2 ∂a 2⌳ ∂a ⫹

(7)

∂h¯ T . ∂t

For the smooth bearing case (i.e. for large value of ⌳) the pressure flow factors fx,fy→1 and shear flow factor fs→0, and thus, the Eqn. (7) reduces to the Reynold’s equation valid for a smooth surface bearing. 2.4. Finite Element formulation The lubricant flow field has been discretized using four-noded quadrilateral isoparametric elements. Using the Galerkins techniques and orthogonality condition for Eqn. (7), the following system equations are derived [18–21] ¯ } ⫹ ⍀{R¯ } ⫹ X˙¯ {R¯ } ⫹ Z˙¯ {R¯ } (8) [F¯ ]{p¯ } ⫽ {Q H

J

XJ

J

冕冕 冋

册

¯ ei ⫽ Q

冕冋 冉

冊 冉

⌫e

冕冕冉

冊

冕冕 冋

冉 冊册

冕冕 冋

冉 冊册

Ae

R¯ eXJi ⫽

Ae

R¯ eZJi ⫽

Ae

1 ⌳h¯ 1 ⫹ erf 2 冑2

1 ⌳h¯ 1 ⫹ erf 2 冑2

冊册

F¯ x ⫽ ⫺

p¯ cosa da db

(10a)

冕冕

p¯ sina da db.

(10b)

⫺l 0

⫺l 0

The resultant fluid film reaction is given by F¯ ⫽ [F¯ 2x ⫹ F¯ 2z ]1/2.

(10c)

2.7.2. Journal center equilibrium position ¯ o, the X and Z For a given vertical external load W components of fluid-film reaction at steady state condition (∂h¯ / ∂t ⫽ 0) becomes as [18]

Nicosa da db

¯ o ⫽ 0. F¯ x ⫽ 0 and F¯ z⫺W

Nisina da db.

2.5. Restrictor flow equation The equation of flow through the capillary restrictor in non-dimensional form is expressed [6,19] as ¯ R ⫽ C¯ S2(1⫺p¯ c). Q

冕冕

l 2p

F¯ z ⫽ ⫺

fs ∂Ni h¯ T ⫹ da db ⌳ ∂a

1 R¯ eHi ⫽ 2

2.7.1. Fluid film reaction (F¯ ) The fluid film reaction components along the X and Z directions are given by

l 2p

fs h¯ ∂p¯ ∂p¯ ⍀ ¯ fx ⫹ fy ⫺ hT ⫹ Nid⌫e 12 ∂a ∂b 2 ⌳ 3

The fluid-film pressure distribution in the clearance space of a journal bearing is obtained by solving Eqns (8) and (9) together with the required boundary conditions. The static and dynamic performance characteristics are computed using the required expressions [6,18–21].

and

h¯ 3 ∂Ni∂Nj ∂Ni∂Nj ⫹ fy fx da db 12 ∂a ∂a ∂b ∂b

Ae

2.7. Performance characteristics

ZJ

where the coefficients of the matrices in the above equation for an eth element are defined as F¯ eij ⫽

1. nodes situated on the external boundary of the bearing have zero pressure, p¯ 兩b ⫽ ⫿ 1.0= 0.0 2. nodes situated on hole have equal pressure, 3. flow of lubricant through the restrictor is equal to the bearing input flow, 4. at the trailing edge of the positive region p¯ ⫽ (∂p¯ / ∂a)= 0.0

(9)

(11a)

The bearing reaction terms F¯ x and F¯ z in Eqn. 11a can be expanded using Taylor’s series expansion about the ith journal center position. Assuming that the alteration in the journal center position is quite small and retains terms only up to first order in the Taylor’s series expansion, the corrections (⌬X¯ J|i, ⌬Z¯ J|i) on the coordinates are obtained. The expressions for (⌬X¯ J|i, ⌬Z¯ J|i) are as given in [18]: The new journal center position co-ordinates (X¯ (iJ ⫹ 1), Z¯ (iJ ⫹ 1)) are given by

2.6. Boundary conditions

X¯ (i+1) ⫽ X¯ iJ ⫹ ⌬X¯ iJ and Z¯ (i+1) ⫽ Z¯ iJ ⫹ ⌬Z¯ iJ J J

The following boundary conditions are used for the lubricant flow field.

where (X¯ iJ, Z¯ iJ) are the co-ordinates of the ith journal center position. The final journal center equilibrium position is estab-

(11b)

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473

3. Solution procedure The analysis of a compensated hydrostatic/hybrid journal bearing system including the effect of surface roughness requires the simultaneous solution of an average Reynold’s equation (Eqn. (7)) along with the restrictor flow equation (Eqn. (9)) as a constraint satisfying the required boundary conditions. Figure 2 shows the overall solution scheme used for the solution of the hole-entry hybrid journal bearing problem considering surface roughness effects. The overall numerical procedure adopted for the computation of performance characteristics comprises of the following steps:

Fig. 2.

Solution scheme.

lished using a suitable iterative scheme. Iterations are continued until the following criterion is not satisfied.

冋

册

((⌬X¯ iJ)2 ⫹ (⌬Z¯ iJ)2)1/2 × 100 ⬍ 0.001. ((X¯ iJ)2 ⫹ (Z¯ iJ)2)1/2

(11c)

2.7.3. Stiffness and damping coefficients The linearized fluid film stiffness coefficients are defined as S¯ ij ⫽ ⫺

∂F¯ i ∂qj

(i ⫽ x,z)

(12)

where qj ⫽ X¯ J, Z¯ J.The linearized fluid film damping coefficients are defined as ∂F¯ i C¯ ij ⫽ ⫺ ¯ (i ⫽ x,z) ∂q˙j where q¯˙ ⫽ X¯˙ , Z¯˙ . j

J

J

(13)

¯ o assuming a 1. Initially for a specified external load W steady state condition (i.e. X¯˙ J ⫽ Z¯˙J ⫽ 0), the value of the nominal fluid-film thickness h¯ is computed for the chosen tentative values of the journal center position X¯ Jand Z¯ Jusing Eqn. (4). 2. Using the value of nominal fluid-film thickness h¯ computed in step 1, the value of the average fluid-film thickness h¯ T is calculated using Eqn. (3). 3. The value of the pressure flow factor fx for transverse and isotropic roughness (i.e. gⱕ1) is computed using Eqn. (5a) and for longitudinal roughness (i.e. g ⬎ 1) it is computed using Eqn. (5b) at every node point. Then fy is computed by taking the inverse value of g i.e. 1 / g (Eqn. (5c)) at all node points. 4. The value of shear flow factor ⌽s is computed using Eqn. (6b) or (6c) depending upon the value of (⌳.h¯ ) at all the node points. The values for the coefficients used in Eqns (5) and (6) for the computation of pressure flow factors in step 3 and shear flow factor in the present step are assigned for various values of surface pattern parameter g, in a subroutine COEFF. 5. Using the value of flow factors (fx, fy and fs), the lubricant flow field system equation (Eqn. (8)), after adjustment for the flow through the capillary restrictor (Eqn. (9)) and modification for boundary conditions is solved for the nodal fluid-film pressure distributions. 6. Using the fluid-film pressure distributions obtained in step 5, the new journal center position (X¯ J, Z¯ J) for the ¯ o) is established using given vertical external load (W Eqn. (11) for the next iteration. 7. Values of nominal fluid-film thickness h¯ and average fluid-film thickness h¯ T are again computed for the next iteration. 8. Steps 3 to 7 are repeated until the criterion for the journal center equilibrium given by Eqn. (11c) is not satisfied or the number of iterations (IEQT) are not exceeded by the maximum number of iterations (INERX). 9. Once the equilibrium journal center position is established, the static and dynamic characteristics are computed using the expressions described earlier.

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4. Results and discussion The performance characteristics of a hole-entry hybrid journal bearing system including the effect of surface roughness presented in this section have been obtained using the analysis and solution algorithm described in the previous section. The validity of the computed results obtained from the developed program is established in two stages. In the first stage, the fluid-film reaction (F¯ o) of a capillary compensated symmetric and asymmetric hole-entry hydrostatic journal bearing having smooth journal and bearing surfaces is computed and compared with the published results of Rowe et al. [16] as shown in Fig. 3(a). These results compare well for a wide range of eccentricity ratios (e) and a maximum deviation of nearly 7% and 7.5% is noted at higher eccentricity ratio for symmetric and asymmetric configurations, respectively. The minor difference between the results may be attributed to the use of different computational schemes. To the best of the authors’ knowledge, there are no results yet reported for the case of non-recessed journal bearings with surface roughness effects. Thus, in the second stage, the fluid-film reaction (F¯ o) for both smooth and rough plain hydrodynamic journal bearing systems is computed by modifying the developed program and comparing the computed results with the published results of Ramesh et al. [13]. The slight difference observed between the two results (Fig. 3(b)) in this case may be due to the use of different numerical techniques and flow field discretization as well as due to the boundary conditions used in the cavitation zone. Moreover, the results computed in the present study are for the case of a plain hydrodynamic journal bearing with conventional Reynold’s boundary conditions whereas those presented by Ramesh et al. [13] are for a submerged bearing with Jacobssen and Floberg boundary conditions at the cavitation zone. In the present work, the results for the bearing performance characteristics have been computed for the generally used bearing geometric and operating parameters as given in Table 1 for both smooth and rough surface hole-entry hydrostatic/hybrid journal bearings. It is assumed that the journal and bearing surfaces have the same surface pattern parameters (gJ ⫽ gb ⫽ g). The effects of surface roughness parameter ⌳ and surface pattern parameter g are presented for the various values of restrictor design parameter C¯ S2 and for the values of ¯ o ⫽ 1.25 (for the symmetric bearing external load W ¯ o ⫽ 1.75 (for the asymmetric bearconfiguration) and W ing configuration). These results are presented in Figs 5–14 and discussed in the following paragraphs. 4.1. Load carrying capacity (F¯ o) Figure 4 shows the effect of the surface roughness parameter (⌳) and the surface pattern parameter (g) on

Fig. 3.

Load versus eccentricity ratio.

the load carrying capacity (F¯ o). To the best of the authors’ knowledge there are no results reported so far which show the influence of the surface roughness parameter (⌳) and surface pattern parameter (g) on F¯ o for the case of non-recessed journal bearings. Thus, the results of F¯ o for a plain hydrodynamic journal bearing were computed by modifying the developed program and plotted in Fig. 4(a). It is observed that the effects of ⌳ and g on F¯ o are of a similar nature to those of the

T. Nagaraju et al. / Tribology International 35 (2002) 467–487

Table 1 Geometric and operating parameters Geometric properties Aspect ratio (l) Land-width ratio (a¯ b) No. of rows of entry holes (k) No. of holes per row (n) Symmetric configuration Asymmetric configuration Surface properties Surface roughness parameter (⌳) Surface pattern parameter (g) Variance ratio (V¯ rj) Operating conditions Speed parameter (⍀) ¯ o) External load (W For symmetric configuration For asymmetric configuration Concentric design pressure ratio (b∗) Restrictor - Capillary Restrictor design parameter (C¯ S2)

1.0 0.25 2 12 6 2-8 1/6,1/3,1,3,6 0.5 0.0 and 1.0 1.25 1.75 0.5 0.05–0.3

475

available published results of Ramesh et al. [13] for the case of a hydrodynamic journal bearing system. Figure 4(b) and (c) show the variation of the load carrying capacity (F¯ o) of a hole-entry journal bearing system versus the surface roughness parameter (⌳) for three surface patterns, namely transverse (g ⫽ 1 / 6), isotropic (g ⫽ 1) and longitudinal (g ⫽ 6) during hydrostatic and hybrid modes of operation, respectively. A horizontal line marked with smooth in each figure indicates the load carrying capacity of smooth surface journal bearings. During the hydrostatic mode of operation, Fig. 4(b), the load carrying capacity of symmetric and asymmetric hole-entry journal bearing configurations increases as the surface roughness parameter ⌳ reduces (i.e. as roughness increases) for transversely (g ⫽ 1 / 6) and isotropically (g ⫽ 1) oriented roughness patterns. While it decreases in the case of a bearing with longitudinally oriented roughness pattern (g ⫽ 6). It may also be noted that as the value of surface roughness parameter ⌳ decreases, the bearing surface becomes more rough. During the hybrid mode of operation, Fig. 4(c), the load carrying capacity increases as ⌳ reduces for longitudinally and isotropically oriented roughness patterns in both configurations. However, in a transversely oriented roughness pattern the value of the load carrying capacity is observed to be reduced for larger surface roughness (⌳ ⬍ 3) compared to that of a smooth surface bearing. Further, in the hydrostatic mode of operation, the transversely oriented roughness pattern provides higher load carrying capacity in both configurations (Fig. 4(b)). This is because of the fact that the circumferential flow of lubricant, which reduces the load carrying capacity of a hydrostatic journal bearing at zero speed, is restricted by the transversely oriented rough surfaces. Whereas in the hybrid mode of operation, the longitudinally oriented roughness pattern (which restricts the axial flow and enhances the circumferential flow in order to develop more hydrodynamic pressure) provides higher load carrying capacity in both configurations (Fig. 4(c)). 4.2. Nominal minimum fluid-film thickness (h¯ min)

Fig. 4. Variation of load carrying capacity.

Figure 5 shows the variation of nominal minimum fluid-film thickness (h¯ min) for symmetric and asymmetric hole-entry journal bearing configurations with surface roughness parameter (⌳) and surface pattern parameter ¯ o) and for a selected (g). At a constant external load (W value of restrictor design parameter (C¯ S2), the value of h¯ min for a rough journal bearing increases as the surface roughness parameter ⌳ reduces (i.e. as roughness increases) during both hydrostatic/hybrid mode of operations of a bearing. Figure 5(a) and (b) indicate the same pattern for both symmetric and asymmetric configurations. Figure 5(c) and (d) depict the variation of h¯ min with three different surface roughness orientations hav-

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Fig. 5. Variation of nominal minimum fluid-film thickness.

ing different values of surface pattern parameter g (i.e. ¯ o) and 1/6, 1/3, 1, 3,6). At a constant external load (W for a chosen value of surface roughness parameter (⌳), the value of h¯ min is significantly affected due to the surface roughness orientation. For a symmetric hole-entry journal bearing configuration operating in hydrostatic mode of operation, the value of h¯ min increases for transversely (g ⬍ 1) and isotropically (g ⫽ 1) oriented roughness pattern for all chosen values of C¯ S2 while it shows this trend only for lower values of C¯ S2 (i.e. for C¯ S2⬍0.1) for the longitudinally (g ⬎ 1) oriented roughness pattern (Fig. 5(c)). For an asymmetric configuration, the value

of h¯ min increases for a transversely and isotropically oriented roughness pattern while it decreases for a longitudinally oriented roughness pattern during hydrostatic mode of operation (Fig. 5(d)). During the hybrid mode of operation, an improved value of h¯ min is obtained for all surface patterns having the value of g=1/6, 1/3, 1, 3, and 6 in both symmetric and asymmetric configurations. The improvement in the value of h¯ min is seen to be large for transversely oriented roughness patterns in both configurations during the hydrostatic mode of operation. Whereas, during the hybrid mode of operation, it is more for longitudinally and isotropically oriented roughness

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Fig. 6.

477

Variation of bearing flow.

patterns in symmetric and asymmetric configurations, respectively. ¯) 4.3. Bearing flow (Q ¯ with Figure 6 shows the variation of bearing flow Q surface roughness parameter (⌳) and surface pattern ¯ o) parameter (g). At a constant value of external load (W ¯ reduces and for a chosen value of C¯ S2, the bearing flow Q as surface roughness parameter (⌳) reduces (i.e. as surface roughness increases), Fig. 6(a) and (b), during hydrostatic and hybrid mode of operations in both symmetric and asymmetric hole-entry journal bearing configurations. Figure 6(c) and (d) show that the value of

¯ reduces for isotropically (g ⫽ 1) and bearing flow Q longitudinally (g ⬎ 1) oriented roughness patterns while it increases for transversely (g ⬍ 1) oriented roughness patterns during hydrostatic and hybrid mode of operations in both bearing configurations. This is due to the fact that the longitudinally oriented rough surfaces restrict the axial flow (i.e. axial flow) permitting more circumferential flow and hence reduce the bearing flow while the opposite action takes place in transversely oriented rough surfaces. However, isotropically oriented rough surfaces restrict circumferential as well as axial flow and provide reduced bearing flow compared to smooth surfaces.

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Fig. 7.

Variation of direct fluid-film stiffness coefficient (S¯ xx).

4.4. Fluid-film stiffness coefficients (S¯ xx, S¯ xz, S¯ zx, S¯ zz) Figures 7–10 show the variation of fluid-film stiffness coefficients. Figure 7(a) shows that at constant value of ¯ oand a chosen value of C¯ S2, the value of external load W the direct stiffness coefficient S¯ xx, in general, increases as the surface roughness parameter decreases (i.e. as the bearing becomes more rough) in a symmetric hole-entry journal bearing configuration during both hydrostatic and hybrid mode of operation. In an asymmetric configuration, the same trend is observed during the hydrostatic mode of operation for all the chosen values of C¯ S2, however, no definite trend is observed during the hybrid mode of operation in which variation is seen to

be more dependent on the C¯ S2 value (Fig. 7(b)). In a symmetric hole-entry journal bearing configuration, Fig. 7(c), the value of the direct fluid-film stiffness coefficient S¯ xx increases for transversely (g ⬍ 1) and isotropically (g ⫽ 1) oriented roughness patterns during hydrostatic and hybrid modes of operation for all chosen values of C¯ S2 while it reduces at higher values of C¯ S2 (i.e. for 0.1 to 0.3) for longitudinally (g ⬎ 1) oriented roughness patterns. For an asymmetric configuration, the value of S¯ xx increases only for isotropically oriented roughness patterns and it reduces for transversely and longitudinally oriented roughness patterns during the hydrostatic mode of operation (Fig. 7(d)). During the hybrid mode of operation, the value of S¯ xx reduces for

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Fig. 8.

479

Variation of cross-coupled fluid-film stiffness coefficient (S¯ xz).

all surface roughness patterns at lower values of C¯ S2. ¯ o and a chosen At a constant value of external load W value of C¯ S2, the cross-coupled stiffness coefficient S¯ xz increases as the surface roughness parameter ⌳ reduces (i.e. as roughness increases) for both configurations (Fig. 8(a) and (b)) while the opposite trend is seen for the value of the cross-coupled stiffness coefficient S¯ zx (Fig. 9(a) and (b)). Figure 8(c) and (d) show that the isotropically (g ⫽ 1) and longitudinally (g ⬎ 1) oriented roughness pattern provides an enhanced value of S¯ xz while the transverse (g ⬍ 1) pattern provides a reduced value in both configurations. The opposite trend is observed for the variation of the cross-coupled stiffness coefficient S¯ zx(Fig. 9(c) and (d)).

Figure 10(a) and (b) shows that for an isotropically (g ⫽ 1) oriented roughness pattern, the value of S¯ zz increases as the surface roughness parameter ⌳ reduces (i.e. as surface roughness increases) for all the chosen values of C¯ S2 in a symmetric bearing for the hybrid mode of operation and in an asymmetric bearing for both hydrostatic and hybrid modes of operations. This trend is observed only for the lower values of C¯ S2(⬍0.175) during the hydrostatic mode of operation in the symmetric configuration. From Fig. 10(c) and (d) it can be noticed that the value of S¯ zz increases for transversely (g ⬍ 1) and isotropically (g ⫽ 1) oriented roughness patterns for all the chosen values of C¯ S2 in symmetric and asymmetric configurations during the hydrostatic mode

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Fig. 9.

Variation of cross-coupled fluid-film stiffness coefficient (S¯ zx).

of operation while the opposite trend is seen for longitudinally (g ⬎ 1) oriented roughness patterns. During the hybrid mode of operation, the isotropic and transverse roughness pattern provides an enhanced value of S¯ zz for symmetric configurations (for all values of C¯ S2), whereas all surface patterns provide an enhanced value of S¯ zz for asymmetric configurations (for the value of C¯ S2 ⬍ 0.225). 4.5. Fluid-film damping coefficients (C¯ xx, C¯ xz, C¯ zx, C¯ zz) Figures 11–13 show the variation of fluid-film damping coefficients with surface roughness parameter (⌳) and surface pattern parameter (g). Figure 11(a) and (b) ¯ o and a show that at a constant value of external load W

chosen value of C¯ S2, the value of the direct fluid-film damping coefficient C¯ xx increases as the surface roughness parameter ⌳ reduces (i.e. as roughness increases) for both configurations during hydrostatic and hybrid modes of operation. From Fig. 11(c) and (d) it may be observed that in both isotropic (g ⫽ 1) and longitudinal (g ⬎ 1) roughness patterns, the value of C¯ xx increases during hydrostatic and hybrid modes of operation in both configurations while the opposite trend is seen for the transverse (g ⬍ 1) roughness pattern under the same operating conditions. In both symmetric and asymmetric journal bearing configurations, the increase in the value of C¯ xx is more significant in the case of the longitudinally oriented roughness pattern during the hybrid mode of operation. Whereas, it is more significant for an iso-

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Fig. 10.

481

Variation of direct fluid-film stiffness coefficient (S¯ zz).

tropically oriented roughness pattern during the hydrostatic mode of operation in asymmetric configuration while the increase in the value of C¯ xx with respect to roughness orientation is dependent on the value of C¯ S2 in the symmetric configuration (see Fig. 11(c)). Figure 12(a) and (b) show that, the value of crosscoupled fluid-film damping coefficients C¯ xz⬵C¯ zx reduces as the surface roughness parameter (⌳) reduces in the symmetric configuration (for all the chosen values of C¯ S2) and in the asymmetric configuration (for 0.1 ⬍ C¯ S2 ⬍ 0.225). The variations in the value of crosscoupled fluid-film damping coefficients are comparatively small for an asymmetric configuration because of ¯ o, the asymmetric the fact that for a given external load W

configuration provides a larger fluid-film thickness compared to the symmetric configuration, consequently the effect of surface roughness declines. In a symmetric hole-entry journal bearing configuration (Fig. 12(c)), the values of cross-coupled fluid-film damping coefficients are observed to increase for transversely oriented (for all values of C¯ S2) and longitudinally oriented (for only lower values of C¯ S2) roughness patterns while it reduces for isotropically oriented roughness patterns for all the chosen values of C¯ S2. However, in an asymmetric holeentry journal bearing configuration (Fig. 12(d)), the values of cross-coupled damping coefficients are found to increase for isotropically and longitudinally oriented roughness patterns for all the chosen values of C¯ S2, while

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Fig. 11.

Variation of direct fluid-film damping coefficient (C¯ xx).

transversely oriented roughness patterns show this trend only for lower values of C¯ S2. Further, in the asymmetric configuration, the values of cross-coupled damping coefficients are seen to be positive for longitudinal roughness patterns for all values of C¯ S2. ¯ o and a chosen At a constant value of external load W value of C¯ S2, the value of the direct fluid-film damping coefficient C¯ zz, increases as the surface roughness parameter ⌳ reduces (i.e. as roughness increases) in both symmetric, Fig. 13(a), and asymmetric, Fig. 13(b) configurations during both hydrostatic and hybrid modes of operation. The isotropically (g ⫽ 1) and longitudinally (g ⬎ 1) oriented roughness patterns seems to provide improved values of C¯ zz while the transverse (g ⬍ 1) pattern provides reduced values during hydrostatic and

hybrid modes of operation for both configurations (Fig. 13(c) and (d)). The improvement in the values of C¯ zz is more significant in the case of the longitudinal roughness pattern compared to the isotropic pattern during hydrostatic and hybrid modes of operation in both configurations. ¯ th) 4.6. Threshold speed (w Figure 14 shows the variation of the threshold speed ¯ th with the surface roughness parameter ⌳ and the surw face pattern parameter g. From Fig. 14(a) and (b) it may be noticed that for an isotropically oriented (g ⫽ 1) roughness pattern, the stability threshold speed margin ¯ th increases as the surface roughness parameter (⌳) w

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Fig. 12.

483

Variation of cross-coupled fluid-film damping coefficient.

reduces, i.e. as the bearing surface roughness increases for the chosen values of restrictor design parameter C¯ S2 for both symmetric and asymmetric configurations. The symmetric and asymmetric hole-entry journal bearing configurations having isotropic (g ⫽ 1) and transverse (g ⬍ 1) roughness patterns show an enhanced stability threshold speed margin (Fig. 14(c) and (d)), while bearings with longitudinal (g ⬎ 1) roughness pattern show a reduced stability margin for 0.1 ⬍ C¯ S2 ⬍ 0.3 (in a symmetric configuration) and for 0.05⬍C¯ S2⬍0.3 (in an asymmetric configuration). Further, it may be observed that the improvement in the stability threshold speed margin is more in the bearing configuration having a transversely oriented roughness pattern as compared to the isotropically oriented roughness pattern for the restrictor design parameter C¯ S2 ⬎ 0.1.

From Figs 5–13, the variation in the performance characteristics due to the surface roughness effect is observed to be small for the case of the hybrid mode of operation of bearing configurations. This is due to the ¯ o), the hybrid jourfact that at a constant external load (W nal bearing operates at a smaller eccentricity ratio (i.e. at higher values of h¯ ) due to the generation of hydrodynamic action. Consequently, the value of (⌳.h¯ ) tends to be greater than 3 and as result of this the flow factors fx,fy→1 and fs→0, as shown in Figs 1 and 2 in Patir and Cheng’s paper [12]. Thus, the average Reynold’s equation (Eqn. (7)) used for a rough bearing reduces to the usual form of Reynold’s equation for a smooth bearing and hence the variation in bearing performance characteristics becomes less. In order to have a better physical insight into the effect

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Fig. 13. Variation of direct fluid-film damping coefficient (C¯ zz).

of surface roughness on the performance of a nonrecessed journal bearing, the percentage changes in some selected bearing characteristics parameters with respect to the corresponding similar bearing with a smooth surface have been computed and shown in Table 2. The percentage change in the bearing performance characteristics have been computed for both symmetric and asymmetric configurations for the values of g= 1/6, 1, 6, ⌳ ⫽ 4, b∗ ⫽ 0.5 and e ⫽ 0.6. From Table 2 it may be observed that for a transverse roughness pattern, the asymmetric configuration shows around a 7% increase in the value of the load carrying capacity while for an isotropic roughness pattern, the symmetric configuration shows around a 4% increase during the hydrostatic mode

of operation. However, during the hybrid mode of operation both configurations show around 13% and 8% increases in the value of the load carrying capacity for the longitudinal and the isotropic roughness patterns, respectively. The maximum reduction in the value of ¯ ) is observed to be of the order of 38% bearing flow (Q in an asymmetric configuration with the longitudinal roughness pattern during the hybrid mode of operation. From Table 2 it may also observed that during the hybrid mode of operation, the asymmetric hole-entry journal bearing configuration shows considerable enhancement (around 94%) in the value of the direct stiffness coefficient S¯ zz (with a longitudinal roughness pattern). For both symmetric and asymmetric hole-entry

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Fig. 14.

485

¯ th). Variation of threshold speed (w

journal bearing configurations, the increase in the value of the direct fluid-film damping coefficient C¯ zz is observed to be of the order of 42% during the hydrostatic mode of operation for the longitudinal roughness pattern. Further, the maximum enhancement in the stability ¯ th) is found to be of the order threshold speed margin (w of 41%, 85% and 131% for transverse, isotropic and longitudinal roughness patterns, respectively, for an asymmetric hole-entry journal bearing configuration.

5. Conclusions On the basis of the results presented, the following conclusions have been drawn:

1. For a hole-entry journal bearing system operating in the hydrostatic mode of operation, the transversely oriented roughness pattern provides a higher load carrying capacity as compared to a corresponding similar bearing with a smooth surface, whereas the longitudinally oriented roughness pattern provides a lesser load carrying capacity. ¯ o), for a hole-entry jour2. At a constant external load (W nal bearing system operating in the hybrid mode of operation, the longitudinally oriented roughness pattern gives an enhanced value of the nominal minimum fluid-film thickness while a transversely oriented roughness pattern provides a lesser value. 3. Inclusion of surface roughness effects in the analysis affects the bearing dynamic coefficients. The trans-

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Table 2 Percentage change in performance characteristics of capillary compensated hole- entry journal bearing, l ⫽ 1.0, a¯ b ⫽ 0.25, e ⫽ 0.6, b∗ ⫽ 0.5, V¯ rj ⫽ 0.5, ⌳ ⫽ 4 PCH

⍀

Symmetric configuration

Asymmetric configuration

% Change in rough bearing

F¯ o p¯ max ¯ Q S¯ xx S¯ zz C¯ xx C¯ zz ¯ th w

0.0 1.0 0.0 1.0 0.0 1.0 0.0 1.0 0.0 1.0 0.0 1.0 0.0 1.0 1.0

% Change =

Smooth bearing g ⫽ 1 / 6

g⫽1

1.44412 3.66549 0.91381 2.32183 1.00608 0.89080 2.40642 4.42313 1.64105 5.21126 13.26992 6.88973 21.52445 13.89805 2.31860

4.14 8.73 3.06 8.37 ⫺4.16 ⫺6.16 4.14 15.91 ⫺6.84 69.37 9.17 ⫺10.68 22.03 12.38 79.15

5.91 0.76 ⫺0.57 ⫺2.17 1.86 1.97 5.95 0.37 18.52 22.83 ⫺0.26 2.49 ⫺1.53 1.25 20.86

% Change in rough bearing g⫽6

Smooth bearing g ⫽ 1 / 6

⫺4.17 13.93 5.22 14.40 ⫺10.05 ⫺13.29 ⫺4.18 18.38 ⫺49.04 66.78 17.12 26.54 42.60 31.27 33.03

1.86025 3.77608 0.91532 2.29850 0.37407 0.15276 1.14624 3.80143 1.09110 4.94713 15.38068 6.77529 21.57041 13.03706 2.16784

7.24 0.95 ⫺0.17 ⫺1.97 ⫺1.14 ⫺0.20 7.94 5.62 33.33 32.57 ⫺0.51 ⫺13.44 ⫺1.60 ⫺1.71 41.02

g⫽1 3.62 8.82 2.99 8.33 ⫺8.49 ⫺24.56 ⫺1.92 20.07 ⫺14.37 71.86 9.61 ⫺9.56 22.09 13.03 85.25

g⫽6 ⫺6.64 13.83 4.74 13.92 ⫺14.26 ⫺38.62 ⫺26.11 19.69 ⫺102.04 94.45 19.02 ⫺10.01 42.95 20.88 131.17

PCH|Rough⫺PCH|Smooth × 100, PCH = Performance characteristic. PCH|Smooth

versely oriented roughness pattern gives enhanced values of direct fluid-film stiffness coefficients S¯ xx and S¯ zz for both symmetric and asymmetric holeentry journal bearing configurations operating in the hydrostatic mode of operation. Whereas the longitudinally oriented roughness pattern gives enhanced values of direct fluid-film stiffness coefficients S¯ xx and S¯ zz as well as direct fluid-film damping coefficients C¯ xx and C¯ zz in the hybrid mode of operation. 4. For a hole-entry journal bearing system operating in the hydrostatic/hybrid mode of operation, the isotropically and longitudinally oriented roughness patterns give a reduced value of lubricant flow require¯ ) as compared to a corresponding similar ment (Q smooth surface bearing, whereas the bearing with transversely oriented roughness pattern requires more lubricant flow. 5. The stability threshold speed margin of a hole-entry journal bearing system computed by considering the effect of surface roughness shows an improved performance. In the case of a symmetric configuration, ¯ th is observed to be of the increase in the value of w the order of 79% for an isotropically oriented roughness pattern when the bearing operates at the concentric designed pressure ratio b¯ ∗ ⫽ 0.5 and eccentricity ratio eⱖ0.6. Whereas in the case of an asymmetric ¯ th is configuration, the improvement in the value of w found to be as high as 131% for a longitudinally oriented roughness pattern. 6. The study undertaken in the present work amply dem-

onstrates that, for a non-recessed hole-entry hydrostatic/hybrid journal bearing system, the inclusion of the surface roughness effect in the analysis is more appropriate for an accurate generation of bearing characteristics data. A designer needs to take a judicious selection of parameters such as ⌳(surface roughness parameter), g (surface pattern parameter), C¯ S2 (restrictor design parameter) in conjunction with bearing configurations (symmetric/asymmetric) so as to get an improved bearing performance. References [1] Ai X, Cheng HS, Hua D. A finite element analysis of dynamically loaded journal bearings in mixed lubrication. Trib. Trans 1998;41(2):273–81. [2] Christensen H., Stochastic models for hydrodynamic lubrication of rough surfaces, IMechE, Part 1, 55, vol. 184, pp. 1013-1026, 1969-1970. [3] Christensen H. and Tonder K., The hydrodynamic lubrication of rough journal bearings, Trans. ASME, J. of Lubr. Tech., Series F, April, 166- 170, 1973. [4] Guha SK. Analysis of steady-state characteristics of misaligned hydrodynamic journal bearings with isotropic roughness effect. Trib. Int 2000;33(1):1–12. [5] Hashimoto H. Surface roughness effects in high-speed hydrodynamic journal bearings. Trans. ASME J. of Trib 1997;119:776–80. [6] Jain SC, Sharma SC, Nagaraju T. Misaligned journal effects in liquid hydrostatic non-recessed journal bearings. Wear 1997;210:67–75. [7] Kato T, Obara S. Improvement in dynamic characteristics of circular journal bearings by means of longitudinal microgrooves. Trib. Trans 1996;39(2):462–8.

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