519

Short Paper Proc. of Int. Colloquiums on Computer Electronics Electrical Mechanical and Civil 2011

3D Surface Roughness and Fluid Inertia Effects on Steady State Characteristics of Hydrodynamic Journal Bearing: Model Formulation E. Sujith Prasad1, T. Nagaraju2, J. Prem Sagar2 1

Department of Mechanical Engineering, T.John institute of technology, Bangalore -560083. (India).

Email: sujith35@gmail.com 2 Department of Mechanical Engineering, P.E.S. College of Engineering, Mandya-571401 (India) Email: tnagghally@yahoo.co.in, premsag.2386@gmail.com Majumdar and Hamrock [5] shown that the average Reynolds equation derived by Patir and Cheng for an incompressible lubricant can be used for a compressible lubricant if the surface structures in both cases are same. Tonder [6] proposed a perturbation technique to determine the flow factors. Lunde and Tonder [7] determine the pure pressure flow factors using finite element method. Li et al [8] modified the Patir and Cheng’s flow factors by using perturbation approach with Green’s function technique for Non-Newtonian lubricant. Nagaraju et al (9) developed a modified average Reynolds equation to include viscosity variation across and along fluid-film of rough bearings and investigated the combined influence of surface roughness and non-Newtonian behavior of the lubricant on hybrid journal bearing performance. However, all the above mentioned models have been confined to neglect the fluid inertia. Among the few studies related to the fluid inertia effect, Constantinescu and Galetuse [10] evaluated the momentum equations for laminar and turbulent flows by assuming that the shape of the velocity profiles is not strongly affected by the presence of inertia forces. Banerjee et al (11) introduced an extended form of the Reynolds equation to include the effect of fluid inertia, adopting an iteration scheme.

Tichy and Bou-Said [12], Bou-Said and Ehret [13] and Kakoty and Majumdar [14, 15] used the method of averaged inertia in which the inertia terms are integrated over the film thickness to account for the inertia effect in their studies. The above studies [10-15] were mainly based on ideally smooth bearing and journal surfaces. To the best of the authors knowledge, no mathematical models have been developed to include the surface roughness effect along with fluid inertia effect in the analysis of journal bearing systems. In the present work, a modified average Reynolds equation and a solution algorithm are developed to include surface roughness and fluid inertia effects in the analysis of lubrication problems. The developed model is used to study the influence of fluid inertia and surface roughness effects on the steady state characteristics such as circumferential pressure, load carrying capacity and stability threshold speed of a hydrodynamic journal bearing system.

Abstract— In the present work, a modified form of average Reynolds equation is developed to include the combined influence of 3D surface roughness and fluid inertia in the analysis of journal bearings. The modified average Reynolds equation is derived starting from the Navier-Stokes equations and flow continuity equation using Patir and Cheng’s (2, 3) flow factors. The local velocity components are expressed in terms of Patir and Cheng’s flow factors and the velocity profiles are assumed to be change very little due to fluid inertia. The developed mathematical model and solution algorithm are expected to be quite useful to the bearing designer.

Index Terms— Convective inertia, Surface roughness, Hydrodynamic lubrication.

I. INTRODUCTION Ever-increasing demand for the hydrodynamic journal bearing systems to operate under high speed and high eccentricity makes it imperative to design this class of bearings accurately. In such cases, the familiar assumptions of a smooth surface can no longer be employed to accurately predict the performance of journal bearing systems as no machining surfaces are perfectly smooth. When the bearings operating under high speed and low viscosity lubricants, the flow may become laminar but fluid inertia forces cannot be neglected. Therefore, it is imperative to include the influence of surface roughness and fluid inertia effects in the design and analysis of journal bearings. From past several years, several concepts and mathematical models have been developed by the various researchers to incorporate the surface roughness effects in the lubrication problems. Notably among these researchers, Christensen and Tonder [1] developed the average Reynolds equation to analyze the effects of surface roughness on the lubrication problems but this model is limited to one-dimensional roughness structures oriented either transversely or longitudinally and is not applicable to the 3D area distributed surface roughness. Later, Patir and Cheng [2, 3] introduced a new concept of flow factors for deriving an average Reynolds equation applicable for any general roughness structure and derived a n average Reynolds equation to include 3d surface roughness effects in the analysis. Teale and Lebeck [4] evaluated the Patir and Cheng’s flow model and derived some flow factors for truncated rough surfaces. These flow factors helps to study the effects of partially worn surface on lubrication problems. © 2011 AMAE DOI: 02.CEMC.2011.01. 519

II. ANALYSIS A. Average fluid-film thickness The geometry of the journal bearing system along with rough surface and co-ordinate system is shown in Fig.1 172

Short Paper Proc. of Int. Colloquiums on Computer Electronics Electrical Mechanical and Civil 2011 The local fluid-film thickness hL in non-dimensional form is

I xx 

expressed as hL  h   The non-dimensional form of average fluid-film thickness hT , which is equal to the expected or mean value of local fluidfilm thickness, is expressed as

 ( h   ) ( )d

I xz 

I yy 

hL

0

u LvL dz and

hL 2 vL dz 0



(7c)

The velocity profile remains parabolic even after fluid inertia is considered (Constantinescu and Galetuse [10]). Then the local velocity profiles and shear stress components are expressed as [14,15]



hT  E[ h   ] 

hL 2 u L dz ;

0

(1a)



Where  ( ) is the probability density function of combined roughness  . For Gaussian distribution of surface heights it is expressed as

 ( ) 

1 2 

2 2 e  2

(1b)

For the fully lubricated region (i.e. for h  3 ) and partially lubricated region (i.e. for h  3 ) of the bearing, the (1a) can be expressed as (Nagaraju et al [16]). h  hT   2  h

   1  erf   h  2   

 1  ( h ) 2 2      2 e 

for  h  3

(2) for  h  3

Where  (  1  ) in the present work is defined as surface roughness parameter and h is the nominal fluid-film thickness. It is same as the fluid-film thickness obtained for a smooth surface journal bearing and is expressed as

h  1  X J cos   Z J sin 

(3)

Fig. 1 Bearing geometry and surface profile

B. Derivation of Modified Average Reynolds Equation To include the fluid film inertia forces, the average Reynolds equation is derived starting from the Navier-Stokes equations and flow continuity equation using Patir and Cheng’s [2,3] flow factors in the following manner. At any local fluid-film thickness, h L as shown in surface profile (Fig.1), the averaged form of momentum and continuity equations, after integration over film thickness, can be written as I xy  q I   x  xx  x y  t

 p L z h   h L   zx z 0L  x 

 q y I xy I yy  P    hL L   zy       t  x  y y  

q x q y hT   0 x y t

z  hL z 0

qx 

 zx  

0

u L dz ; q y 

u L vL ;  zy   z z

© 2011 AMAE DOI: 02.CEMC.2011.01. 519



hL

0

v L dz

(8a)

 z2 z  vL  6Vm  2   h   L hL  h

 zx 0L 

12U m hL

(8b) hL

and  zy 0 

12Vm hL

(8c)

(4)

Where U m and V m are the mean velocities due to pressure induced flow in x and y directions and are expressed as

(5)

Um 

(6)

and  zy

hL2 pL hL2 p L and Vm  Substitution for 12 x 12 y hL 0

h

 zx 0L

from (8c) into momentum equations (4) and (5)

and proper simplification yields

Where hL

 z2 z  Uz u L  6U m  2   h   L hL  hL

Um 

(7a)

Vm 

(7b)

173

hL2 pL hL  12  x 12 

 q x I xx I xy     t  x  y   

hL2 p L hL  q y I xy I yy     12  y 12   t x y

   

(9a)

(9b)

Short Paper Proc. of Int. Colloquiums on Computer Electronics Electrical Mechanical and Civil 2011 C. Inertia Functions

The integration of respective velocity components 8(a) and 8(b) across the local film thickness and substitution for U m and Vm from 9(a) and 9(b) yields the local fluid flow per unite width of bearing surface in the x and y direction as h 3 p L hL2 qx   L  Gx   UhL 12 x 12 2

Similar to the mean flow q x and q y , the mean or expected velocity components can also be obtained by modifying the Poiseuille and Couette terms in the expression of local velocity components 8(a) and 8(b) using flow factors and are expressed in non-dimensional form as

(10a)

h3 pL hL2 qy   L  Gy 12 x 12

 

u  6U m ( z 2  z )  z 

(10b)

q y I xy I yy q x I xx I xy   and G y  t  x  y t x y

(14a)

q x   x

h3 p UhT U s hT2    G x  12  x 2 2 12 

q y   y

h 3 p hT2  Gy 12 y 12 

 

U m  x

Vm   y

(11b)



h 2 p 12 



Re* hT Gx 12

(15a)



Re* hT G y 12

(15b)

 6   U m  U m        h   U m   5 T  2  2         1  1  s     3   h2    T   

 2  2   s      6    Vm 1    U m  s  hT  U m     s    2  2     3   hT       5  6   h 6  U m      Vm  U m     T   Vm hT   Vm  s     5    2    5

(16a)

(12)

 hT Re*   3  3 hT G x  hT G y  t 12    



12 

12   G x   U m hT   hT  s   5

 h 3 p   hT   s  y    12   2  2   



h 2 p

Now, the local velocities in (7) can be replaced by mean velocities and inertia functions are obtained from (10c).

Neglecting time derivative terms for steady state analysis, they can be expressed as

(11a)

Substitution of (11a) and (11b) into the continuity equation (6), after considerable manipulation, yields the modified average Reynolds equation in terms of flow factors, average fluid-film thickness and convective inertia function. It can be expressed in non-dimensional form as   h 3 p   x    12   

(14b)

Similarly, the expected values of pressure induced mean velocity components are by modifying 9(a) and 9(b) and are expressed as

(10c)

The functions Gx and G y are having temporal (time dependent) and convective inertia terms. In steady state analysis of journal bearings all time dependent terms are set equal to zero. The expected or mean flow per unite width can be obtained in the same line of Patir and Cheng [2,3] using the flow factors and are expressed as



s z hT

v  6 yVm ( z 2  z )

Where, Gx 

 



 6  V  6      h G y   hT  U m     s  m  Vm  U m    T 2  2      5 2     5  U    V 6    12  6  h m     V m hT   Vm  s   Vm hT  m   V m2  T 5    2    5   5  

The pressure flow factors (  x ,  y ) in (12) are expressed as

(16b)

[2] III. SOLUTION PROCEDURE

 x  1  Ce r h for   1

 r for  >1  y  h,     x  h, 1 

Initially, the pressure distribution and mean velocity components (15) are obtained by solving modified average Reynolds equation (12) for inertia-less solution. Then, the inertia functions are updated using (16). and Reynolds equation is solved iteratively using these updated inertia functions. The iterative process is terminated when the difference in nodal pressure in the successive iteration becomes less than 0.1%. Once the pressure field is converged, the load carrying capacity and stability threshold speed of the bearing are computed using relevant expressions given in Nagaraju et al [16].

x  1  C h

The shear flow factor

(13a)

 s  is expressed as [3]





 s  2V rj  1  s

(13b)

Where

  e h  h 

 s  A1  h

1

2

2

3

 s  A2e  0.25 h  for

© 2011 AMAE DOI: 02.CEMC.2011.01. 519

for h  5

 h >5

IV. RESULTS AND DISCUSSIONS

(13c)

The modified average Reynolds equation and a computer code developed in the present work is used to study the 174

Short Paper Proc. of Int. Colloquiums on Computer Electronics Electrical Mechanical and Civil 2011 influence of surface roughness and fluid inertia effects on circumferential press distribution, load carrying capacity and stability threshold speed of a hydrodynamic journal bearing.

The validity of the results obtained in the present work has been established by computing the load carrying capacity of smooth and rough bearings using the geometric and effect operating conditions of test hole-entry hybrid journal bearing used by Xu [19] with and without considering fluid inertia effect and compared with his experimental result. The load capacity of rough bearing that computed with fluid inertia was found to be in good agreement with the experimental result as shown in Fig.2. The significance of the modified average Reynolds equation is verified by computing numerical results for the circumferential pressure distribution, load carrying capacity and stability threshold speed of the bearing.

These results are computed for the geometric and operating conditions shown in each graph. In the present work the roughness is considered on either journal surface or bearing surface. The former case is termed as

Fig. 3 Circumferential pressure distribution

As seen from Fig. 4, the load carrying capacity ( Fo ) of both smooth and rough bearings increases due to fluid inertia.

The stationary roughness provides enhanced load carrying capacity than that of a smooth bearing for entire range of eccentricity ratio while moving roughness shows opposing trend. From Fig. 5, it can be seen that the stability threshold speed (  th ) of both smooth and rough bearings decreases due to fluid inertia. The moving roughness which was observed to provide reduced load carrying capacity is providing enhanced stability threshold speed while the stationary roughness which was providing enhanced load capacity is observed to provide reduced stability threshold speed. The stability threshold speed is also observed to be reduces as eccentricity ratio increases in smooth bearing and stationary type rough bearing while it is slightly increases in a bearing having moving type roughness.

Fig. 2 Load carrying capacity vs eccentricity ratio

moving roughness ( V rj  1 ) and latter case is termed as stationary roughness ( V rj  0 ) Figure 3 shows the influence of fluid inertia and roughness characteristics of opposing surfaces (i.e. variance ratio, Vrj ) on circumferential fluid-film pressure distribution. As seen from Fig.3, the circumferential pressure of both smooth and rough bearings slightly increases due to fluid inertia. The stationary roughness (i.e. rough bearing and smooth journal, Vrj  0 ) provides the enhanced fluidfilm pressure as compared to smooth bearing while the moving roughness (i.e. rough journal and smooth bearing,

Vrj  1 ) provides reduced fluid-film pressure. The influence

Fig. 4 load carrying capacity of bearings

of fluid inertia is seen to be insignificant in moving roughness as compared to smooth and stationary rough bearings.

© 2011 AMAE DOI: 02.CEMC.2011.01. 519

175

Short Paper Proc. of Int. Colloquiums on Computer Electronics Electrical Mechanical and Civil 2011

( j , b )  2 (X J , ZJ )

= Journal center coordinates, ( X J , Z J ) c

 = Circumferential coordinate, ( X R J )  = Axial coordinate, ( Y R J )

0.5 x  = Surface pattern parameter,  0.5 y

 = Combined roughness height, (    J   b ),  m

 =  /c  = Eccentricity ratio, e c  = Surface roughness parameter, (c  )

Fig. 5 Stability threshold speed of bearing

 = Aspect ratio, L D

CONCLUSIONS

 0.5 x, y = 0.5 correlation lengths of the x and y profile,

A modified average Reynolds equation developed in this paper can be used to study the effects of surface roughness parameters such as asperity height distribution, roughness orientation and roughness characteristics of opposing surfaces considering fluid inertia effects. The bearing configurations which having moving type roughness or stationary type roughness shows opposite trend between load carrying capacity and stability threshold speed.

 = Dynamic viscosity of lubricant, N.sec.m-2  = Density of lubricant, Kg. m-3

 = RMS value of combined roughness,  2   2 , m J b



2 2  = Speed parameter,  J  r R J c p s



 0

D = Journal diameter, mm e = Journal eccentricity, mm

Subscripts and Superscripts b = Bearing J = Journal = Non-dimensional parameter

Fo = Fluid film reaction, Fo p s R J2

h = Nominal fluid-film thickness, mm hL = Local fluid-film Thickness, mm

REFERENCES

hT = Average fluid-film thickness, mm

[1] Christensen H. and Tonder K, “The Hydrodynamic Lubrication of Rough Bearing Surfaces of Finite Width”, ASME Jour. of Lubr.

Tech., vol. 93(3), pp. 324-330, 1971.

[2] Patir N. and Cheng H. S., “An Average Flow Model for Determining Effect of Three-Dimensional Roughness on Partial Hydrodynamic Lubrication”, ASME Jr. of Lub. Tech., vol. 100, pp.

12-17, 1978.

[3] Patir N. and Cheng H. S., “Application of Average Flow Model to Lubrication between Rough Sliding Surfaces”, ASME Jr. of Lub. Tech., vol. 101, pp. 220-230, 1979.

[4] Teale.J.N. and Lebeck.A.O., “An Evaluation of the Average Flow Model [2] for Surface Roughness Effects in Lubrication”, ASME Jour. of Tribo. Trans., vol. 102, pp. 360-367, July 1980.

[5] Majumdar.B.C and Hamrock.B.J. “Extenstion of the PatirCheng Flow Simulation of a Rough Surface Bearing to a Compressible Lubricant”. Jour.Mech.Engg.Sc. vol. 24(4), pp. 209-214, 1982.

[6] Kristian Tonder. “The Hydrodynamic Lubrication of Rough Surfaces Based on a new perturbation Approach”. ASME Jour. of Tribo. Trans., vol. 106, pp. 440-447, October 1984.

[7] Lunde.L and Tonder.K. “Pressure and Shear Flow in a Rough Hydrodynamic Bearing, Flow Factor Calculation”. ASME Jour. of Tribo. Trans., vol. 119, pp. 549-555, July 1997.

= (h, hL , hT ) c

R J = Radius of journal, mm

p = Pressure, ( p) p s p s = Supply pressure, N.mm-2

t

c 2 j 

= Time,  j t

2 u , v = Fluid velocity components, (u , v )( rRJ c p s )

 R 

j   U m , Vm = Mean velocity components, U m , Vm  2   c ps 

(V rj , Vrb )

= Variance ratio of journal and bearing,

© 2011 AMAE DOI: 02.CEMC.2011.01. 519

= Shear stress,  R J cp s 

2 erf (x ) = Error function, 2  exp( y )dy

c = Radial clearance, mm

Re* = Modified Reynolds number,



x

NOMENCLATURE

h , h L , hT

m

176

Short Paper Proc. of Int. Colloquiums on Computer Electronics Electrical Mechanical and Civil 2011 [8] Li, W.-L., Weng, C.-I. and Hwang.C.C. “An Average Reynolds Equation for Non-Newtonian Fluid with Application to the Lubrication of the Magnetic-Head-Disk Interface”, STLE Tribol.Trans., vol. 40(1), pp. 111-119, 1997.

[9] Nagaraju T., Sharma Satish C. and Jain S. C., “Performance of Externally Pressurized Non-Recessed Roughened Journal Bearing System Operating with Non-Newtonian Lubricant”, STLE Tribo.

Trans., vol. 46(3), pp.404-413, 2003.

[10] Constantinescu V. N. and Galetuse S., “On the Possibilities of Improving the Accuracy of the Evaluation of Inertia Forces in Laminar and Turbulent Films”, ASME Jr. of Lub. Tech., vol. 96(1), pp. 69-79, 1974.

[11] Banerjee, M. B., Shandil, R. G., Katyal, S. P., Dube, G. S., Pal, T. S., and Banerjee, K. “A Nonlinear theory of hydrodynamic lubrication”. J. Math. Analysis applic., Vol 117, pp 48-56, 1986.

© 2011 AMAE DOI: 02.CEMC.2011.01. 519

[12] Tichy J. and Bou-Said B., “Hydrodynamic Lubrication and Bearing Behavior with Impulsive loads”. STLE Tribo. Trans., vol.

34(4), pp. 505-512, 1991.

[13] Bou-Said B. and Ehret P., “Inertia and Shear-Thinning effects on Bearing Behavior with Impulsive Loads”, ASME Jr. of Trib., vol.

116, pp. 535-540, 1994.

[14] Kakoty S. K. and Majumdar B. C., “Effect of Fluid Inertia on Stability of Oil Journal Bearing”. ASME Jr. of Tribo, Vol 122, pp 741-745, October 2000.

[15] Kakoty S. K. and Majumdar B. C., “Effect of Fluid Inertia on the Dynamic Coefficients and stability Oil Journal Bearings”. Pro. of Instn. of Mech. Engirs., Vol 214, pp 229-242, 2000.M. Young, The Technical Writer’s Handbook. Mill Valley, CA: University Science, 1989.

[16] Nagaraju T., Sharma Satish C. and Jain S. C., “Influence of Surface Roughness Effects on the Performance of Non-Recessed Hybrid Journal Bearings”, Tribo. Int., vol. 35(7), pp.467-487, 2002.

[17] Xu S., “Experimental Investigation of Hybrid Bearings”, STLE Tribo. Trans., vol. 37(2), pp.285-292, 1994.

177