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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: Performance Analysis E. Sujith Prasad1, T. Nagaraju2 and Yashavanth Kumar K. H2 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, yashu_kh@yahoo.co.in

Abstract— In the present work, the influence of surface roughness and fluid inertia effects on steady state characteristics such as load carrying capacity and stability threshold speed of journal bearing systems is studied using a modified form of average Reynolds equation developed in the authors previous paper [19]. Solution of the modified Reynolds equation is obtained using finite element method. The effects of the roughness height, orientation and characteristics of surface roughness of opposing surfaces on steady-state characteristics of journal bearing systems are studied including fluid inertia effect. The developed mathematical models, solution algorithm and the results obtained in this work are expected to be quite useful to the bearing designers.

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. Based on this concept, Hashimoto [4] studied the combined influence of surface roughness and turbulent flow of lubricant and Ramesh, et al. [5] studied the combined influence of roughness and thermal effects on the performance of hydrodynamic journal bearings. Nagaraju, et al. [6,7] studied the individual and combined influence of surface roughness and bearing flexibility effects on the steady state performance characteristics of hybrid journal bearing. Later, Nagaraju et al [8] 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 nonNewtonian behavior of the lubricant on hybrid journal bearing performance. Sharma et al [9] extend this study to predict the combined influence of surface roughness and thermal effects. Wang et al [10] investigated the combined influence of bearing shell deformation due to hydrodynamic and contact pressure as well as thermal deformation on the bearing performance. Li et al [11] studied the effect of surface roughness on the static performance of hydrodynamic journal bearing operating with non-Newtonian lubricant. However, none of the above mentioned studies considered the influence of fluid film inertia in the analysis of 3D surface roughness of hydrodynamic journal bearing systems. Among the few studies related to the fluid inertia effect, Constantinescu and Galetuse [12] 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 [13] introduced an extended form of the Reynolds equation to include the effect of fluid inertia, adopting an iteration scheme. Chen and Chen [14] obtained the steady state characteristics of finite bearings including the inertia effect. Tichy and Bou-Said [15], Bou-Said and Ehret [16] and Kakoty and Majumdar [17, 18] 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 [11-18] were mainly based on ideally smooth bearing and journal surfaces. In the present work, a modified average Reynolds equation developed in the authors previous paper [19] is used to study the influence of surface roughness and fluid inertia effects on the load carrying capacity and threshold speed of the journal bearing.

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

I. INTRODUCTION From past several years, the incorporation of many physical effects into the analysis of fluid-film bearings has provided much more realistic performance data. In particular, the familiar assumptions of a smooth surface can no longer be employed to accurately predict the performance characteristics of journal bearing systems, since no machining surfaces are perfectly smooth. The fluid-film thickness in the journal bearing systems is only of the order of few micrometers and the height of the surface roughness asperities is generally of the same order as the mean separation of the bearing surfaces. This means that considerable deviations in the nominal fluid-film thickness does exist under these conditions and consequently the performance characteristics of journal bearing systems gets affected by the surface roughness considerably. Further, in the bearings operating under high speed and low viscosity lubricants, it is possible to arrive at a situation where flow is laminar but fluid inertia forces cannot be neglected. In view of the above, it is imperative to include the combined influence of surface roughness and fluid inertia effects in the analysis of journal bearing systems. From last few decades, considerable amount of research works related to the effects of surface roughness on the performance of journal bearing systems have been reported in the literature. 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 © 2011 AMAE DOI: 02.CEMC.2011.01. 520

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Short Paper Proc. of Int. Colloquiums on Computer Electronics Electrical Mechanical and Civil 2011 The influence of fluid inertia and surface roughness parameters such as asperity height distribution, roughness orientation and roughness characteristics of opposing surfaces on the steady state characteristics of a hydrodynamic journal bearing system is studied.

 r

for  >1

 x  1  C h





 y  h,    x  h, 1



(4a)

The shear flow factor s  is expressed as [3]

II. ANALYSIS.





 s  2V rj  1  s A. Average fluid-film thickness ( hT )

Where

The geometry of the journal bearing system along with rough surface and co-ordinate system is shown in Fig.1.

 s  A1  h

 

(4b)

   2

e  2  h  3  h for h  5

For the fully lubricated region (i.e. for h  3 ) and partially

 s  A2 e  0.25 h 

lubricated region (i.e. for h  3 ) of the bearing, the (1) can be expressed as (Nagaraju et al [6]) h  2  h   1    e ( h ) 2  1  erf   hT   2   2    2  h 

for  h >5 (4c) The pressure induced mean velocity components are

for h  3

U m  x

(1) for h  3

where ( 1  ) in the present work is defined as surface

Vm   y

roughness parameter and h is the nominal fluid-film thickness and is expressed as

h  1  X J cos   Z J sin 

h 2 p 12  h 2 p 12 

Re* hT Gx 12

(5a)

Re* hT G y 12

(5b)



The inertia function G x and G y appeared in (3) are expressed as

(2)

 6   U m  U m       h 12 s  U m   5  T G x   U m hT   hT    2 2      5   1  1 s     3   hT2       2  2    s   6  s       Vm   1  h    U m    hT  5 U m  2   2 s    3      T 

(6a)

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

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

III. SOLUTION PROCEDURE

Fig. 1: Bearing geometry and surface profile

Initially, the pressure distribution and pressure induced mean velocity components (5) are obtained by solving modified average Reynolds equation (3) for inertia-less solution. Then, the inertia functions are updated using (6) 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%. Using these converged pressure fields, the load carrying capacity and stability threshold speed are computed using relevant expressions given in Nagaraju et al [6].

B. Derivation of Modified Average Reynolds Equation The modified average Reynolds equation derived in the previous paper [19] is expressed as  

 h 3 p    h 3 p       x 12     y 12      

 2      

R* hT h   s   T - e  2  t 12   hT3G x  h 3G y   T 







(3)



IV.RESULTS AND DISCUSSIONS Steady state analysis of a hydrodynamic journal bearing system is conducted numerically using modified from of the average Reynolds equation developed in the present work. The validity of the results obtained from the modified average Reynolds equation and corresponding computer code

The pressure flow factors (  x ,  y ) are expressed as [2]

(6b)

for   1

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Short Paper Proc. of Int. Colloquiums on Computer Electronics Electrical Mechanical and Civil 2011 developed in the present work has been established by computing the load carrying capacity of smooth bearing and rough bearing with transverse roughness pattern (   1 / 6 ) using the geometric and operating conditions of Xu’s test hole-entry hybrid journal bearing (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 effect was found to be in good agreement with the experimental result as shown in Fig.2.

that having two-sided roughness, Vrj  0.5 (both journal and bearing rough). The load carrying capacity of the bearing increases when the inertia effect is considered in both smooth and rough bearings. The value of load carrying capacity of a bearing that having two-sided roughness increases as the surface roughness parameter () reduces (i.e. as combined roughness height increases) for transverse (=1/6), isotropic (=1) and longitudinal (=6) roughness patterns irrespective of the consideration of fluid inertia effect.

Fig. 3 Load carrying capacity Fig. 2 Load carrying capacity vs eccentricity TABLE 1 BEARING GEOMETRIC AND OPERATING PARAMETERS

Figure 4 show the effects of surface roughness parameter, roughness orientation and roughness characteristics of opposing surfaces on the load carrying capacity of square (   1 ), short (   0.5 ) and long (   1.5 ) bearings. As seen from Figs. 4(a) and 4(c), higher load carrying capacity is observed in the stationary roughness, V rj  0 (i.e. rough bearing and smooth journal) case for transverse, and longitudinal roughness patterns as compared to two-sided ( Vrj  0.5 ) and moving roughness, Vrj  1 (i.e. smooth bearing and rough journal) cases in a square as well as long bearings. For the stationary roughness case, the transverse roughness pattern () is seen to provide the largest

The results showing the effects of surface roughness and fluid inertia on load carrying capacity ( Fo ) and stability threshold speed ( th ) of hydrodynamic journal bearing are computed and presented for the generally used geometric and operating parameters of a typical bearing listed in Table 1. These results are presented in Figs. 3 through 5. The corresponding results for smooth bearing are also shown in each graph by a horizontal line marked with ‘s’ for the sake of comparison. The following paragraphs detail the influence of roughness orientations (  ) and variance ratio ( Vrj ) on the above mentioned bearing performance characteristic parameters. Fig. 3 shows the effects of surface roughness parameter (  ), roughness orientation (  ) and fluid film inertia on the load arrying capacity ( Fo ) of a square bearing,   1 180 © 2011 AMAE DOI: 02.CEMC.2011.01. 520

enhancement in the value of Fo in a square bearing and long bearing. This is due to the fact that, the circumferential flow is dominant in square and long bearings and the stationary roughness case produces resistance to the velocity induced flow of the lubricant in circumferential at the converging section of the bearing clearance. Consequently the fluid-film pressure at the converging section increases and hence the stationary roughness provides higher load carrying capacity. As the transverse roughness pattern that having longer asperities in axial direction restrict more lubricant than that of a longitudinal roughness, it provides maximum enhancement. On the other hand, the moving roughness case scrapes the lubricant at the converging section and increase the dominant circumferential flow and hence the moving roughness, especially with transverse roughness pattern shows reduced value of load carrying capacity than that of a corresponding similar smooth bearing. However, in a short bearing the axial flow is dominant. Hence the longitudinal roughness pattern with longer asperities in circumferential direction restricts the axial flow and provides maximum enhancement in the value of load carrying capacity as shown in Fig. 4(b). For a moving roughness case, the

Short Paper Proc. of Int. Colloquiums on Computer Electronics Electrical Mechanical and Civil 2011 and stationary roughness on stability threshold speed are exactly opposite to that of a load carrying capacity of the bearing as seen from Figs. 4.

transverse roughness pattern provides reduced value of Fo as compared to a corresponding smooth bearing for square, short and long bearing systems. As there is no additional flow transport for a two-sided roughness case, the improvement in the load carrying capacity from transverse and longitudinal roughness patterns is mainly attributed to the restriction on dominant pressure induced flow imposed by the roughness patterns. Figure 5 show the effects of surface roughness parameter, roughness orientation and roughness characteristics of opposing surfaces on stability threshold speed of square (   1 ), short (   0.5 ) and long (   1.5 ) bearings. From Figs. 5(a) to 5(c), it can be seen that the moving roughness with transverse roughness pattern provided maximum enhancement in the value of stability threshold speed than that of a corresponding smooth bearing in all i.e. square, short and long bearings.

Fig. 5 Stability threshold speed (a) square bearing (b) short bearing (c) Long bearing

CONCLUSIONS From the results presented in the previous section, the following conclusions are draw. 1. For both inertia less and inertia analysis of a square bearing, the longitudinal roughness provides maximum enhancement in the value of load carrying capacity as compared to corresponding smooth bearing when both journal and bearing surfaces are rough i.e in two-sided roughness. 2. The influence of roughness orientation on load carrying capacity of the bearing is mainly depends on the bearing

Fig. 4 Load carrying capacity (a) square bearing (b) short bearing (c) Long bearing

The stationary roughness with transverse pattern provides reduced stability threshold speed than that of the corresponding smooth bearing. These trends of the moving © 2011 AMAE DOI: 02.CEMC.2011.01. 520

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



land width ratio, . 3. For square and long bearings, the transverse type of roughness pattern in stationary roughness case provides maximum enhancement in the value of load carrying capacity. In short bearing, the longitudinal type roughness pattern in stationary roughness shows this trend. 4. Irrespective of the land width ratio of the bearing, the moving roughness with transverse type roughness pattern provided reduced load carrying capacity than that of a corresponding smooth bearing. However, it provides maximum enhancement in the value of stability threshold speed.

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



REFERENCES [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] Hashimoto, H. “Surface roughness effects in high speed hydrodynamic journal bearings”. ASME Jr. of Trib., vol. 119, pp. 776-780, 1997. [5] Ramesh J., Majumdar B. C. and Rao N. S., “Thermohydrodynamic Analysis of Submerged Oil Journal Bearing Considering Surface Roughness Effects”, ASME Jr. of Trib., vol. 119, pp. 100-106, 1997. [6] 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. [7] Nagaraju T., Sharma Satish C. and Jain S. C., “Study of Orifice Compensated Hole-Entry Hybrid Journal Bearing Considering Combined Influence of Surface Roughness and Flexibility Effects”, Tribo. Int., vol. 39, pp. 715-725, 2006. [8] 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. [9] Sharma Satish C., Nagaraju T. and Jain S. C., “Performance of Orifice Compensated Hole-Entry Hybrid Journal Bearing System Considering Surface Roughness and Thermal Effects”, STLE Tribo. Trans., vol. 47(4), pp.557-566, 2004. [10] Wang Q, Shi F. and Lee S. C., “A Mixed-TEHD Model for Journal Bearing Conformal Contacts – Part II: Contact, Film Thickness and Performance Analysis”, ASME Jr. of Trib., vol. 120, pp. 206-213, 1998. [11] Li W.L., Weng C.I. and Lue J.I., “ Surface Roughness Effects in Journal Bearings with Non-Newtonian Lubricants”, Trib. Trans., vol. 39(4), pp. 819-826, 1996. [12] 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. [13] 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. [14] Chen, C. H., Chen, C. K. “The influence of fluid inertia on the operating characteristics of finite journal bearings Wear”, Vol. 131, pp 229-240, 1989. [15] Tichy J. and Bou-Said B., “Hydrodynamic Lubrication and Bearing Behavior with Impulsive loads”. STLE Tribo. Trans., vol. 34(4), pp. 505-512, 1991. [16] Bou-Said B. and Ehret P., “Inertia and Shea-Thinning effects on Bearing Behavior with Impulsive Loads”, ASME Jr. of Trib., vol. 116, pp. 535-540, 1994.

= Radial clearance, mm

Fo = Fluid film reaction, Fo p s R J2

h = Nominal fluid-film thickness, mm hT = Average fluid-film thickness, mm

h , hT

= (h, hT ) c

R J = Radius of journal, mm

p = Pressure, ( p) p s p s = Supply pressure, N.mm-2 Re* = Modified Reynolds number, t

c 2 j 

= Time,  j t

(V rj , Vrb ) = Variance ratio of journal and bearing,

( j , b )  2  = 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  )

 = Aspect ratio, L D

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

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

= Shear stress,  R J cp s 

erf (x ) = Error function,

NOMENCLATURE



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Short Paper Proc. of Int. Colloquiums on Computer Electronics Electrical Mechanical and Civil 2011 [17] 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. [18] 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.

[19] Sujith Prasad E., Nagaraju T. and Prem Sagar J., “3D Surface Roughness and Fluid Inertia Effects on Steady State Characteristics of Hydrodynamic Journal Bearing: Model Formulation”, Proceedings of Int. Joint Colloquiums, CEMC-2011 Paper ID CEMC-ME519, Muvatupuzha, Kerala, India, Sept. 20-21, 2011 (Accepted) [20] Xu S., “Experimental Investigation of Hybrid Bearings”, STLE Tribo. Trans., vol. 37(2), pp.285-292, 1994.

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