ANNALS of Faculty Engineering Huneoara International Journal of Engineering Tome XIV [2016] Fascicule 2 [May] ISSN: 1584-2665 [print; online] ISSN: 1584-2673 [CD-Rom; online] a free-accessmultiisciplinarypublication of thefaculty of Engineering Huneoara 1.Girishkumar C. PANCHAL, 2. Himanshu C. PATEL, 3. G.M. DEHERI INFLUENCE OF SURFACE ROUGHNESS THROUGH A SERIES OF FLOW FACTORS ON THE PERFORMANCE OF A LONGITUDINALLY ROUGH FINITE SLIDER BEARING 1. Department of Mathematics, Government Engineering College, Ganhinagar, Gujarat State, INDIA 2. Gujarat University, Ahmeaba, Gujarat State, INDIA 3. Department of Mathematics, Saraar Patel University, Vallabh Viyanagar, Gujarat State, INDIA ABSTACT: Efforts have been mae to analyse the influence of roughness parameters on the pressure an loa carrying capacity in a rough finite plane slier bearing for longituinally rough surfaces by taking account of the influence of surface roughness through a series of flow factors which has been introuce by Patir (1978). The associate stochastically average Reynols type equation is solve with appropriate bounary conitions. Expressions are obtaine for pressure an loa carrying capacity numerically. The results are presente graphically. In aition it is easily seen that the increment in the measure of longituinal roughness causes the ecrease in loa carrying capacity of the bearing. Keywors: average Reynols equation, finite slier bearing, flow factor, longituinal roughness 1.INTRODUCTION The stuy of roughness effect is very important in a bearing system. It is well known that the bearing surfaces after having some run-in an wear evelop roughness. There are many stuies ealing with their investigations were confine to slier an journal bearings various film shapes by Pinkus (1961) an Hamrock (1994). The effect of surface roughness was iscusse by many investigators viz. Davies (1963), Burton (1963), Michell (1950) an Toner (1972). It has gaine an increasing attention after the introuction of stochastic concept an Stochastic Reynols equation by Christensen (1969,1972), Christensen an Toner (1971,1972) governing the mean pressure in bearings having transverse an longituinal roughness. Christensen an Toner s approach forme the base of the analysis to stuy the effect of surface roughness in a number of investigationsby Prakash an Tiwari (1982), Guha (1993), Gupta an Deheri (1996), Anharia et al.(1997). The surface roughness effects on the ynamic characteristics of slier bearing with finite with were theoretically stuie by Chiang, Hsiu-Lu, et al.(2005)an observe that the steay loa-carrying capacity, ynamic stiffness an amping coefficient were increase as the effects of transverse roughness increase while the influences of the isotropic an longituinal roughness ha a reverse tenency.the effect of surface roughness on the performance of hyroynamic slier bearings was stuie by Anharia et al.(2001).the effect of longituinal roughness on magnetic flui base squeeze film was stuie by Anharia an Deheri (2010). The effect of longituinal surface roughness on the behaviour of slier bearing with squeeze film forme by a magnetic flui was analyse by Deheri et al.(2004).the effect of surface roughness on the performance of a magnetic flui base parallel plate porous slier bearing was observe by Patel an Deheri (2011). Slip velocity an roughness effect on magnetic flui base infinitely long bearings was analyse by Patel et al. (2014). Patir an Cheng (1978) moifie the averagereynols equation for rough surfaces. They efine pressure an shear flow factors, 227 Fascicule 2
ANNALS of Faculty Engineering Huneoara International Journal of Engineering which were obtaine inepenently by numerical flow simulation using ranomly generate or measure surface roughness profiles. In this paper, we analyse the influence of surface roughness parameters an flow factorwhich is strongly epenent on the surface pattern parameter (γ) on the longituinally rough slier bearing. 2.ANALYSIS Patir an Cheng (1978, 1978) evelope Average Reynols equation which tookaccount of the surface topography. The estimation of the average film thickness (Mean gap) was escribe by, h T = (h + δ)f(δ)δ = E(h T ) (1) h where, h T = h + δ (2) The mean pressure in a rough slier bearing is governe by the average Reynols equation (Patir(1978))is given by, x φ h 3 p x 12µ x + y φ h 3 p y 12µ y = U h T + Uσ φ s (3) 2 x 2 x Assuming that the flow of lubricant is steay an in X-irection only an U1=U, U2=0. Moreover for longituinally rough surface (γ > 1, the variations in roughness heights in X-irection is negligible (Figure2), so the effect of φ s may be treate as negligible (Patir (1978)). Equation (3) turns out to be, φ h 3 p x x 12µ h T x (4) = U x 2 For a rough plane slier bearing as shown in Figure3, one consiers h = h m + m(l x) (5) δ is assume to be stochastic in nature an is governe by the probability ensity function f(δ), c < δδ < cc, where c ismaximum eviation from the mean film thickness. Then the varianceα, the stanar eviation σ an the skewness parameter ε which is the symmetry of the ranom variable δ are escribe by Deheri et al.(2004)in terms of the expecte values as : c E(R) = R f(δ)δ (6) c E(δ) = α (7) E[(δ α) 2 ] = σ 2 (8) an E[(δ α) 3 ] = ε (9) It is to be note that while α an εcan assume both positive an negative values, σ is always positive. Chiang, Hsiu-Lu, et al. (2005) presente the approximation to f(δ) as, f(δ) = 32 δ2 1 35c c 2 3 c δ c (10) Thus h T can be approximate as, Figure 1. Surface roughness an film geometry 0 elsewhere Figure 2. Longituinally rough surface Figure 3. Bearing geometry 228 Fascicule 2
ISSN: 1584-2665 [print]; ISSN: 1584-2673 [online] h T = 13 8 h h Now as per the average process as iscusse by Anharia et al.(2001),equation(4) reuces to, φ m(h) 1 (p) x x 12µ = U x 2 x [n(h) 1 ] (11) where(p) is expecte value of the mean pressure level p an m(h) = h 3 [1 3αh 1 + 6h 2 (σ 2 + α 2 ) 20h 3 (ε + 3σ 2 α + α 3 )] (12) while n(h) = h 1 [1 αh 1 + h 2 (σ 2 + α 2 ) h 3 (ε + 3σ 2 α + α 3 )] (13) The non-imensional form of equation(11) is foun to be, where, X φ X M(h ) 1 P X = 6 X [N(h ) 1 ] (14) h = h, X = x, m = ml, P = h m (15) h m l h m µul M(h ) = h 3 1 3α h 1 + 6h 2 σ 2 + α 2 20h 3 (ε + 3σ 2 α + α 3 ) an N(h ) = h 1 1 α h 1 + h 2 σ 2 + α 2 h 3 (ε + 3σ 2 α + α 3 ) Patir(1978)establishe the experimental relation for φ x which is as uner, φ x = 1 + C H r ( for γ > 1) (16) φ X = 1 + C (h H m ) r ( for γ > 1) (17) where H = h, H σ m = h m.(18) σ an the constants C an r are given as functions Table-1. Relation between,c, r an H of γ(patir (1978)) in Table-1. γγ C r HH Many of the investigators ha observe that if 3 0.225 1.5 H > 0.5 H = h is very large (H 6) the smooth film 6 0.520 1.5 H > 0.5 σ theory is applicable an so the roughness effects 9 0.870 1.5 H > 0.5 are not that important. If H > 3, the roughness effect is significant. If H < 3, the roughness effect increases further, which is calle partial lubrication regime ue to the presence of rough surface contacts. If H < 0.5, this assumption may not be justifie because a very large portion of the nominal area remains in contact. Subject to the following bounary conitions: P = 0, at X = 0 an 1 (19) P X = 0, at which the mean gap is maximum, sayq (Constant) Equation (14) leas to, X P (X) = 1 1 [ 6 N(h ) 1 Q ]X (20) 0 φ X M(h ) 1 where, Q 1 = 6 M(h ) 1 X M(h ) X (21) 0 φ X N(h ) 0 φ X The imensionless loa carrying capacity per unit with is given by, W = w.h m 2 = 1 P X (22) µul 2 0 3. RESULTS AND DISCUSSION Figure 4 6 ealing with the effect of variance on the loa carrying capacity establishe that the variance (+ve) ecreases the loa carrying capacity while the variance (-ve) causes increase the loa carrying capacity. It is interesting to see that the loa carrying capacity enhances ue to an increasing stanar eviation (Figure 7-9) which oes not happen in the case of transverse roughness pattern. The trens of loa carrying capacity with respect to skewness run almost similar to that of variance (Figure 10-12). However loa ecreases as γ increases. 2 p 229 Fascicule 2
ANNALS of Faculty Engineering Huneoara International Journal of Engineering w* vs α*( γ=6,n*=3.8,hm=3,ε*=-0.025 ) σ*=0 σ*=0.025 6 4 σ*=0.05 σ*=0.075 σ*=0.1 2-0.1-0.05 0 0.05 0.1 α* Figure 4. Variation of loa carrying capacity with respect to α* w* vs α* ( n*=3.8,hm=3, σ*=0.1, ε*=- 2 0.025) γ=3 γ=6 6 γ=9 4 2-0.1-0.05 0 0.05 0.1 Figure 6. Variation of loa carrying capacity with respect to α* α* Figure 8. Variation of loa carrying capacity with respect to σ* 6 4 2 w* vs σ* ( γ=6,n*=3.8,hm=3, ε*=-0.025 ) α*=-0.05 α*=-0.025 α*=0 α*=0.025 α*=0.05 0 0.05 0.1 0.15 w* vs ε* ( γ=6,n*=3.8,hm=3, α*=-0.05) 74 34 0.294 Figure 10. Variation of loa carrying capacity with respect to ε* σ* σ*=0 σ*=0.025 σ*=0.05 σ*=0.075 σ*=0.1 0.254-0.04-0.02 0 0.02 0.04 ε* w* vs α* ( γ=6,n*=3.8,hm=3,σ*=0.1 ) 0.5 0.2 0.1-0.1-0.05 0 0.05 0.1 α* ε*=-0.025 ε*=-0.01 ε*=0 ε*=0.01 ε*=0.025 Figure 5. Variation of loa carrying capacity with respect to α* 5 5 0.25 0.2 w* vs σ* ( γ=6,n*=3.8,hm=3, α*=- 0.05 ) ε*=-0.025 ε*=-0.01 ε*=0 ε*=0.01 ε*=0.025 0 0.05 0.1 0.15 Figure 7. Variation of loa carrying capacity with respect to σ* 1 9 7 Figure 9. Variation of loa carrying capacity with respect to σ* Figure 11. Variation of loa carrying capacity with respect to ε* σ* w* vs σ* ( n*=3.8,hm=3, α*=-0.05, ε*=- 0.025) γ=3 γ=6 γ=9 σ* 0 0.05 0.1 0.15 w* vs ε* ( γ=6,n*=3.8,hm=3, σ*=0.1) 6 α*=-0.05 α*=-0.025 9 α*=0 2 α*=0.025 α*=0.05 0.25 0.18-0.04-0.02 0 ε* 0.02 0.04 230 Fascicule 2
w* vs ε*( n*=3.8,hm=3, σ*=0.1, α*=- 1 0.05) γ=3 7 γ=6 γ=9 3 0.29 ISSN: 1584-2665 [print]; ISSN: 1584-2673 [online] The positive effect of stanar eviation associate with roughness is isplaye in Figures 7-9. It is clearly observe that the rate of increase in the loa carrying capacity with respect to the stanar eviation is more for large positively skewe roughness. Thus the trio, negatively skewe roughness, stanar eviation an variance (-ve) may result in an enhance performance irrespective of what γ is. 4. CONCLUSION This article establishes that the roughness must be accore top priority while esigning this type of bearing systems. It is more crucial from bearing s life perio point of view. 0.25-0.04-0.02 0 ε* 0.02 0.04 Figure 12. Variation of loa carrying capacity with respect to ε* Nomenclature: h T Average film thickness (Mean gap) (m) f(δ) Frequency ensity function of combine roughness amplitue δ(m 1 ) m Inclination of slier bearing l Length of slier bearing (m) w carrying capacity (N) W carrying capacity (Dimensionless) h T Local film thickness (m) p Local pressure (N m 2 ) p Mean pressure level (N m 2 ) P Mean pressure level (Dimensionless) h m Minimum film thickness at the trailing ege of slier bearing (m) H m Minimum film thickness Roughness ratio h m σ h Nominal film thickness (m) H Nominal film thickness Roughness ratio h σ U 1, U 2 Velocities of surfaces in X-Direction (m s) σ Composite rms roughness given by Gaussian istribution of heights σ 2 1 + σ 2 2. (m) ρ Density of lubricant (Kg m 3 ) φ x, φ y Pressure flow factors δ = δ 1 + δ 2 Ranom roughness amplitues of the two surfaces measure from their mean level (m) φ s Shear flow factor σ 1, σ 2 Stanar eviations of the surfaces (m) µ Viscosity of lubricant (Kg m. s) References 1. Anharia, P. I., an G. M. Deheri. "Effect of Longituinal Roughness on Magnetic Flui Base Squeeze Film between Truncate Conical Plates." Flui Dynamics & Materials Processing 7.1 (2010): 111-124. 2. Anharia, P. I., Gupta, J. L., &Deheri, G. M. (1997). Effect of longituinal surface roughness on hyroynamic lubrication of slier bearings. BOOK-INSTITUTE OF MATERIALS, 668, 872-880. 3. Burton, R. A. (1963). Effects of two-imensional, sinusoial roughness on the loa support characteristics of a lubricant film. Journal of Fluis Engineering, 85(2), 258-262. 4. Chiang, H. L., Chou, T. L., Hsu, C. H., Hsu, C. H., & Lin, J. R. (2005). Surface roughness effects on the ynamic characteristics of finite slier bearings. Journal of CCIT, 34(1), 1-11. 5. Christensen, H. (1972). A theory of mixe lubrication. Proceeings of the Institution of Mechanical Engineers, 186(1), 421-430. 6. Christensen, H. (1969). Stochastic moels for hyroynamic lubrication of rough surfaces. Proceeings of the Institution of Mechanical Engineers, 184(1), 1013-1026. 7. Christensen, H., &Toner, K. (1971). The hyroynamic lubrication of rough bearing surfaces of finite with. Journal of Tribology, 93(3), 324-329. 8. Tøner, K., & Christensen, H. (1972). Waviness an roughness in hyroynamic lubrication. Proceeings of the Institution of Mechanical Engineers, 186(1), 807-812. 231 Fascicule 2
ANNALS of Faculty Engineering Huneoara International Journal of Engineering 9. Davies, M. G. (1963). The generation of pressure between rough flui lubricate, moving, eformable surfaces. Lubr. Eng, 19, 246. 10. Deheri, G. M., Anharia, P. I., & Patel, R. M. (2004). Longituinally rough slier bearings with squeeze film forme by a magnetic flui. Inustrial Lubrication an Tribology, 56(3), 177-187. 11 Guha, S. K. (1993). Analysis of ynamic characteristics of hyroynamic journal bearings with isotropic roughness effects. Wear, 167(2), 173-179. 12. Gupta, J. L., &Deheri, G. M. (1996). Effect of roughness on the behavior of squeeze film in a spherical bearing. Tribology Transactions, 39(1), 99-102. 13. Hamrock, B. J. 1994, Funamentals of Flui Film Lubrication, McGraw-Hill. New York. 14. Michell, A. G. M. (1950). Lubrication: its principles an practice. Blackie. 15. Anharia, P. I., Gupta, J. L., &Deheri, G. M. (2001). Effect of surface roughness on hyroynamic lubrication of slier bearings. Tribology transactions, 44(2), 291-297. 16. Patel, J. R., &Deheri, G. (2014, January). Slip Velocity an Roughness Effect on Magnetic Flui Base Infinitely Long Bearings. In Proceeings of International Conference on Avances in Tribology an Engineering Systems (pp. 97-109). Springer Inia. 17. Patel, N. D., &Deheri, G. M. (2011). Effect of surface roughness on the performance of a magnetic flui base parallel plate porous slier bearing with slip velocity. Journal of the Serbian society for Computational Mechanics, 5(1), 104-118. 18. Patir, N. (1978). Effects of surface roughness on partial film lubrication using an average flow moel base on numerical simulation. University Microfilms, Chapter-2. 19. Patir, N., & Cheng, H. S. (1978). An average flow moel for etermining effects of three-imensional roughness on partial hyroynamic lubrication. Journal of Tribology, 100(1), 12-17. 20. Patir, N., & Cheng, H. S. (1979). Application of average flow moel to lubrication between rough sliing surfaces. Journal of Tribology, 101(2), 220-229. 21. Pinkus, O., &Sternlicht, B. (1961). Theory of hyroynamic lubrication. McGraw-Hill. 22. Prakash, J., & Tiwari, K. (1982). Lubrication of a porous bearing with surface corrugations. Journal of Tribology, 104(1), 127-134. 23. Toner, K. C. (1972, December). Surface istribute waviness an roughness. In First worl conference in Inustrial Tribology (Vol. 3, pp. 1-8). ANNALS of Faculty Engineering Huneoara International Journal of Engineering copyright UNIVERSITY POLITEHNICA TIMISOARA, FACULTY OF ENGINEERING HUNEDOARA, 5, REVOLUTIEI, 331128, HUNEDOARA, ROMANIA http://annals.fih.upt.ro 232 Fascicule 2