ME 575 Hydrodynamics of Lubrication

1. ME 575Hydrodynamics of Lubrication By Parviz Merati, Professor and Chair Department of Mechanical and Aeronautical Engineering Western Michigan University Kalamazoo, Michigan

2. ME 575Hydrodynamics of LubricationFall 2001 • An overview of principles of lubrication • Solid friction • Lubrication • Viscosity • Hydrodynamic lubrication of sliding surfaces • Bearing lubrication • Fluid friction • Bearing efficiency • Boundary lubrication • EHD lubrication

3. ME 575Hydrodynamics of Lubrication • Movieon “Lubrication Mechanics, an Inside Look” • General Reynolds equation • Hydrostatic bearings • Thrust bearings • Homework #1 • Journal bearings • Homework #2 • Hydrodynamic instability • Thermal effects on bearings • Viscosity • Density

4. ME 575Hydrodynamics of Lubrication • Viscosity-pressure relationship • Laminar flow between concentric cylinders • Velocity profile • Pressure • Mechanical Seals • Moment of the fluid on the outer cylinder • Homework #3

5. Solid Friction • Resistance force for sliding • Static • Kinetic • Causes • Surface roughness (asperities) • Adhesion (bonding between dissimilar materials) • Factors influencing friction • Frictional drag lower when body is in motion • Sliding friction depends on the normal force and frictional coefficient, independent of the sliding speed and contact area

6. Solid Friction • Effect of Friction • Frictional heat (burns out the bearings, ignites a match) • Wear (loss of material due to cutting action of opposing • Engineers control friction • Increase friction when needed (using rougher surfaces) • Reduce friction when not needed (lubrication)

7. Lubrication • Lubrication • Prevention of metal to metal contact by means of an intervening layer of fluid or fluid like material • Lubricants • Mercury, alcohol (not good lubricants) • Gas (better lubricant) • Petroleum lubricants or lubricating oil (best) • Viscosity • Resistance to flow • Lubricating oils have wide variety of viscosities • Varies with temperature

8. Lubrication • Hydrodynamic lubrication (more common) • A continuous fluid film exists between the surfaces • Boundary lubrication • The oil film is not sufficient to prevent metal-to-metal contact • Exists under extreme pressure • Hydrodynamic lubrication • The leading edge of the sliding surface must not be sharp, but must be beveled or rounded to prevent scraping of the oil from the fixed surface • The block must have a small degree of free motion to allow it to tilt and to lift slightly from the supporting surface • The bottom of the block must have sufficient area and width to float on the oil

9. Lubrication • Fluid Wedge • The convergent flow of oil under the sliding block develops a pressure-hydrodynamic pressure-that supports the block. The fluid film lubrication involves the ‘floating” of a sliding load on a body of oil created by the “pumping” action of the sliding motion. • Bearings • Shoe-type thrust bearings (carry axial loads imposed by vertically mounted hydro-electric generators) • Journal bearings (carry radial load, plain-bearing railroad truck where the journal is an extension of the axle, by means of the bearings, the journal carries its share of the load) • In both cases, a tapered channel is formed to provide hydrodynamic lift for carrying the loads

10. Fluid Friction • Fluid friction is due to viscosity and shear rate of the fluid • Generates heat due to viscous dissipation • Generates drag, use of energy • Engineers should work towards reducing fluid friction • Flow in thin layers between the moving and stationary surfaces of the bearings is dominantly laminar  = shear stress Z = viscosity dU/dy = shear rate

11. Fluid Friction • Unlike solid friction which is independent of the sliding velocity and the effective area of contact, fluid friction depends on both • Unlike solid friction, fluid friction is not affected by load • Partial Lubrication (combination of fluid and solid lubrication) • Insufficient viscosity • Journal speed too slow to provide the needed hydrodynamic pressure • Insufficient lubricant supply

12. Overall Bearing Friction • A relationship can be developed between bearing friction and viscosity, journal rotational speed and load-carrying area of the bearing irrespective of the lubricating conditions F = Frictional drag N = Journal rotational speed (rpm) A = Load-carrying area of the bearing f = Proportionality coefficient

13. Overall Bearing Friction • Coefficient of friction (friction force divided by the load that presses the two surfaces together)  is the coefficient of friction and is equal to F/L. L is the force that presses the two surfaces together. P is the pressure and is equal to L/A.

14. Overall Bearing Friction • ZN/P Curve • The relationship between  and ZN/P depends on the lubrication condition, i.e. region of partial lubrication or region of full fluid film lubrication. Starting of a journal deals with partial lubrication where as the ZN/P increases,  drops until we reach a full fluid film lubrication region where there is a minimum for . Beyond this minimum if the viscosity, journal speed, or the bearing area increases,  increases.

15. Analysis • Proper bearing size is needed for good lubrication. • For a given load and speed, the bearing should be large enough to operate in the full fluid lubricating region. The bearing should not be too large to create excessive friction. An oil with the appropriate viscosity would allow for the operation in the low friction region. If speed is increased, a lighter oil may be used. If load is increased, a heavier oil is preferable. • Temperature-Viscosity Relationship • If speed increases, the oil’s temperature increases and viscosity drops, thus making it better suited for the new condition. • An oil with high viscosity creates higher temperature and this in turn reduces viscosity. This, however, generates an equilibrium condition that is not optimum. Thus, selection of the correct viscosity oil for the bearings is essential.

16. Boundary Lubrication • Viscosity Index (V.I) is value representing the degree for which the oil viscosity changes with temperature. If this variation is small with temperature, the oil is said to have a high viscosity index. A good motor oil has a high V.I. • Boundary Lubrication • For mildly severe cases, additives known as oiliness agents or film-strength additives is applicable • For moderately severe cases, anti-wear agents or mild Extreme Pressure (EP) additives are used • For severe cases, EP agents will be used

17. Boundary Lubrication • Oiliness Agents • Increase the oil film’s resistance to rupture, usually made from oils of animals or vegetables • The molecules of these oiliness agents have strong affinity for petroleum oil and for metal surfaces that are not easily dislodged • Oiliness and lubricity (another term for oiliness), not related to viscosity, manifest itself under boundary lubrication, reduce friction by preventing the oil film breakdown. • Anti-Wear Agents • Mild EP additives protect against wear under moderate loads for boundary lubrications • Anti-wear agents react chemically with the metal to form a protective coating that reduces friction, also called as anti-scuff additives.

18. Boundary Lubrication • Extreme-Pressure Agents • Scoring and pitting of metal surfaces might occur as a result of this case, seizure is the primarily concern • Additives are derivatives of sulfur, phosphorous, or chlorine • These additives prevent the welding of mating surfaces under extreme loads and temperatures • Stick-Slip Lubrication • A special case of boundary lubrication when a slow or reciprocating action exists. This action is destructive to the full fluid film. Additives are added to prevent this phenomenon causing more drag force when the part is in motion relative to static friction. This prevents jumping ahead phenomenon.

19. EHD Lubrication In addition to full fluid film lubrication and boundary lubrication, there is an intermediate mode of lubrication called elaso-hydrodynamic (EHD) lubrication. This phenomenon primarily occurs on rolling-contact bearings and in gears where NON-CONFORMING surfaces are subjected to very high loads that must be borne by small areas. -The surfaces of the materials in contact momentarily deform elastically under extreme pressure to spread the load. -The viscosity of the lubricant momentarily increases drastically at high pressure, thus increasing the load-carrying ability of the film in the contact area.

20. Reynolds Equation • In bearings, we like to support some kind of load. This load is taken by the pressure force generated in a thin layer of lubricant. A necessary condition for the pressure to develop in a thin film of fluid is that the gradient of the velocity profile must vary across the thickness of the film. Three methods are available. • Hydrostatic Lubrication or an Externally Pressurized Lubrication- Fluid from a pump is directed to a space at the center of bearing, developing pressure and forcing fluid to flow outward. • Squeeze Film Lubrication- One surface moves normal to the other, with viscous resistance to the displacement of oil. • Thrust and Journal Bearing- By positioning one surface so it is slightly inclined to the other and then by relative sliding motion of the surfaces, lubricant is dragged into the converging space between them.

21. Reynolds Equation • Use Navier-Stokes equation and make the following assumptions • The height of the fluid film h is very small compared with the length and the span (x and z directions). This permits to ignore the curvature of the fluid film in the journal bearings and to replace the rotational with the transnational velocities.

22. Reynolds Equation • Since the fluid layer is thin, we can assume that the pressure gradient in the y direction is negligible and the pressure gradients in the x and z directions are independent of y • Fluid inertia is small compared to the viscous shear • No external forces act on the fluid film • No slip at the bearing surfaces • Compared with u/y and w/y, other velocity gradient terms are negligible

23. Reynolds Equation B.C. y = 0.0, u = U1 , v = V1 , w = W1 y = h, u = U2 , v = V2 , w = W2 Integrating the x component of the above equations would result in the following equation.

24. Reynolds Equation Integrating the z-component

25. Reynolds Equation • u and w have two portions; • A linear portion • A parabolic portion

26. Reynolds Equation • Using continuity principal for a fluid element of dx, dz, and h, and using incompressible flow, we can write the following relationship Where,

27. Reynolds Equation Fluid moving into the fluid element in the Y direction is q1

28. Reynolds Equation The last two terms are nearly always zero, since there is rarely a change in the surface velocities U and W.

29. Reynolds Equation in Cylindrical Coordinate System R1 and R2 are the radial velocity of the two surfaces T1 and T2 are the tangential velocity of the two surfaces V1 and V2 are the axial velocity of the two surfaces

30. Hydrostatic Bearings • Lubricant from a constant displacement pump is forced into a central recess and then flows outward between bearing surfaces. The surfaces may be cylindrical, spherical, or flat with circular or rectangular boundaries. • If the pad is circular as shown in the following figure,

31. Hydrostatic Bearings Total Load P The hydrostatic pressure required to carry this load is p0.

32. Hydrostatic Bearings What is the volumetric flow rate of the oil delivery system? Using Reynolds Equation for rectangular system, and substituting x with r, and considering that U1 and U2 are zero, the following relationship can be obtained for radial component of the flow velocity ur.

33. Hydrostatic Bearings What is the power required for the bearing operation? A = Cross sectional area of the pump delivery line V = Average flow velocity in the line  = Mechanical efficiency

34. Hydrostatic Bearings What is the required torque T if the circular pad is rotated with speed n about its axis ? The tangential component of the velocity is represented by Wt and the shear stress is shown by 

35. Thrust Bearings • There should be a converging gap between specially shaped pad or tilted pad and a supporting flat surface of a collar. The relative sliding motion forces oil between the surfaces and develop a load-supporting pressure as shown in the following figure. • Using the Reynolds Equation and using h/z = 0, for a constant viscosity flow, the following equation is obtained

36. Thrust Bearings This equation can be solved numerically. However if we assume that the side leakage w is negligible, thus p/z is negligible, then the equation can be solved analytically

37. Thrust Bearings Total load can be found by integrating over the surface area of the bearing. Flat Pivot Flat pivot is the simplest form of the thrust bearing where the fluid film thickness is constant and the pressure at any given radius is constant. There is a pressure gradient in the radial direction. The oil flows on spiral path as it leaves the flat pivot.

38. Thrust Bearings What is the torque T required to rotate the shaft? Shear stress is represented by 

39. Thrust Bearings What is the pressure in the lubricant layer? Pressure varies linearly from the center value of p0 to zero at the outer edge of the flat pivot. If we define an average pressure as pav

40. Thrust Bearings What is the viscous friction coefficient?

41. Thrust Bearings Pressure Variation in the Direction of Motion

42. Thrust Bearings Integrating and using the following boundary condition

43. Thrust Bearings • As the attitude of the bearing surface a is reduced, pressure magnitude decreases in the fluid film and the point of maximum pressure approaches the middle of the bearing surface. For a = 0, the pressure remains constant.

44. Thrust Bearings What are the total load and frictional force on the slider? Define P and F' as the load and drag force per unit length perpendicular to the direction of motion. q is the shear stress and is defined by the following equation

45. Thrust Bearings Coefficient of friction f is defined by the following relationship.

46. Thrust Bearings If  is the angle in radians between the slider and the bearing pad surface, then the following equations based on the equilibrium conditions of the film layer exist. Since  is very small, film layer thickness h and e are small relative to the bearing length B, sin    , and cos   1. It is also safe to assume that Fr is small compared with Q.

47. Thrust Bearings Critical value of  occurs when Fr =0. This will result in  is the angle of friction for the slider. When  > , Fr becomes negative. This is caused by reversal in the direction of flow of the oil film . The critical value of a is thus obtained by using the following relationship. Thus the range of acceptable variation for a is 0 < a <0.86

48. Homework 1 For a thrust bearing, plot non-dimensionalized pressure along the breath of the bearing for several values of the bearing attitude defined by a=e/h, ( 0  a  0.86). In addition, plot non-dimensionalized maximum pressure, load per unit length measured perpendicular to the direction of motion, tangential pulling force, and virtual friction coefficient versus the bearing attitude. For each plot, please discuss your findings and provide conclusions. Note: Please refer to figure 5.11 and sections 5.4.2, 5.4.3, and 5.4.4 of your notes for additional information.

49. Journal Bearings In a plain journal bearing, the position of the journal is directly related to the external load. When the bearing is sufficiently supplied with oil and external load is zero, the journal will rotate concentrically within the bearing. However, when the load is applied, the journal moves to an increasingly eccentric position, thus forming a wedge-shaped oil film where load-supporting pressure is generated.

50. Journal Bearings Oj = Journal or the shaft center Ob = Bearing center e = Eccentricity The radial clearance or half of the initial difference in diameters is represented by c which is in the order of 1/1000 of the journal diameter.  = e/c, and is defined as eccentricity ratio If  = 0, then there is no load, if  = 1, then the shaft touches the bearing surface under externally large loads.