ME 575 Hydrodynamics of Lubrication By Parviz Merati

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

ME 575 Hydrodynamics of Lubrication Fall 2001 l 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

ME 575 Hydrodynamics of Lubrication l l l l l Movie on “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

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

Solid Friction l Resistance for sliding – Static – Kinetic l Causes – Surface roughness (asperities) – Adhesion (bonding between dissimilar materials) l 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

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

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

Lubrication l Hydrodynamic lubrication (more common) – A continuous fluid film exists between the surfaces l Boundary lubrication – The oil film is not sufficient to prevent metal-to-metal contact – Exists under extreme pressure l 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

Lubrication l 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. l 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

Fluid Friction l 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 d. U/dy = shear rate

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

Overall Bearing Friction l 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

Overall Bearing Friction l 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.

Overall Bearing Friction l 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.

Analysis l 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. l 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.

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. l 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

Boundary Lubrication l 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. l 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.

Boundary Lubrication l 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 l 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.

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.

Reynolds Equation l 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.

Reynolds Equation l 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.

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

Reynolds Equation B. C. y = 0. 0, u = U 1 , v = V 1 , w = W 1 y = h, u = U 2 , v = V 2 , w = W 2 Integrating the x component of the above equations would result in the following equation.

Reynolds Equation Integrating the z-component

Reynolds Equation l u and w have two portions; – A linear portion – A parabolic portion

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

Reynolds Equation Fluid moving into the fluid element in the Y direction is q 1

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

Reynolds Equation in Cylindrical Coordinate System R 1 and R 2 are the radial velocity of the two surfaces T 1 and T 2 are the tangential velocity of the two surfaces V 1 and V 2 are the axial velocity of the two surfaces

Hydrostatic Bearings l 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. l If the pad is circular as shown in the following figure,

Hydrostatic Bearings Total Load P The hydrostatic pressure required to carry this load is p 0.

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 U 1 and U 2 are zero, the following relationship can be obtained for radial component of the flow velocity ur.

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

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

Thrust Bearings l 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

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

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.

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

Thrust Bearings What is the pressure in the lubricant layer? Pressure varies linearly from the center value of p 0 to zero at the outer edge of the flat pivot. If we define an average pressure as pav

Thrust Bearings What is the viscous friction coefficient?

Thrust Bearings Pressure Variation in the Direction of Motion

Thrust Bearings Integrating and using the following boundary condition

Thrust Bearings l 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.

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

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

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.

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

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.

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.

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.

Journal Bearings What is the lubricant’s film thickness h? Using the above figure, the following relationship can be obtained for h The maximum and minimum values for h are r = Journal radius r+c = Bearing radius

Journal Bearings Using Reynolds equation and assuming an infinite length for the bearing, i. e. , p/ z = 0, and U = U 1+U 2 , the following differential equation is obtained. Reynolds found a series solution in 1886 and Sommerfeld found a closed form solution in 1904 which is widely used.

Journal Bearings Modern bearings are usually shorter, the length to diameter ratio is often shorter than 1. Thus, the z component cannot be neglected. Ocvirk in 1952 showed that he could safely neglect the parabolic pressure induced part of the U component of the velocity and take into account the z variation of pressure. Thus, the following simplified equation can be obtained. If there is no misalignment of the shaft and bearing, h and h/ x are independent of z, then the above equation can be easily integrated with the following boundary conditions for a journal of length l.

Journal Bearings Ocvirk Solution of the Short Bearing Approximation Thus, axial pressure distribution is parabolic.

Journal Bearings At which angle the maximum pressure occur? m=? To find m’ p/ =0. What is the total load that is developed within the bearing? The oil film experiences two forces, one from the bearing, the other from the journal. The bearing force P passes through the center point of the bearing, the journal force P passes through the journal center.

Journal Bearings The hydrodynamic pressure force is always normal to the bearing and journal surfaces. In order to find the total load, the pressure force over the bearing surface must be integrated. Since the oil film is stationary, the resultant of the external forces and moments, i. e. bearing and journal forces and moments exerted on the oil film, must be zero. The total load P carried by the bearing is calculated by the following equation. Where is defined as the attitude angle and is the angle between the line of force and the line of centers. The two components of the load normal and parallel to the line of centers are represented by P sin and P cos .

Journal Bearings Journal Load and the Attitude Angle

Journal Bearings With an increasing load, will vary from 0 to 1 and the attitude angle vary from 90 degrees to zero. The path of the journal center Oj as the load and eccentricity are increased is shown in the following figure.

Homework 2 Non-dimensionalize the hydrodynamic pressure and load of equations 5. 48 and 5. 51 of your notes, respectively. These are the Ocvirk equations for short journal bearings. Plot this non-dimensionalized pressure versus at z = 0. 0 for eccentricity ratios = 0. 1, 0. 3, 0. 5, 0. 7, and 0. 9. Plot the location and magnitude of the maximum pressure with respect to at z = 0. 0. Plot the non-dimensionalized load P and the attitude angle versus . For each plot, please discuss your findings and provide conclusions.

Hydrodynamic Instability l Synchronous whirl – Caused by periodic disturbances outside the bearing such that the bearing system is excited into resonance. Shaft inertia and flexibility, stiffness and damping characteristics of the bearing films, and other factors affect this instability. The locus of the shaft center called the whirl orbit increases at the critical shaft speed where there is resonance. It is usual procedure to make the bearings such that the critical speeds do not coincide with the most commonly used running speeds. This may be done either by increasing the bearing stiffness so that the critical speeds are very high, or reducing the stiffness so that the critical speeds are quickly passed through and normal operation takes place where the attenuation is large. Stiffness can be increased by reducing the bearing clearance. Introduction of extra damping by mounting the bearing housings in rubber “O” rings or metal diaphragms are other methods to suppress the synchronous whirl.

Hydrodynamic Instability l Half-Speed whirl – This is induced in the lubricant film itself and is called “half-speed whirl”. This is because due to existence of the attitude angle , the reaction force from the lubricant on the shaft has a component normal to the line connecting the centers of the shaft and the bearing. This component causes the shaft to move in a circumferential direction, i. e. , at the same time as the shaft moves around its center, the shaft center rotates about the bearing center. If the whirl takes place at the half the rotational speed of the shaft, this will coincide with the mean rotational speed of the lubricant. Because, the lubricant, on the average, does not have a relative velocity with respect to the shaft, the hydrodynamic lubrication fails. Extra damping, axial groves on the bearing housing, partial bearing are some of the techniques to get rid of this instability.

Hydrodynamic Instability

Thermal Effects on Bearings We have assumed that fluid viscosity and density remains constant in deriving the Reynolds equations. In reality due to viscous dissipation because of the large existing shear stress, the lubricant’s temperature rises and thus the fluid density and viscosity change. Since the fluid is unable to expand due to restriction, fluid pressure increases as the temperature increases. This is called Thermal Wedge. Consider the General R. E. with the viscosity and density variation in the sliding direction.

Thermal Effects on Bearings After integrating the above equation, In this equation, A and B are constants. The variation of density with temperature can be approximated by the following relationships.

Thermal Effects on Bearings Contribution of viscosity variation for liquids compared with density variation is negligible since viscosity increases with pressure and decreases with temperature. Thus, we can assume that viscosity remains constant in the sliding direction. Using the following boundary conditions, the pressure variation due to temperature variation for a parallel bearing can be obtained.

Thermal Effects on Bearings Where, For mineral oil,

Thermal Effects on Bearings Thus, for a rise in temperature of 100 C , ' =0. 93. The dimensionless pressure p' is

Thermal Effects on Bearings p´max is about 0. 011 and for a plane-inclined slider, p´max is about 0. 042. The parallel surface bearing has a load capacity approximately 1/3. 5 that of the corresponding inclined slider. It is rare that the temperature rise is 100 C , usually the temperature rise due to viscous dissipation is in the order of 2 -20 C and under these conditions, it is safe to assume that the effect of temperature is negligible.

Viscosity-Pressure Relationship In some situations where extreme pressures can occur such as in the restricted contacts between gear teeth and between rolling elements and their tracks, viscosity relationship with pressure is represented by the following equation. Where 0 and are reference viscosity and the pressure exponent of the viscosity, respectively. In order to integrate R. E. , we have to introduce parameter q defined as

Viscosity-Pressure Relationship The differential equation that is obtained as the result of this substitution, looks like a normal R. E. with viscosity term being 0. This equation can then be integrated and pressure can be obtained from the following relationship. Although load remains finite, pressure is tending to approach an infinite value between two disks rolling with some degree of sliding as shown in the following figure. This does not happen in reality. In reality, large pressures produce deformation of the bodies which distribute the pressure over a finite area. This is called “Elasto-hydrodynamic” lubrication or EHD.

Laminar Flow Between Concentric Cylinders Using Navier-Stokes equations in cylindrical systems and the following simplifications, Vr =0 Vz = 0 v = u The r-component is The component is

Laminar Flow Between Concentric Cylinders B. C. for velocity B. C. for pressure

Laminar Flow Between Concentric Cylinders For the case of mechanical seals where the inner cylinder is rotating and the outer cylinder is stationary, i. e. 2 = 0

Laminar Flow Between Concentric Cylinders If the inner cylinder is at rest , 1 = 0, the moment of the fluid on a length L of the outer cylinder is described by

Laminar Flow Between Concentric Cylinders Viscosity can be calculated from this equation if the moment on the outer cylinder is measured.

Homework 3 Calculate and plot pressure ratio p/p 1, and velocity ratio u/(R 1 1) versus the radial location (r-R 1)/(R 2 -R 1) for the flow between concentric cylinders for water, oil, and sodium iodide solution. The radii of the inner and outer cylinders are R 1 = 0. 031 m and R 2 = 0. 046 m, respectively. p 1 is the pressure at the inner cylinder surface and r is the radial location. The outer cylinder is stationary and the inner cylinder is rotating at 1, 200 rpm. Density of water, oil and sodium iodide solution (67% by volume) are 1, 000, 880, and 1, 840 Kg/m 3, respectively. Assume that p 1 is atmospheric pressure. Although the flow at this rotational speed is turbulent, the time average of the flow velocity and pressure are close to the laminar flow values. For each plot, please discuss your findings and provide conclusions.

Thank You