ME%20575%20Hydrodynamics%20of%20Lubrication

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

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ME 575Hydrodynamics of Lubrication

• 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

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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

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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

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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)

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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

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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

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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

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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

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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

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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

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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.

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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.

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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 oils 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.

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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

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Boundary Lubrication

• Oiliness Agents
• Increase the oil films 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.

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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.

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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.

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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.

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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.

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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

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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.

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Reynolds Equation

• Integrating the z-component

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Reynolds Equation

• u and w have two portions
• A linear portion
• A parabolic portion

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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,

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Reynolds Equation
Fluid moving into the fluid element in the Y
direction is q1
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Reynolds Equation
The last two terms are nearly always zero, since
there is rarely a change in the surface
velocities U and W.
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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
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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,

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

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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 ?

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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

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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

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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.
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Thrust Bearings

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

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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

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Thrust Bearings

• What is the viscous friction coefficient?

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Thrust Bearings

• Pressure Variation in the Direction of Motion

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Thrust Bearings

• Integrating and using the following boundary
  condition

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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.

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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

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Thrust Bearings

• Coefficient of friction f is defined by the
  following relationship.

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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.

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Thrust Bearings

• Critical value of ? occurs when Fr 0. This will
  result in
• ? is the angle of friction for the slider. When
  ? gt ?, 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 lt a lt0.86

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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
  ae/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.

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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.

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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.

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Journal Bearings

• What is the lubricants 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
• rc Bearing radius

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Journal Bearings

• Using Reynolds equation and assuming an infinite
  length for the bearing, i.e., ?p/? z 0, and U
  U1U2 , 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.

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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.

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Journal Bearings

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

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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.

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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 ?.

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Journal Bearings
Journal Load and the Attitude Angle
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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.

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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.

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Hydrodynamic Instability

• 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.

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Hydrodynamic Instability

• 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.

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Hydrodynamic Instability
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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
  lubricants 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.

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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.

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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.

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Thermal Effects on Bearings

• Where,
• For mineral oil,

67
Thermal Effects on Bearings

• Thus, for a rise in temperature of 100 C?, ?'
  0.93. The dimensionless pressure p' is

68
Thermal Effects on Bearings

• pmax is about 0.011 and for a plane-inclined
  slider, pmax 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.

69
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

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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.

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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

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Laminar Flow Between Concentric Cylinders

• B.C. for velocity
• B.C. for pressure

73
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
74
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

75
Laminar Flow Between Concentric Cylinders

• Viscosity can be calculated from this equation if
  the moment on the outer cylinder is measured.

76
Homework 3

• Calculate and plot pressure ratio p/p1, and
  velocity ratio u/(R1?1) versus the radial
  location (r-R1)/(R2-R1) for the flow between
  concentric cylinders for water, oil, and sodium
  iodide solution. The radii of the inner and
  outer cylinders are R1 0.031 m and R2 0.046
  m, respectively. p1 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/m3,
  respectively. Assume that p1 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.

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Thank You