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Hydrodynamic BearingsHydrodynamic Bearings --TheoryTheory

Lecture 25Lecture 25

Engineering 473Engineering 473

Machine DesignMachine Design

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Lubrication ZonesLubrication ZonesBoundary

LubricationMixed-film

Lubrication

Hydrodynamic

Lubrication

CoefficientofFric

tion

Bearing Parameter

psiarea),jected(force/propressurep

rev/secspeed,rotationaln

sec/in-lb,viscositydynamic

p

n

ParameterBearing

2

Contact between

journal and bearing

Boundary LubricationBoundary Lubrication

MixedMixed--film Lubricationfilm Lubrication

Intermittent contact

Hydrodynamic LubricationHydrodynamic Lubrication

Journal rides on a fluid

film. Film is created by the

motion of the journal.

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Stable/Unstable LubricationStable/Unstable Lubrication

Boundary

LubricationMixed-film

Lubrication

Hydrodynamic

Lubrication

Coef

ficientofFrictio

n

Bearing Parameter

p

n

Hydrodynamic Lubrication

is often referred to as stable

lubrication.

If the lubrication temperature

increases, the viscosity

drops. This results in alower coefficient of friction,

that causes the lubrication

temperature to drop. => Self

Correcting.

Mixed-film lubrication is unstable an increase in lubrication

temperature causes further increases in lubrication temperature.

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Newtonian FluidNewtonian FluidA Newtonian fluid is any fluid whose shear stress and

transverse rate of deformation are related through the

equation.

dy

du =

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Dynamic ViscosityDynamic Viscosity

dy

du =

reyn

in

seclb

insec

inin

lb

2

f2

f

=

=

UnitsUnits

ipsips SISI

2

2

m

secN

msec

mm

N

=

Other common units are discussed in the text.

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Pumping ActionPumping ActionWhen dry, friction will cause the

journal to try to climb bearing

inner wall.

When lubricant is introduced, theclimbing action and the viscosity of

the fluid will cause lubricant to be

drawn around the journal creating a

film between the journal and bearing.The lubricant pressure will push the

journal to the side.

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Analysis AssumptionsAnalysis Assumptions

1. Lubricant is a Newtonian fluid

2. Inertia forces of the lubricant arenegligible

3. Incompressible

4. Constant viscosity

5. Zero pressure gradient along thelength of the bearing

6. The radius of the journal is large

compared to the film thickness

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Analysis GeometryAnalysis Geometry

Actual GeometryActual Geometry Unrolled GeometryUnrolled Geometry

From boundary layer theory, the pressure

gradient in the y direction is constant.

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XX--Momentum EquationMomentum Equation

pdydzdxdzdy

y

dxdzdydzdx

dx

dpp0Fx

++

+==

y

dx

dp

=

y

u

=

2

2

y

u

dx

dp

=

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XX--Momentum EquationMomentum Equation(Continued)(Continued)

( ) ( )xCyxCydx

dp

2

1u 21

2 ++=

0u0,y == ( ) 0xC2 =

-Uuh(x),y == ( )( )

( )dx

dp

2

xh

xh

UxC1 =

( )( )( )

yxh

Uyxhy

dx

dp

2

1u 2 =

Note that h(x) and

dp/dx are not known

at this point.

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Mass Flow RateMass Flow Rate

( )

( )( )( )

( )

( ) ( )

=

=

=

2

xUh

dx

dp

12

xh

m

dyyxh

Uyxhydxdp

21m

udym

3

xh

0

2

xh

0

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Conservation of MassConservation of Mass

Conservation of Mass RequiresConservation of Mass Requires

0dxmd =

( ) ( )

=

2

xUh

dx

dp

12

xhm

3

( ) 0dxdh

2U

dxdp

12xh

dxd

3

=

( )dx

dhU6

dx

dp

xh

dx

d3

=

ReynoldsReynolds EquationEquation

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h(x) Relationshiph(x) Relationship

e

( ) ( )

( )

( )

( )

+=

+=

=

+=

=

D

2xcos1cxh

1ch

1ch

cos1ch

ce

r

rmax

rmin

r

r

cr= radial clearance

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SommerfeldSommerfeld SolutionSolution

( )

dx

dh

U6dx

dp

xh

dx

d3

=

( )

+= D

2x

cos1cxh r

A. Sommerfeld solved these

equations in 1904 to find the

pressure distribution around thebearing.

It is known as a long bearing

solution because there is no flowin the axial direction.

( )( )( ) o222rp

cos12cos2sin6

cUrp +

+++=

r is the journal radius, is a chosen design parameter.

0

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OcvirkOcvirk ShortShort--Bearing SolutionBearing SolutionA short bearing allows lubricant flow in the longitudinal

direction, z, as well as in the circumferential direction, x.

( ) ( )x

h6U

dz

dp

xh

zdx

dp

xh

x

33

=

( )32

2

2

r cos1

sin3z

4

l

rc

Up

+

= 0

The Ocvirk solution (1955) neglects the first term

as being small compared to the axial flow.

GoverningGoverning

EquationEquation

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Short & Long BearingShort & Long Bearing

ComparisonsComparisons

%

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AssignmentAssignmentUse Matlab to plot the pressure distribution predicted by the

Sommerfeld equation for a journal bearing having a

clearance ratio of 0.0017, journal radius of 0.75 in, of 0.6,=2.2reyn, shaft rotational speed=20 rev/sec, and po=o.

First, generate the plot only for the range equals 0 to .

Second, generate the plot for the range equals 0 to 2.What happens to the pressure distribution from to 2. Is

this physically possible? Discuss what would happen to thelubricant if this pressure distribution occurred.