hydostatic bearing systems

8/10/2019 hydostatic bearing systems

1/71

Fundamentals of Fluid Film Lubrication

Hamrock, Schmid & JacobsonISBN No. 0-8247-5371-2

Hydrostatic

BearingSystems

Figure 13.1 Formation of fluid inhydrostatic bearing system. (a) Pumpoff; (b) pressure build up; (c) pressure

times recess area equals normalapplied load; (d) bearing operating; (e)increased load; (f) decreased load.[From Rippel (1963)].

WzBearing runner

Bearing pad

Recess pressure,pr= 0 Flow,

q= 0Supply pressure,ps= 0

Bearingrecess

Manifold

Restrictor

(a)

Wz

pr0

ps0

(b)

p=pl

p=pl

Wz

q= 0

(c)

Wz

p=pr

ho

q

p=ps

(d)

Wz + Wz

p=pr + pr

ho ho

q

p=ps

(e)

Wz Wz

p=pr pr

ho + ho

q

p=ps

(f)


2/71



Circular Step Pad & Pressure

wz

ro

ri

h0

p = 0pr

q

wz

dr

pr

r

p = 0

Figure 13.2 Radial-flow hydrostaticthrust bearing with circular step pad.

Figure 13.3 Pressure distribution inradial-flow hydrostatic thrust bearing.


3/71



Pad CoefficientsSill

Recess

ri

ro

7

6

5

4

3

2

1

Flowc

oefficient,qb;pow

ercoefficient,Hb

0 .2 .4 .6 .8 1.0

Ratio of recess radius to bearing radius, ri/ro

0

.2

.4

.6

.8

1.0

Loadcoefficient,ab

Hb

ab

qb

Figure 13.4 Chart for determiningbearing pad coefficients for circular step

thrust bearing. [From Rippel (1963)].

ab=1

(ri/ro)2

2ln(ro/ri)

qb= !

3[1

(ri/ro)2

]

Hb= 2! ln(ro/ri)

3 [1 (ri/ro)2]2

Load coefficient:

Flow coefficient:

Power coefficient:


4/71



Annularrust Pad Bearing

Flow Flow

r1

r2

r3

r4

Sills

Recess

Figure 13.5 Configurations of annular thrust pad bearing.[From Rippel (1963)].


5/71



Pad Coefficients40

32

24

16

8

0

Flowc

oefficient,qb;powerc

oefficient,Hb

Loadcoefficient,ab

1.2

.8

.4

.2 .4 .6 .8 1.0

(r3- r2)/(r4- r1)

qbHbab

r1

r4

3/4

1/2

1/4

Figure 13.6 Chart for determining bearing padcoefficients for annular thrust pad bearings. [FromRippel (1963)].

ab= 1

2r2

4 r

2

1

r

2

4 r

2

3

ln(r4/r3)

r2

2 r

2

1

ln(r2/r1)

qb= !

6ab

1

ln(r4/r3)+

1

ln(r2/r1)

Hb =qb

ab

Load coefficient:

Flow coefficient:

Power coefficient:


6/71



Rectangular Hydrostatic PadB

L b

l

pr

= 0

Figure 13.7 Rectangular hydrostatic pad.


7/71



Pad Coefficients

b

B8

7

6

5

4

3

2

1

Flowc

oefficient,qb;powercoefficient,Hb

1.4

1.2

1.0

.8

.6

.4

.2

0

Loadcoefficient,ab

.2 .4 .6 .8 1.0

(a)

2b/B

ab(numerical)

ab(eq. (14.20))

Hb

qb

0 .2 .4 .6 .8 1.02b/B

B

bL

Hb

qb

Hb

ab

B/L = 2

B/L = 4

(b)

b

Figure 13.8 Pad coefficients. (a) Squarepad; (b) rectangular pad with B= 2Land b=l.

Hb =qb

ab

Load coefficient:

Flow coefficient:

Power coefficient:

ab =1

2

1+

Ar

As

= 1

b

B

L+2b

BL

qb =1

6ab

Bb

+

L

b


8/71



Compensated Hydrostatic Bearingswz

Runner

Bearingpad

Capillarycompensatingelement

Supplymanifold

From pump

dc

qcpr

lc

ps

h0

Figure 13.9 Capillary-compensatedhydrostatic bearing. [From Rippel(1963)].

wzRunner

Orifice

Supplymanifold

From pump

Bearingpad

dppr

ps

h0

do

~

Figure 13.10 Orifice-compensatedhydrostatic bearing. [From Rippel(1963)].


9/71



Flow-Valve Compensation

To bearing recess

Variable orifice forcontrolling flowset point

From

pump

pr

ps

Figure 13.10 Constant-flow-valve

compensation in hydrostatic bearing.[From Rippel (1963)].


10/71



Compensating Element

Ranking


11/71



Speed vs. Load

Solution

Asymptotic

Complete High-speed asymptote

Infinitelength

Incompressible low-speed asymptote

Finite length

Loadcomponentalong

lineofcenters,wz

Relative surface velocity, ub

Figure 14.1 Effect of speed on load for self-acting, gas-lubricatedbearings. [From Ausman (1961).]


12/71



Rectangular-Steprust Bearing

Feedgroove

Step

Ridge

b

xy

hs

hr

ls

lr

lg

ub

Figure 14.2 Rectangular-step thrustbearing. [From Hamrock (1972).]

b

N0(ls+ lr+ lg)

ro ri

Figure 14.3 Transformation ofrectangular slider bearing into circularsector bearing.


13/71



Optimum Step

Parameters

.6

.5

.4

.3

.2

.1

0

g

Ha

6

5

4

3

2

1

0

.6

.5

.4

.3

.2

.1

0

6

5

4

3

2

1

0

10-1 100 101 102 103

g

Ha

Steplocationp

arameter,g

Filmt

hicknessratio,Ha;

length-to-widthratio,

Dimensionless bearing number,a=60ubb

pahr2

(a)

(b)

Figure 14.4 Effect of dimensionless

bearing number on optimum stepparameters. (a) For maximum

dimensionless load-carrying capacity;(b) for maximum dimensionless

stiffness. [From Hamrock (1972).]


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Load-Carrying

Capacity &Stiffness

4

100

10-1

10-2

10-3

Length-to-widthratio,

Step locationparameter,

g

Filmthickness

ratio,Ha

Optimal

0.918

.915

Optimal Optimal

0.555

.577

1.693

1.470

W Kg

(a)

4

100

10-1

10-2

10-3

10310210110010-1

(b)

W

Kg

Dimensionlessload-carryingcapacity,

W;dimensionlessstiffness,Kg

Figure 14.5 Effect of dimensionlessbearing number on dimensionless load-

carrying capacity and dimensionlessstiffness. (a) For maximumdimensionless load-carrying capacity;

(b) for maximum dimensionlessstiffness. [From Hamrock (1972).]


15/71



Spiral-Grooverust Bearing

hs

ri

hr

ro

p = pa

p = pa

rm

r

ri

a

r

g

Figure 14.6 Spiral-groove thrust bearing. [FromMalanoski and Pan (1965).]


16/71



Spiral-Grooverust BearingCharacteristics

1.3

1.2

1.1

0

Groovefactor,Gf

.02 .04 .06 .08

Inverse of number of grooves, 1/N0

(a)

Radiusratio,

r

0.7

.6

.5

.4

All

1.0

.8

.6

.4

.2

0 10 20 30 40 50

(b)

Dimensionless bearing number, s

Dimensionlessload,W

!

0 10 20 30 40 50

1.0

.8

.6Dimensionless

torque,Tq

s

(d)

1.5

1.0

.5

0 10 20 30 40 50s

(c)Dimensionlessstiffness

coefficient,K

!

4

3

2

1

(e)Dimen

sionlessflow,Qm

0 10 20 30 40 50s

5

4

3

2

20

18

16

1.8

1.71.6

1.50 10 20 30 40 50

s

(f)

Filmt

hickness

ratio,Ha

Groove

angle,

a,

deg

Groovew

idth

ratio,a

0 10 20 30 40 50s

(g)

Groovelength

fraction,Rg

.8

.6

.4

.2

Unstable

Stable

Figure 14.7 Charts for determiningcharacteristics of spiral-groove thrust

bearings. (a) Groove factor; (b) load; (c)stiffness; (d) torque; (e) flow; (f) optimal

groove geometry; (g) groove lengthfactor. [From Reiger (1967).]


17/71



Pressure Perturbation Solution

100

80

60

40

20

0

Altitudeangle,

,

deg

.1 .3 .5 .7 1 1.5 2 3 5 10 !

1.0

.8

.6

.4

.2

0

Dimensionless bearing number,j

Loadparameter,

wz

/(pa

rb)

.5

1.0

2.0

3.0

5.0

!

0.5

1.0

2.0

3.05.0!

Attitude angleLoad

Width-to-diameter

ratio,j

Figure 15.1 Design chart for radially loaded, self-acting, gas-lubricated journal bearings (isothermal first-order perturbationsolution.) [From Ausman (1959).]


18/71



LinearizedphSolution1.4

1.2

1.0

.8

.6

.4

.2

0 1.0.8.6.4.2

Dimensionlessload,wr/

pa

b2r

Eccentricity ratio,

Linearized ph

First-order perturbation

Experimental data (from Sternlicht and Elwell, 1958)

Computer solutions}

Effect of dimensionless load on eccen-

tricity ratio for finite-length, self-acting,

gas-lubricated journal bearing. Dimen-sionless bearing number Lj, 1.3; width-

to-diameter ratio lj, 1.5. [From Ausman

(1961)]


19/71



Pivoted-Pad Bearings

Pivot

Bearing

ShaftO'

O

e

r + c

r

hi

hp

h0

wc

p

p

p

Figure 15.3 Geometry of individualpivoted-pad bearing. [From Gunter et al.(1964)]

Pivot circle

Pivot

Shaft

hp,3

e'

r + c'

yx

r

hp,1

hp,2

'

ppwt

Pivoted partialjournal bearing

Figure 15.4 Geometry of pivoted-padjournal bearing with three pads. [FromGunter et al. (1964)]


20/71



Pivoted-Pad Perfor-

mance Parameters

.9

.8

.7

.6

.5

.4

.3

.2

.1

0

Dimensionlesslo

ad,

Wr=wr/par

b

(a)

Angle betweenline of centers

and padleading edge,

p,deg

Converging-divergingfluid film Converging

fluid film

Eccentricityratio,

0.65

.60

.55

.50

.45

Optimumpivotlocation

= 0.70

= 0.90

= 0.80

100 90 85.580

70

60

50

40

30

20

100

1.5

1.4

1.3

1.2

1.1

1.0

.9

.8

.7

.6

.5

.4

.3

.2(b)

.58 .60 .62 .64 .66 .68 .70 .72 .74 .76

Dimensionlesspivotfilmt

hickness,

Hp=hp

/c

Dimensionless pivot location, p/p

p,deg

0

0.60.70.80.90

0.65.55.50.45

10

20

30

40

50

60

7080

85.590100

.58 .60 .62 .64 .66 .68 .70 .72 .74 .76


.9

.8

.7

.6

.5

.4

.2

.3Dimensionlessoutletfilm

thickness,

H0=h0

/c

.58 .60 .62 .64 .66 .68 .70 .72 .74 .76


0.60 .70.80

0.45

.50

.55

.65

.90

(c)

p,deg10

20

30

4050607080

85.590

Figure 15.5 Charts for determining load

coefficient, pivot film thickness, andtrailing-edge film thickness. Bearing

radius-to-length ratio r/b, 0.6061; angu-

lar extent of pad ap, 95.5; dimension-

less bearing number Lj, 3.5. (a) Dimen-

sionless load; (b) dimensionless pivot

film thickness. [From Gunter et al.

(1964)]


21/71



Herringbone-Groove Journal

Bearing

R

b1/2

a

b

hr h

s

lr

ls

Figure 15.6 Configuration of concentric herringbone-groovejournal bearing.


22/71



Parameters for Herringbone

Bearing1.0

.8

.6

.4

.2

Width-to-diameter

ratio,j

1/41/2

1

2

28

26

24

22

20

18

34

32

30

28

26

24

22

20

18

Grooveangle,

a,

deg

(d)

0 40 80 120 1600 40 80 120 160

1.0

.8

.6

.4

.2

Groovelengthratio,g

(c)

Dimensionless bearing number, j

Figure 15.7 Charts for determiningoptimal herringbone-journal-bearinggroove parameters for maximum radialload. Top plots are for grooved memberrotating; bottom plots are for smooth

member rotating. (a) Optimal filmthickness ratio; (b) optimal groove widthratio. [From Hamrock and Fleming(1971)]


23/71



Parameters for Herringbone

Bearing (cont.)

2.8

2.6

2.4

2.2

2.0

Width-to-diameter ratio,

j

1/41/2

12

.6

.5

.4

.3

.2

(b)

0 40 80 120 160

.6

.5

.4

.3

.2

Groovewidthratio,b


2.8

2.6

2.4

2.2

2.0

3.0

1.80 40 80 120 160

(a)

Filmt

hicknessratio,

Ha

Figure 15.7 Concluded. (c) Optimalgroove length ratio; (d) optimal groove

angle.


24/71



Load-Carrying Capacity

16

12

8

4

0(a)

Width-to-diameter

ratio,j

1/41/212

16

12

8

4

0

(b)

40 80 120 160


Appliednormalload,wz,

N

Figure 15.8 Chart for determiningmaximum normal load-carrying capacity.

(a) grooved member rotating; (b)smooth member rotating. [From

Hamrock and Fleming (1971)]


25/71



Stability of Herringbone-

Groove Bearings2010

8

6

4

2

1

.8

.6

.4

.2

.1

.08

.06

.04

.02

Dimensionlessstabilityparameter,M

1 2 4 6 8 10 20 40 60 80


Width-to-diameter

ratio,j

1/41/2

12

Stable

Smooth member rotating

UnstableGroovedmemberrotating

Figure 15.9 Chart for determiningmaximum stability of herringbone-

groove bearings. [From Fleming andHamrock (1974).]


26/71



Foil Bearing

pTT

(a) (b)

z

x

hmin

T

T

Inletregion

Centralregion

Exitregion

h0

u0

O

r

Figure 15.10 (a) Schematic illustration of a foil bearing; (b) free-bodydiagram of a section of foil.


27/71



Pressure in Foil Bearing

Dimensionless position,

6 4 64 206 4 64 20

0.2

0.4

0.6

0.8

1.0

0

-0.2 Dimension

lesspressure,p/(T/r)

1

2

3

4

5

0

7

6

Dimension

lessfilmt

hickness,

h/h0

Film thickness

Pressure

Inletregion

Exitregion

Figure 15.11 Pressure distribution and film thickness in a foilbearing. [From Bhushan (2002).]


28/71



Lubrication of Rigid Cylinderz

h

ub

r

h0

Circular

Parabolic

(a)

ua

z

wza

wa

fa

fb

wxa

wz= w

zb

(b)

Figure 16.1 Lubrication of a rigid cylinder near a plane. (a) Coordinatesand surface velocities; (b) forces.


29/71



Cavitation Fingersuh

m

2

Air

Air

Air

Figure 16.2 Cavitation fingers.


30/71



Effect of Leakage

1.0

.8

.6

.4

.2

0

Normalloadratio

,(wz

)infinite

(wz

)finite

1.0.8.6.4.2

Width-to-diameter ratio, j= b/2r

Ratio of radius

to centralfilm thickness,

h

106

105

104

103

102

Figure 16.4 Effect of leakage ontangential load component.

0

Tangentialload

ra

tio,(wx

)infinite

(wx

)finite

1.0.8.6.4.2

Width-to-diameter ratio, j= b/2r

Ratio of radius

to central

film thickness,h

106

105

104

103

102

1.0

.8

.6

.4

.2

Figure 16.3 Side-leakage effect onnormal load component.


31/71



Contact

GeometrySolid a

Solid b

u h0

rax

rbx

ua

ub

x

x

Sax

Sbx

(a-1)

Solid a

Solid b

ray

rby

Say

Sby

h0

y

y

(a-2)

x

(b-1)

x

u

Rx

Sx

h0

Syy

y

h0

Ry

(b-2)

Figure 16.5 Contact geometry. (a) Tworigid solids separated by a lubricant film:

(a-1) y=0 plane; (a-2)x=0 plane. (b)

Equivalent system of a rigid solid near aplane separated by a lubricant film: (b-1)

y=0 plane; (b-2)x=0 plane. [FromBrewe et al. (1979)].


32/71



Boundary Conditions & Nodal

Structure(a)

(b)

(c)

Pressure

Dimensionless coordinate, X = x/Rx

Figure 16.6 Effect of boundary conditions. (a)Solution using full Sommerfeld boundaryconditions; (b) solution using half Sommerfeldboundary condition; (c) solution using Reynoldsboundary conditions. [From Brewe et al.(1970)].

(0,0)

Y

X

(-XE, YE)

(-XE, 0)

(-XE, -YE)

Figure 16.7 Variable nodal structureused for numerical calculations. [FromBrewe et al. (1979)].


33/71



Hydrodynamic Li

2.0

1.50 5 10 15 20 25 30 35 40

Radius ratio, r= Ry/Rx

L() = /2

Redu

cedhydrodynamic

lift,

L

Parabolic approximationFull circular filmKapitza's analysis (1955)

Figure 16.8 Effect of radius ratio on reduced hydrodynamic lift.

[From Brewe et al. (1979)].


34/71

Fundamentals of Fluid Film LubricationHamrock, Schmid & JacobsonISBN No. 0-8247-5371-2

Pressure for Two

Radius Ratios

Rolling

direction

Inlet

Outlet

Cavitation

boundary

X

Y

P

(a)

Rolling

direction

Cavitation

boundary

Inlet

Outlet

X

Y

P

(b)

Figure 16.9 Three-dimensional repre-

sentation of pressure distribution as

viewed from outlet region for two radiusratios ar. (a) ar= 1.00; (b) ar= 36.54.

[From Brewe et al. (1979)].


35/71


Pressure Contours

(a) (b) (c)

Pressure contours for three radius ratios ar. (a) ar= 25.29; (b) ar = 8.30; (c)

ar= 1.00. [From Brewe et al. (1979)].


36/71


Comparison of Fully Flooded

and Starved Contact P

Cavitation

boundary

Rolling

direction

Inlet

Outlet

XY

P

Cavitationboundary

Inlet

boundary

X

Y

Inlet Outlet

Hin= 1.00

Hmin

= 1 x 10-4

X

Rx

(a)

Inlet Outlet

Hin= 0.001

Hmin

= 1 x 10-4

X

Rx

(b)

Figure 16.11 Three-dimensional representationof pressure distributions for

dimensionless minimum film

thickness Hminof 1.0 x 10

-4

.(a) Fully flooded condition;(b) starved condition. [From

Brewe and Hamrock.(1982)].


37/71


Comparison of Fully Flooded

and Starved Contact

Inlet Outlet

Hin

= 1.00

Hmin

= 1 x 10-3

X

Rx

(a)

Inlet Outlet

Hin

= 0.002 Hmin

= 1 x 10-3

X

Rx

(b)

Cavitation

boundary

Rolling

direction

Inlet

OutletX

Y

P

Cavitationboundary

Inletboundary

P

X

Y

Figure 16.11 Three-dimensional representation of

pressure distributions fordimensionless minimum film

thickness Hminof 1.0 x 10-3

. (a)Fully flooded condition; (b)

starved condition. [From Breweand Hamrock. (1982)].


38/71


Pressure Contours - Starved

Cavitation

boundary

Cavitation

boundaryCavitationboundary

Inletmeniscus

boundary

Inlet

meniscus

boundary

(a) (b) (c)

Figure 16.13 Isobaric contour plots for three fluid inlet levels for dimensionless

minimum film thickness Hminof 1.0 x 10

-4

. (a) Fully flooded condition: dimensionlessfluid inlet level Hin, 1.00; dimensionless pressure, where dP/dX=0, Pm, 1.20 x 106;dimensionless load-speed ratio W/U, 1153.6. (b) Starved condition; Hin, 0.004; Pm=1.19 x 106; W/U= 862.6. (c) Starved condition: Hin =0.001; Pm= 1.13 x 106; W/U=567.8. [From Brewe and Hamrock. (1982)].


39/71


Inlet Level Effect

1.0

.9

.8

.7

.6

.5

.40 .2 .4 .6 .8 1.0

Filmt

hicknessreductionfactor,

s

Dimensionless fluid inlet level, Hin

Dimensionless

minimum

film thickness,

HminCritically

starved region

5 x 10-5

5 x 10-410-3

10-4

Figure 16.14 Effect of fluid inlet level on film thickness reduction factor inflooded conjunctions. [From Brewe and Hamrock (1982)].


40/71


Lubricant Flow

ua

r

Inlet Hin

Pressure

Hmin

Cavitated

region

Dimensionless coordinate, X = x/Rx

wa

Figure 16.15 Lubricant flow for arolling-sliding contact and correspondingpressure buildup. [From Ghosh et al.(1985)].


41/71


Effect of Velocity3

2

1

0

Dynamicloadra

tio,

f

10-2 10-1 100

Dmensionless normal velocity parameter, |w|

Normalapproach

Normalseparation

Effect of dimensionless normal velocity

parameter on dynamic load ratio. Di-

mensionless central film thickness Hmin,1.0 x 10-4; radius ratio ar, 1.0; dimen-

sionless fluid inlet level Hin, 0.035.

[From Ghosh et al. (1985)].

Rolling


42/71


Pressure

Distributions

Rolling

direction

X

Y

Cavitation

boundary

XY

XY X Y

X

Y

(a) (b)

(c) (d)

(e)

Figure 13.2 Radial-flow hydrostaticthrust bearing with circular step pad.

Pressure distribution in contact for vari-

ous values of dimensionless normal ve-

locity parameter bw. (a) bw= -1.0; (b) bw= -0.5; (c) bw; (d) bw= 0.25; (e) bw=

0.75. Dimensionless central film thick-

ness, Hmin, 1.0 x 10-4; radius ratio ar,

1.0; dimensionless fluid inlet level Hin,

0.0006. [From Ghosh et al. (1985)].


43/71


Performance Parameters

3

2

1

010-2 10-1 100

Dmensionless normal velocity parameter, |w|

4

Dynam

icpeakpressure,

s

Normalapproach(

w= -1.0)

Normalseparation(

w= 0.75)

Effect of dimensionless normal velocity

parameter on dynamic peak pressureratio. Dimensionless central film thick-

ness Hmin, 1.0 x 10-4; radius ratio ar,

1.0; dimensionless fluid inlet level Hin,

0.035. [From Ghosh et al. (1985)].

3.0

2.8

2.6

Dynamic

loadratio,

t

Normal

approach

(= -1.0)

Normalseparation

(= 0.75)

Radius ratio, r

10-1 100 101 102

.46

.42

.38

Dynamicloadratio,

t

Figure 16.19 Effect of radius ratio on

dynamic load ratio. Dimensionlesscentral film thickness Hmin, 1.0 x 10-4;dimensionless fluid inlet level Hin, 0.035.[From Ghosh et al. (1985)].


44/71


Peak Pressure vs. Radius Ratio

4.2

3.8

3.4

.22

.20

.18

Normalapproach

(= -1.0)

Normal

separation(= 0.75)

Radius ratio, r

10-1 100 101 102Dimension

lesspeakpressure

ratio,

s

Dimension

lesspeakpressure

ratio,

s

Figure 16.20 Effect of radius ratio on dynamic peak pressure ratio.Dimensionless central film thickness Hmin, 1.0 x 10-4; dimensionless fluid inletlevel Hin, 0.035. [From Ghosh et al. (1985)].


45/71


Contact Geometry

wz

y

x

y

x

Solid a

Solid b

rby

ray

rbx

rax

wz

Figure 17.1 Geometry of contactingelastic solids. [From Hamrock andDowson (1981).]


46/71


Radii of Curvature

(a)

(b)

(c)

rax= r

raxr

ray= r ray ray ray

raxrax raxBarrel shapeSphere Cylinder Conic frustum Concave shape

ray

rbx

rbx

rbyRi

Thrust Radial inner Radial outer

rbyrby

rbx

Rirby

Thrust Cylindrical inner Cylindrical outer

rbx

rbxrbx

rbyrby

Figure 17.2 Sign designations for radii of curvature of various machineelements. (a) Rolling elements; (b) ball bearing races; (c) rolling bearingraces.


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Pressure Distribution

Dx2

pm

x

y

p

Dy2

Figure 17.3 Pressure distribution inellipsoidal contact.

p= pm

1

2x

Dx

2

2y

Dy

21/2

pm =6wz

!DxDy

Pressure:

Maximum pressure:


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Ellipticity Parameter andElliptic Integrals

0 4 8 12 16

Radius ratio, r

20 24 3228

Ellipticintegrals,

and

Elliptic integral

Elliptic integralEllipticity parameter k

e

Ellipticityparameter,k

e

1

2

3

4

5

2

4

6

8

10

Figure 17.4 Variation of ellipticity parameter and elliptic integrals of first andsecond kinds as function of radius ratio. [From Hamrock and Brewe (1983).]


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Hertz Contact Summary

Dy = 2

6k2EwzR

!E

1/3

Dx= 2

6EwzR

!kE

1/3

!m = F

9

2ER

wz

"kE

21/3

E =

2

(1!2a)/Ea+(1!2

b)/Eb

Contact dimensions:

Maximum elastic deformation:

Effective elastic modulus:


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Elliptic Integrals


51/71


Effect of Radius Ratio onSubsurface Stress


52/71


Simplified Equations for EllipticIntegrals

Dy

2

Dx

2

x

y

Dy2

Dx2

x

y


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Fundamentals of Fluid Film LubricationHamrock, Schmid & Jacobson

ISBN No. 0-8247-5371-2

Conformity

a

b

(a) (b)

(c)

Figure 17.5 Three degrees ofconformity. (a) Wheel on rail; (b) ball onplane; (c) ball-outer-race contact. [FromHamrock and Brewe (1983).]

l l l


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ISBN No. 0-8247-5371-2

Calculation of ElasticDeformation

Table 18.1: Three ways of calculating elastic deformations. [From Houpert and Hamrock (1986); OK denoted Okamura(1982); HJ denotes Hamrock and Jacobson (1984); HH denotes Houpert and Hamrock (1986).]

Nmax Xmin Xendm/H 1

Hm/Hm

OK HJ HH OK HJ HH

51 -1.0 1.0 1.9 103 2.7 103 8.6 107 9.4 103 5.5 103 2.7 103

51 -3.6 1.4 4.9 103 1.1 102 1.0 105 3.4 102 1.6 102 8.6 103151 -1.0 1.0 6.3 104 5.1 104 5.1 108 4.0 103 1.4 103 6.4 104

151 -3.6 1.4 1.6 103 2.0 103 5.4 107 1.2 102 4.4 103 2.1 103

301 -1.0 1.0 3.1 104 1.8 104 8.1 109 2.2 103 5.6 104 2.5 104

301 -3.6 1.4 7.8 104 7.1 104 9.0 108 6.0 103 1.8 103 8.6 104

661 -3.6 1.4 2.2 106 2.7 103 2.6 104 51 a -1.0 a1.0 6.5 107 2.7 104aNonumiform


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ISBN No. 0-8247-5371-2

Load Components

HrHr,end= mHr,m

Xr,end Xr,maxr,min 0

1 2 3... N Nmaxl

Xr

Figure 18.1 Sketch to illustrate

calculations ofXr,endand N. [FromHoupert and Hamrock (1986).]

-dh

z

x

dx u

a

wbx'

wbz'

wb'

fb'

fa'

wb'

wa'

waz'

h

Figure 18.2 Load components and

shear forces. [From Hamrock andJacobson (1984).]

fil l


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ISBN No. 0-8247-5371-2

Profiles at EarlyIterations

1.50

1.25

1.00

.75

.50

.25

0

Iteration

0

1

14

h/2hm Pr

1.50

1.25

1.00

.75

.50

.25

0

h/2hm

Pr

Dimensionlesspressure,

Pr;d

imensionlessfilms

hape,

h/2

hm

Dimensionlessxcoordinate,

Xr= 2x/D

x

-3 -2 -1 0 1 2

Figure 18.3 Pressure profiles and filmshapes at iterations 0, 1, and 14 with

dimensionless speed, load, and materialparameters fixed at U= 1.0 x 10-11, W=

2.045 x 10-5, and G=5007. [FromHoupert and Hamrock (1986).]


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ISBN No. 0-8247-5371-2

Detail of Spike Location


Xr= 2x/D

x

(a)

.8

.6

.4

.2

0

Dimensionless

pressure,Pr=p/pH

5.50 10-2

5.25

5.00

4.75

4.50(b)

.90 .92 .94 .96 .98 1.00

Dimensionlessfilm

thickness,Hr=4hR/Dx2

100

0

-100

-200

-300

-400(a)

Dimensionlesspressure

gradient,dP

r/dX

r

.90 .92 .94 .96 .98 1.00

.3

0

-.3

-.6

-.9

-1.2


Xr= 2x/D

x

Dimensionlessfilm

thickness,

Hr-m

Hr,m

/

(b)

Figure 18.6 Pressure and film thicknessprofiles in region 0.9 !Xr!1.0. (a)Dimensionless pressure; (b) dimensionlessfilm thickness. [From Hamrock et al. (1988).]

Figure 18.7 Pressure gradient and Hr-

rmHr,m/rprofiles in region 0.9 !Xr!

1.0. [From Hamrock et al. (1988).]


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ISBN No. 0-8247-5371-2

Compressibility Effect2.5

2.0

1.5

1.0

.5

0

Dimensio

nlesspressure,

Pr=p/pH


Xr= 2x/Dx

-2 -1 0 1 2

Figure 18.8 Dimensionless pressure

and film thickness profiles for anincompressible fluid. Viscous effectswere considered. [From Hamrock et al.(1988).]

Dimensionlessxcoordinate, Xr= 2x/D

x

2.5

2.0

1.5

1.0

.5

0-2 -1 0 1 2

Dimensio

nlesspressure,

Pr=p/pH

Figure 18.9 Dimensionless pressure

and film thickness profiles for acompressible fluid. Viscous effects wereconsidered. [From Hamrock et al.(1988).]


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ISBN No. 0-8247-5371-2

Detail of Spike Location6.5 x 10-2

6.0

5.5

5.0

4.5

Dimension

lessfilmt

hickness,

Hr;

dimensionlessvariable,m

Hr,m

/

.90 .92 .94 .96 .98 1.00


Xr= 2x/Dx

Hr,icand (Hr,m)ic

(Hr,m)ic

Hr,ic

Hr,c

mHr,m

c

mHr,m

c

Hr,cand

Figure 18.10 Dimensionless film thickness and rm

Hr,m

/rprofiles for

compressible and incompressible fluids in region 0.9 !Xr!1.0. Viscous

effects were considered. [From Hamrock et al. (1988).]


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ISBN No. 0-8247-5371-2

Pressure as a Function of Load

Dimensionlessxcoordinate, Xe= x / Rx


Pe

=

p/E'

1.00 x 10-2

.75

.50

.25

0-.04 -.02 0 .02 .04

Dimensionless

load,

W'

2.04

4

6

13

30

50 x 10-5

Figure 18.11 Variation of dimensionless pressure in elastohydrodynamicallylubricated conjunction for six dimensionless loads with dimensionless speedand materials parameters held fixed at U=1.0 x 10-11and G=5007. [From Panand Hamrock (1989).]


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ISBN No. 0-8247-5371-2

Speed Effects

Dimensionlessxcoordinate, Xe= x / Rx

-.04 -.02 0 .02 .04

.5 x 10-2

.4

.3

.2

.1

0


Pe=

p/E'

Dimensionless

speed,

U

50 x 10-12

13

1

Figure 18.12 Variation ofdimensionless pressure in

elastohydrodynamically lubricatedconjunction for three dimensionless

speeds with dimensionless load andmaterials parameters held fixed at W =1.3 x 10-4and G=5007. [From Pan and

Hamrock (1989).]


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ISBN No. 0-8247-5371-2

Spike AmplitudeTable 18.2: Effect of dimensionless load, speed, and materials parameters on dimensionless pressure spike amplitude. [FromPan and Hamrock (1989)]

Dimensionless Dimensionless Dimensionless Dimensionless Curve-fit

load speed materials pressure spike dimensionless

parameter, parameter, parameter amplitude pressure spike Error, Results

amplitude percent,

W = wz

ERxU =

0u

ERxG = E Pe,s =

psk

EPe,s =

psk

E

Pe,s Pe,s

Pe,s

100

0.2045 104 1.0 1011 5007 0.22781 102 0.23218 102 -1.9161 Load

0.4 104

1.0 1011

5007 0.27653 102

0.26285 102

4.94730.6 104 1.0 1011 5007 0.30755 102 0.28332 102 7.8771

1.3 104 1.0 1011 5007 0.33890 102 0.32689 102 3.5431

3.0 104 1.0 1011 5007 0.36150 102 0.38159 102 -5.5564

5.0 104 1.0 1011 5007 0.44791 102 0.41941 102 6.3637

1.3 104 0.1 1011 5007 0.17489 102 0.17354 102 0.7707 Speed

1.3 104 .25 1011 5007 0.20763 102 0.22327 102 -7.5346

1.3 104 .5 1011 5007 0.24706 102 0.27016 102 -9.3501

1.3 104 .75 1011 5007 0.28725 102 0.30203 102 -5.1445

1.3 104 1.0 1011 5007 0.33890 102 0.32689 102 3.5431

1.3 104 3.0 1011 5007 0.42776 102 0.44219 102 -3.3744

1.3 104 5.0 1011 5007 0.49218 102 0.50889 102 -3.3946

2.6 104 2.0 1011 2504 0.33549 102 0.34291 102 -2.2107 Materials

1.3 104 1.0 1011 5007 0.33890 102 0.32689 102 3.5431

0.8667 104 0.6667 1011 7511 0.30760 102 0.31788 102 -3.3421


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ISBN No. 0-8247-5371-2

Spike LocationTable 18.3: Effect of dimensionless load, speed, and materials parameters on dimensionless pressure spike location. [FromPan and Hamrock (1989)]


load speed materials pressure spike dimensionless

parameter, parameter, parameter location pressure spike Error, Results

location percent,

W = wz

ERxU =

0u

ERxG =E Xe,s =

xs

RxXe,s =

xs

Rx

Xe,s

Xe,s

Xe,s

100

0.2045 104 1.0 1011 5007 0.48561 102 0.52467 102 -8.0429 Load

0.4 104 1.0 1011 5007 0.82948 102 0.78778 102 5.0254

0.6 104

1.0 1011

5007 1.08220 102

1.00722 102

6.92841.3 104 1.0 1011 5007 1.70980 102 1.60922 102 5.8827

3.0 104 1.0 1011 5007 2.68600 102 2.67116 102 0.5524

5.0 104 1.0 1011 5007 3.49820 102 3.64033 102 -4.0630

1.3 104 0.1 1011 5007 1.76520 102 1.68895 102 4.3201 Speed

1.3 104 .25 1011 5007 1.74670 102 1.65675 102 5.1495

1.3 104 .5 1011 5007 1.73000 102 1.63281 102 5.6178

1.3 104 .75 1011 5007 1.71620 102 1.61896 102 5.6655

1.3 104 1.0 1011 5007 1.70980 102 1.60922 102 5.8827

1.3 104 3.0 1011 5007 1.65310 102 1.57251 102 4.8747

1.3 104 5.0 1011 5007 1.62860 102 1.55574 102 4.4739

2.6 104 2.0 1011 2504 2.42060 102 2.28843 102 5.4601 Materials

1.3 104

1.0 1011

5007 1.70980 102

1.60922 102

5.88270.8667 104 0.6667 1011 7511 1.38400 102 1.30969 102 5.3689


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ISBN No. 0-8247-5371-2

FilmicknessTable 18.4: Effect of dimensionless load, speed, and materials parameters on dimensionless film thickness. [From Pan andHamrock (1989)]


load speed materials minimum film dimensionless

parameter, parameter, parameter, thickness minimum film Error, Results

thickness, percent,

W =

wz

ERxU =

0u

ERxG =E He,min =

hmin

RxHe,min=

hmin

Rx

He,min

He,min

He,min

100

0.2045 104 1.0 1011 5007 0.19894 104 0.20030 104 -0.6840 Load

0.4 104 1.0 1011 5007 0.18404 104 0.18382 104 0.1189

0.6 104

1.0 1011

5007 0.17558 104

0.17452 104

0.60121.3 104 1.0 1011 5007 0.15093 104 0.15808 104 -4.7367

3.0 104 1.0 1011 5007 0.13185 104 0.14203 104 -7.7228

5.0 104 1.0 1011 5007 0.14067 104 0.13304 104 5.4221

1.3 104 0.1 1011 5007 0.03152 104 0.03198 104 -1.4585 Speed

1.3 104 .25 1011 5007 0.06350 104 0.06040 104 4.8801

1.3 104 .5 1011 5007 0.10472 104 0.09771 104 6.6896

1.3 104 .75 1011 5007 0.13791 104 0.12947 104 6.1204

1.3 104 1.0 1011 5007 0.15093 104 0.15808 104 -4.7367

1.3 104 3.0 1011 5007 0.34963 104 0.33884 104 3.0857

1.3 104 5.0 1011 5007 0.48807 104 0.48301 104 1.0359

2.6 104 2.0 1011 2504 0.16282 104 0.15786 104 3.0462 Materials

1.3 104 1.0 1011 5007 0.15093 104 0.15808 104 -4.73670.8667 104 0.6667 1011 7511 0.16573 104 0.15821 104 4.5362


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ISBN No. 0-8247-5371-2

Variation in Film Shape

.5 x 10-4

.4

.3

.2

.1

0

Dimensionlessfilmt

hickness,

He

/14

=h/(14Rx

)

Dimensionless

load,

W'50 x 10

-5

30

2.04

46

13

Dimensionlessxcoordinate, Xe= x / R

x

-.04 -.02 0 .02 .04

Figure 18.13 Variation ofdimensionless film shape in

elastohydrodynamically lubricatedconjunction for six dimensionless

loads with dimensionless speedand materials parameters held

fixed at U=1.0 x 10-11and G=5007.[From Pan and Hamrock (1989).]


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ISBN No. 0-8247-5371-2

Central FilmicknessTable 18.5: Effect of dimensionless load, speed, and materials parameters on dimensionless central film thickness. [From Panand Hamrock (1989)]


load speed materials central dimensionless

parameter, parameter, parameter film thickness central Error, Results

film thickness percent,

W = w

z

ERxU =

0u

ERxG =E He,c =

hc

RxHe,c =

hc

Rx

He,c

He,c

He,c

100

0.2045 104 1.0 1011 5007 0.23436 104 0.23490 104 -0.2318 Load

0.4 104 1.0 1011 5007 0.21242 104 0.21015 104 1.0679

0.6 104 1.0 1011 5007 0.20003 104 0.19647 104 1.7785

1.3 104 1.0 1011 5007 0.16607 104 0.17281 104 -4.0564

3.0 104 1.0 1011 5007 0.13997 104 0.15041 104 -7.4577

5.0 104 1.0 1011 5007 0.14777 104 0.13818 104 6.4897

1.3 104 0.1 1011 5007 0.03508 104 0.03512 104 -0.1153 Speed

1.3 104 .25 1011 5007 0.06796 104 0.06621 104 2.5726

1.3 104 .5 1011 5007 0.11135 104 0.10697 104 3.9368

1.3 104 .75 1011 5007 0.14810 104 0.14161 104 4.3803

1.3 104 1.0 1011 5007 0.16607 104 0.17281 104 -4.0564

1.3 104 3.0 1011 5007 0.38079 104 0.36960 104 2.9395

1.3 104 5.0 1011 5007 0.53012 104 0.52632 104 0.7174

2.6 104 2.0 1011 2504 0.18627 104 0.17965 104 3.5561 Materials

1.3 104 1.0 1011 5007 0.16607 104 0.17281 104 -4.0564

0.8667 104 0.6667 1011 7511 0.17767 104 0.16893 104 4.9183


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ISBN No. 0-8247-5371-2

Film Shape for Different Speeds

Dimensionlessxcoordinate, Xe= x / R

x

-.04 -.02 0 .02 .04

1 x 10-4

.8

.6

.4

.2

0

50 x 10-12

D

imensionlessfilmt

hicknes

s,

He

/4=h/(4Rx

) Dimensionless

speed,

U

13

1

Figure 18.14 Variation ofdimensionless film shape for three

dimensionless speeds withdimensionless load and materials

parameters fixed at W = 1.3 x 10-4and

G=5007. [From Pan and Hamrock(1989).]

L i f Mi i Fil


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ISBN No. 0-8247-5371-2

Location of Minimum Filmickness

Table 18.6: Effect of dimensionless load, speed, and materials parameters on dimensionless location of minimum film thickness.[From Pan and Hamrock (1989)]


load speed materials location of dimensionless

parameter, parameter, parameter, minimum location of Error, Results

film thickness percent,

W = w

z

E

RxU =

0u

E

RxG =E Xe,min =

xmin

RxXe,min =

xmin

RxXe,min Xe,min

Xe,min 100

0.2045 104 1.0 1011 5007 0.60900 102 0.62598 102 2.7883 Load

0.4 104 1.0 1011 5007 0.92850 102 0.90398 102 -2.6410

0.6 104 1.0 1011 5007 1.11681 102 1.12884 102 1.0774

1.3 104 1.0 1011 5007 1.76850 102 1.72427 102 -2.5011

3.0 104 1.0 1011 5007 2.72800 102 2.72634 102 -0.0609

5.0 104 1.0 1011 5007 3.53610 102 3.60682 102 2.0000

1.3 104 0.1 1011 5007 1.79940 102 1.76874 102 -1.7038 Speed

1.3 104 .25 1011 5007 1.78670 102 1.75091 102 -2.0032

1.3 104 .5 1011 5007 1.77760 102 1.73754 102 -2.2538

1.3 104 .75 1011 5007 1.777030 102 1.72976 102 -2.2898

1.3 104 1.0 1011 5007 1.76850 102 1.72427 102 -2.5011

1.3 104 3.0 1011 5007 1.73760 102 1.70344 102 -1.9657

1.3 104 5.0 1011 5007 1.72300 102 1.69385 102 -1.6920

2.6 104 2.0 1011 2504 2.5139 102 2.45687 102 -2.2686 Materials

1.3 104 1.0 1011 5007 1.76850 102 1.72427 102 -2.5011

0.8667 104 0.6667 1011 7511 1.43360 102 1.40172 102 -2.2241


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ISBN No. 0-8247-5371-2

Center of PressureTable 18.7: Effect of dimensionless load, speed, and materials parameters on dimensionless center of pressure. [From Panand Hamrock (1989)]

Dim ensionless Dim ensionless Dim ensionless Dim ensionless Curve-fit

load speed materials center of dimensionless

parameter, parameter, parameter, pressure center of Error, Results

pressure percent,

W = w

z

ERxU =

0u

ERxG =E Xe,cp =

xcp

RxXe,cp =

xcp

Rx

Xe,cp

Xe,cp

Xe,cp

100

0.2045 104 1.0 1011 5007 1.00670 105 1.02309 105 -1.6284 Load

0.4 104 1.0 1011 5007 0.57148 105 0.51651 105 9.6191

0.6 104 1.0 1011 5007 0.33977 105 0.34169 105 -0.5670

1.3 104 1.0 1011 5007 1.4402 105 0.15541 105 -7.9060

3.0 104 1.0 1011 5007 0.06233 105 0.06628 105 -6.3391

5.0 104 1.0 1011 5007 0.04330 105 0.03939 105 9.0425

1.3 104 0.1 1011 5007 0.03231 105 .003577 105 -10.6959 Speed

1.3 104 .25 1011 5007 0.05883 105 0.06417 105 -9.0826

1.3 104 .5 1011 5007 0.09300 105 0.09987 105 -7.3812

1.3 104 .75 1011 5007 0.12202 105 0.12935 105 -6.0051

1.3 104 1.0 1011 5007 0.14402 105 0.15540 105 -7.9060

1.3 104 3.0 1011 5007 0.29264 105 0.31324 105 -7.0380

1.3 104 5.0 1011 5007 0.38774 105 0.43392 105 -11.9105

2.6 104 2.0 1011 2504 0.14662 105 0.15295 105 -4.3162 Materials1.3 104 1.0 1011 5007 0.14402 105 0.15541 105 -7.9060

0.8667 104 0.6667 1011 7511 0.15127 105 0.15686 105 -3.6961


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Mass Flow Rate

Table 18.8: Effect of dimensionless load, speed, and materials parameters on dimensionless mass flow rate. [From Pan andHamrock (1989)]


load speed materials mass flow dimensionless

parameter, parameter, parameter, rate mass flow Error, Results

rate percent,

W = w

z

ERxU=

0u

ERxG =E mHm mHm

mHm Hm

mHm

100

0.2045 104 1.0 1011 5007 0.26694 104 0.26760 102 -0.2445 Load

0.4 104 1.0 1011 5007 0.24844 104 0.24508 102 1.3511

0.6 104 1.0 1011 5007 0.23767 104 0.23241 102 2.2152

1.3 104 1.0 1011 5007 0.20291 104 0.21002 102 -3.5031

3.0 104 1.0 1011 5007 0.17557 104 0.18823 102 -7.2087

5.0 104 1.0 1011 5007 0.18788 104 0.17604 102 6.3004

1.3 104 0.1 1011 5007 0.04288 104 0.04268 104 -0.4590 Speed

1.3 104 .25 1011 5007 0.08350 104 0.08047 104 3.1072

1.3 104 .5 1011 5007 0.13607 104 0.13000 104 4.4608

1.3 104 .75 1011 5007 0.18096 104 0.17211 104 4.8921

1.3 104 1.0 1011 5007 0.20291 104 0.21002 104 -3.5031

1.3 104 3.0 1011 5007 0.46515 104 0.44918 104 3.4323

1.3 104 5.0 1011 5007 0.64744 104 0.63965 104 1.2028

2.6 104 2.0 1011 2504 0.22212 104 0.21325 104 3.9954 Materials

1.3 104

1.0 1011

5007 0.20291 104

0.21002 104

-3.50310.8667 104 0.6667 1011 7511 0.21995 104 0.20816 104 5.3607

Documents

hydostatic bearing systems