30079_21a.pdf

CHAPTER 21

LUBRICATION OF MACHINE

ELEMENTS

Bernard J. Hamrock

Department of Mechanical Engineering

Ohio State University

Columbus, Ohio

SYMBOLS

508

21.3 ELASTOHYDRODYNAMIC

LUBRICATION

556

21.3.1 Contact Stresses and

Deformations

21.1 LUBRICATION

FUNDAMENTALS

558

512

21.3.2 Dimensionless Grouping

566

21.1.1 Conformal and

Nonconformal Surfaces

21.3.3 Hard-EHL Results

568

512

21.3.4 Soft-EHL Results

572

21.1.2 Bearing Selection

513

21.3.5 Film Thickness for Different

Regimes of Fluid-Film

Lubrication

21.1.3 Lubricants

516

21.1.4 Lubrication Regimes

518

573

21.1.5 Relevant Equations

520

21.3.6 Rolling-Element Bearings

576

21.2 HYDRODYNAMIC AND

HYDROSTATIC

LUBRICATION

21.4 BOUNDARYLUBRICATION

616

21.4.1 Formation of Films

618

523

21.2.1 Liquid-Lubricated

Hydrodynamic Journal

Bearings

21.4.2 Physical Properties of

Boundary Films

619

524

21.4.3 Film Thickness

621

21.4.4 Effect of Operating

Variables

21.2.2 Liquid-Lubricated

Hydrodynamic Thrust

Bearings

621

21.4.5 Extreme-Pressure (EP)

Lubricants

530

21.2.3 Hydrostatic Bearings

536

623

21.2.4 Gas-Lubricated

Hydrodynamic Bearings

545

By the middle of this century two distinct regimes of lubrication were generally recognized. The first

of these was hydrodynamic lubrication. The development of the understanding of this lubrication

regime began with the classical experiments of Tower, 1 in which the existence of a film was detected

from measurements of pressure within the lubricant, and of Petrov, 2 who reached the same conclusion

from friction measurements. This work was closely followed by Reynolds' celebrated analytical

paper 3 in which he used a reduced form of the Navier-Stokes equations in association with the

continuity equation to generate a second-order differential equation for the pressure in the narrow,

converging gap of a bearing contact. Such a pressure enables a load to be transmitted between the

surfaces with very low friction since the surfaces are completely separated by a film of fluid. In such

a situation it is the physical properties of the lubricant, notably the dynamic viscosity, that dictate

the behavior of the contact.

The second lubrication regime clearly recognized by 1950 was boundary lubrication. The under-

standing of this lubrication regime is normally attributed to Hardy and Doubleday, 4 - 5 who found that

very thin films adhering to surfaces were often sufficient to assist relative sliding. They concluded

that under such circumstances the chemical composition of the fluid is important, and they introduced

the term "boundary lubrication." Boundary lubrication is at the opposite end of the lubrication

Mechanical Engineers' Handbook, 2nd ed., Edited by Myer Kutz.

spectrum from hydrodynamic lubrication. In boundary lubrication it is the physical and chemical

properties of thin films of molecular proportions and the surfaces to which they are attached that

determine contact behavior. The lubricant viscosity is not an influential parameter.

In the last 30 years research has been devoted to a better understanding and more precise definition

of other lubrication regimes between these extremes. One such lubrication regime occurs in noncon-

formal contacts, where the pressures are high and the bearing surfaces deform elastically. In this

situation the viscosity of the lubricant may rise considerably, and this further assists the formation

of an effective fluid film. A lubricated contact in which such effects are to be found is said to be

operating elastohydrodynamically. Significant progress has been made in our understanding of the

mechanism of elastohydrodynamic lubrication, generally viewed as reaching maturity.

This chapter describes briefly the science of these three lubrication regimes (hydrodynamic, elas-

tohydrodynamic, and boundary) and then demonstrates how this science is used in the design of

machine elements.

SYMBOLS

A p

total projected pad area, m 2

a b

groove width ratio

a f

bearing-pad load coefficient

total conformity of ball bearing

semiminor axis of contact, m; width of pad, m

length ratio, b s lb r

b g

length of feed groove region, m

b r

length of ridge region, m

b s

length of step region, m

dynamic load capacity, N

C 1

load coefficient, FIp 0 Rl

radial clearance of journal bearing, m

pivot circle clearance, m

c b

bearing clearance at pad minimum film thickness (Fig. 21.16), m

c d

orifice discharge coefficient

distance between race curvature centers, m

material factor

D x

diameter of contact ellipse along x axis, m

D y

diameter of contact ellipse along y axis, m

diameter of rolling element or diameter of journal, m

d a

overall diameter of ball bearing (Fig. 21.76), m

d b

bore diameter of ball bearing, m

d c

diameter of capillary tube, m

inner-race diameter of ball bearing, m

d 0

outer-race diameter of ball bearing, m

d 0

diameter of orifice, m

modulus of elasticity, NYm 2

1 - vlY 1

/1 - v 2 a

I , NYm 2

effective elastic modulus, 2 I

\ E 0

E b /

metallurgical processing factor

elliptic integral of second kind

eccentricity of journal bearing, m

applied normal load, N

load per unit length, N/m

lubrication factor

elliptic integral of first kind

F c

pad load component along line of centers (Fig. 21.41), N

F e

rolling-element-bearing equivalent load, N

F r

applied radial load, N

F 5

pad load component normal to line of centers (Fig. 21.41), N

F t

applied thrust load, N

/ race conformity ratio

f c coefficient dependent on materials and rolling-element bearing type (Table 21.19)

G dimensionless materials parameter

G speed effect factor

G f groove factor

g e dimensionless elasticity parameter, W 8 ^IU 2

g v dimensionless viscosity parameter, GW 3 JU 2

H dimensionless film thickness, hiR x

H misalignment factor

H a dimensionless film thickness ratio, h s lh r

H b pad pumping power, N m/sec

H 0 power consumed in friction per pad, W

H f pad power coefficient

H min dimensionless minimum film thickness, h min /R x

fi mi n dimensionless minimum film thickness, H min (W/U) 2

Hp dimensionless pivot film thickness, h p /c

H t dimensionless trailing-edge film thickness, h t lc

h film thickness, m

h t film thickness ratio, H 1 Ih 0

h { inlet film thickness, m

h t leading-edge film thickness, m

/z mi n minimum film thickness, m

h 0 outlet film thickness, m

h p film thickness at pivot, m

h r film thickness in ridge region, m

h s film thickness in step region, m

h t film thickness at trailing edge, m

/Z 0 film constant, m

J number of stress cycles

K load deflection constant

K dimensionless stiffness coefficient, cK p lpJ^l

K a dimensionless stiffness, —c dW/dc

K p film stiffness, N/m

K 1 load-deflection constant for a roller bearing

K 1 5 load-deflection constant for a ball bearing

K 00

dimensionless stiffness, cKplpJtl

ellipticity parameter, D y ID x

k c

capillary tube constant, m 3

orifice constant, m 4 /N 1/ 2 sec

k 0

fatigue life

L 0

adjusted fatigue life

L 1 0

fatigue life where 90% of bearing population will endure

L 5 0

fatigue life where 50% of bearing population will endure

bearing length, m

1 0

length of capillary tube, m

l r

roller effective length, m

l t

roller length, m

l v

length dimension in stress volume, m

1 1

total axial length of groove, m

probability of failure

stability parameter, mp a h 5 r /2R 5 /rf

number of rows of rolling elements

mass supported by bearing, N sec 2 /m

m p preload factor

N rotational speed, rps

N R Reynolds number

n number of rolling elements or number of pads or grooves

P dimensionless pressure, piE'

P d diametral clearance, m

P e free endplay, m

p pressure, N/m 2

p a ambient pressure, N/m 2

P 1 lift pressure, N/m 2

/? ma x maximum pressure, N/m 2

p r recess pressure, N/m 2

p s bearing supply pressure, N/m 2

Q volume flow of lubricant, m 3 /sec

Q dimensionless flow, 3r]Q/7rp a h 3 r

Q 0 volume flow of lubricant in capillary, m 3 /sec

Q 0 volume flow of lubricant in orifice, m 3 /sec

Q s volume side flow of lubricant, m 3 /sec

q constant, TT/2 - 1

q f bearing-pad flow coefficient

R curvature sum on shaft or bearing radius, m

R groove length fraction, (R 0 - R 8 )/(R 0 - R 1 )

R g groove radius (Fig. 21.60), m

R 0 orifice radius, m

R x effective radius in x direction, m

Ry effective radius in 3; direction, m

R 1 outer radius of sector thrust bearing, m

R 2 inner radius of sector thrust bearing, m

r race curvature radius, m

r c roller corner radius, m

S probability of survival

Sm Sommerfeld number for journal bearings, r]Nd 3 l/2Fc 2

Sm t Sommerfeld number for thrust bearings, j]ubl 2 IFhl

s shoulder height, m

T tangential force, N

f dimensionless torque, 6 T r lirpJ(R\ + R%) h r h c

T 0 critical temperature

T r torque, N m

U dimensionless speed parameter, urj Q /E'R x

u mean surface velocity in direction of motion, m/sec

v elementary volume, m 3

N dimensionless load parameter, FIE'R 2

W dimensionless load capacity, F/pJ(b r + b s + b g )

W 00 dimensionless load, l.5G f F/irp a (R^ - R 2 ,)

X, Y factors for calculation of equivalent load

;c,v,z coordinate system

x distance from inlet edge of pad to pivot, m

a. radius ratio, RyIR x

a a offset factor

a b groove width ratio, b s l(b r + b s )

a p angular extent of pad, deg

a r radius ratio, R 2 IR 1

(3 contact angle, deg

/3' iterated value of contact angle, deg

p a groove angle, deg

fi f free or initial contact angle, deg

P p angle between load direction and pivot, deg

F curvature difference

y groove length ratio, I 1 Il

A rms surface finish, m

8 total elastic deformation, m

e eccentricity ratio, elc

TJ absolute viscosity of lubricant, N sec/m 2

r\ k kinematic viscosity, Wp, m 2 /sec

Tfo viscosity at atmospheric pressure, N sec/m 2

6 angle used to define shoulder height, deg

0 dimensionless step location, OJ(B 1 + O 0 )

O g angular extent of lubrication feed groove, deg

0 1 angular extent of ridge region, deg

0 0 angular extent of step region, deg

A film parameter (ratio of minimum film thickness to composite surface roughness)

A c dimensionless bearing number, 3>j]a)(R* - R 2 ^Ip 0 H 2

A 7 dimensionless bearing number, 6rja)R 2 /p a c 2

A, dimensionless bearing number, 6rjul/p a h 2

A length-to-width ratio

A a length ratio, (b r + b s + b g )ll

X b (1 + 2/Sa)- 1

IJL coefficient of friction, TIF

v Poisson's ratio

£ pressure-viscosity coefficient of lubricant, m 2 /N

g p angle between line of centers and pad leading edge, deg

p lubricant density, N sec 2 /m 4

PQ density at atmospheric pressure, N sec 2 /m 4

cr ma x maximum Hertzian stress, N/m 2

shear stress, N/m 2

T 0

maximum shear stress, N/m 2

attitude angle in journal bearings, deg

4> p

angle between pad leading edge and pivot, deg

angular location, deg

ifj t

angular limit of if/, deg

*l/ s

step location parameter, b s l(b r + b s + b g )

angular velocity, rad/sec

a} B

angular velocity of rolling-element race contact, rad/sec

a) b

angular velocity of rolling element about its own center, rad/sec

a) c

angular velocity of rolling element about shaft center, rad/sec

a> d

rotor whirl frequency, rad/sec

lo d

whirl frequency ratio, (o d /a)j

o)j

journal rotational speed, rad/sec

Sub-

scripts

a solid a

b solid b

EHL elastohydrodynamic lubrication

elastic

hydrodynamic lubrication

inner

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