30079_21c.pdf

Fig. 21.50 Chart for determining optimal film thickness. (From Ref. 28.) (a) Grooved member

rotating, (b) Smooth member rotating.

6. Calculate

R 1 = { AcP 0 } 112

h r [3T 7 K - Co 0 )[I - (K 2 /*,) 2 ]]

If R 1 Ih,. > 10,000 (or whatever preassigned radius-to-clearance ratio), a larger bearing or

higher speed is required. Return to step 2. If these changes cannot be made, an externally

pressurized bearing must be used.

7. Having established what a r and A c should be, obtain values of K 00 , Q, and T from Figs. 21.62,

21.63, and 21.64, respectively. From Eqs. (21.29), (21.30), and (21.31) calculate K pt Q, and

T r .

8. From Fig. 21.65 obtain groove geometry (b, /3 a , and H 0 ) and from Fig. 21.66 obtain R g .

21.3 ELASTOHYDRODYNAMICLUBRICATION

Downson 3 1 defines elastohydrodynamic lubrication (EHL) as "the study of situations in which elastic

deformation of the surrounding solids plays a significant role in the hydrodynamic lubrication pro-

cess." Elastohydrodynamic lubrication implies complete fluid-film lubrication and no asperity inter-

action of the surfaces. There are two distinct forms of elastohydrodynamic lubrication.

1. Hard EHL. Hard EHL relates to materials of high elastic modulus, such as metals. In this

form of lubrication not only are the elastic deformation effects important, but the pressure-viscosity

Fig. 21.51 Chart for determining optimal groove width ratio. (From Ref. 28.) (a) Grooved mem-

ber rotating, (b) Smooth member rotating.

effects are equally as important. Engineering applications in which this form of lubrication is dom-

inant include gears and rolling-element bearings.

2. Soft EHL Soft EHL relates to materials of low elastic modulus, such as rubber. For these

materials that elastic distortions are large, even with light loads. Another feature is the negligible

pressure-viscosity effect on the lubricating film. Engineering applications in which soft EHL is

important include seals, human joints, tires, and a number of lubricated elastomeric material machine

elements.

The recognition and understanding of elastohydrodynamic lubrication presents one of the major

developments in the field of tribology in this century. The revelation of a previously unsuspected

regime of lubrication is clearly an event of importance in tribology. Elastohydrodynamic lubrication

not only explained the remarkable physical action responsible for the effective lubrication of many

machine elements, but it also brought order to the understanding of the complete spectrum of lubri-

cation regimes, ranging from boundary to hydrodynamic.

A way of coming to an understanding of elastohydrodynamic lubrication is to compare it to

hydrodynamic lubrication. The major developments that have led to our present understanding of

hydrodynamic lubrication 1 3 predate the major developments of elastohydrodynamic lubrication 32 ' 3 3

Fig. 21.52 Chart for determining optimal groove length ratio. (From Ref. 28.) (a) Grooved mem-

ber rotating, (b) Smooth member rotating.

by 65 years. Both hydrodynamic and elastohydrodynamic lubrication are considered as fluid-film

lubrication in that the lubricant film is sufficiently thick to prevent the opposing solids from coming

into contact. Fluid-film lubrication is often referred to as the ideal form of lubrication since it provides

low friction and high resistance to wear.

This section highlights some of the important aspects of elastohydrodynamic lubrication while

illustrating its use in a number of applications. It is not intended to be exhaustive but to point out

the significant features of this important regime of lubrication. For more details the reader is referred

to Hamrock and Dowson. 1 0

21.3.1 Contact Stresses and Deformations

As was pointed out in Section 21.1.1, elastohydrodynamic lubrication is the mode of lubrication

normally found in nonconformal contacts such as rolling-element bearings. A load-deflection rela-

tionship for nonconformal contacts is developed in this section. The deformation within the contact

is calculated from, among other things, the ellipticity parameter and the elliptic integrals of the first

and second kinds. Simplified expressions that allow quick calculations of the stresses and deforma-

tions to be made easily from a knowledge of the applied load, the material properties, and the

geometry of the contacting elements are presented in this section.

Elliptical Contacts

The undeformed geometry of contacting solids in a nonconformal contact can be represented by two

ellipsoids. The two solids with different radii of curvature in a pair of principal planes (x and y)

Fig. 21.53 Chart for determining optimal groove angle. (From Ref. 28.) (a) Grooved member

rotating. (D) Smooth member rotating.

passing through the contact between the solids make contact at a single point under the condition of

zero applied load. Such a condition is called point contact and is shown in Fig. 21.67, where the

radii of curvature are denoted by r's. It is assumed that convex surfaces, as shown in Fig. 21.67,

exhibit positive curvature and concave surfaces exhibit negative curvature. Therefore if the center of

curvature lies within the solids, the radius of curvature is positive; if the center of curvature lies

outside the solids, the radius of curvature is negative. It is important to note that if coordinates x and

y are chosen such that

I + -U-U-L

(21 . 33 )

T 0x

r bx

r ay

r by

coordinate x then determines the direction of the semiminor axis of the contact area when a load is

applied and y determines the direction of the semimajor axis. The direction of motion is always

considered to be along the x axis.

Fig. 21.54 Chart for determining maximum radial load capacity. (From Ref. 28.) (a) Grooved

member rotating, (b) Smooth member rotating.

The curvature sum and difference, which are quantities of some importance in the analysis of

contact stresses and deformations, are

< 2135 >

r - "(K- i)

i-H

where

< 21 - 37 )

F = f + f

*'•*>

K x r ax T bx

5-r + r

r ay

*by

« = TT

(21.38)

K x

Equations (21.36) and (21.37) effectively redefine the problem of two ellipsoidal solids approaching

one another in terms of an equivalent ellipsoidal solid of radii R x and R y approaching a plane.

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