oil film thickness and pressur distributioe n in elastohydrodynamic elliptica contactl s · 2016....

- 1 -

OIL FILM THICKNESS AND PRESSURE DISTRIBUTION

IN ELASTOHYDRODYNAMIC ELLIPTICAL CONTACTS

BY

ALI MOSTOFI, MSc, DIC

A THESIS SUBMITTED FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

OF THE

UNIVERSITY OF LONDON

SEPTEMBER 1981

DEPARTMENT OF MECHANICAL ENGINEERING

IMPERIAL COLLEGE OF SCIENCE AND TECHNOLOGY

UNIVERSITY OF LONDON

- 2 -

ABSTRACT

The pressure distribution, contact arc extent, and deflection are

determined for the elastic contact of a long roller and a cylindrical hole

in an infinite body. It is shown that these variables depend on the

degree of conformity of the two bodies and that Hertz's theory should hot

be used when the arc dimensions are of the same order as the radii of

curvature, as its assumptions are then violated. However, for the case

of circular arc gears, the conformity is such that use of Hertz's theory

is quite adequate. When the hole circumference available is reduced,

pressure discontinuities can occur where the arc terminates.

A numerical method is then developed in which the three-dimensional

elastostatic pressure distribution and contact area shape in frictionless

concentrated contact problems between two elastic bodies of any arbitrary

profiles can be found.

In order to avoid geometrical singularities, an element is

constructed to follow the pressure curve in the vicinity of the

discontinuity zone. It is shown that using this type of element, the

numerical elastostatic results become very accurate.

A general numerical solution to the elastohydrodynamic EHL point

contact problem is presented for moderate loads and material parameters.

Isobars, contours, and regression formulae describe how pressure and oil

film thickness vary with geometry, material properties, load and squeeze

velocity, when the rolling velocity vector is at various angles to the

static contact ellipse long axis. In addition, the EHL behaviour under

pure spin is examined. The theoretical predictions of film thickness

compare favourably with other numerical solutions to the point contact

problem, as well as with experimental results which use the optical

interferometry method to find film thickness and shape.

- 3 -

The EHL numerical method is also applied to a finite cylindrical

roller with axially profiled ends rolling over the surface of an elastic

half space. Convergence was again obtained for moderate loads and

material parameters. Isobars, contours, and section graphs show pressure

variation and film shape. The maximum EHL pressures occur near the

start of the profiling and can exceed the pressure concentrations predicted

by elastostatic theory.

- 4 -

ACKNOWLEDGEMENTS

I would like to acknowledge my greatest indebtedness to my supervisor,

Dr Ramsey Gohar, for his help and constant encouragement throughout the

research period. ~

I am also indebted to Dr P.B. Macpherson for his encouragement and

for providing the funds from Westland Helicopters Limited and Ministry of

Defence, whom I am also grateful to, which enabled me to perform the work

outlined in this thesis.

My thanks to Mrs E.A. Hall for typing this manuscript.

- 5 -

CONTENTS

Page

Title Page 1

Abstract 2

Acknowledgements 4

Contents 5

List of Figures 11

List of Tables 15

Notation to Chapter 1 16





CHAPTER 1: PRESSURE DISTRIBUTION BETWEEN CLOSELY CQNFORMAL

CONTACTING SURFACES 27

1.1 Introduction 27

1.2 Theory 28

1.2.1 Radial deflection of a long roller due to

concentrated line forces 28

1.2.2 Radial deflection of a circular hole due to

concentrated line forces 29

1.3 Compression of Two Closely Conformal Cylindrical

Bodies 36

1.4 Method of Solution 43

1.5 Pressure Distribution from Hertz's Theory 46

1.6 Total Approach of the Bodies from Hertz's Theory 47

1.7 Results 48

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Page

1.8 Importance of Nodal Points Density 50

1.9 Conclusion 50

1.10 References 60

CHAPTER 2: ELASTOSTATIC PRESSURE DISTRIBUTION 61

2.1 Introduction 61

2.2 Introduction to the Model of Novikov Gears 63

2.3 . Theory 64

2.4 Compression of Two Elastic Bodies Having Arbitrary

Profiles 69


2.6 Approximation of the Overestimated Contact Boundary

Shape of a Sphere Indenting Inside a Cone 72

2.7 Results 75

2.8 Discussion 76

2.9 References 83

CHAPTER 3: SINGULARITY ELEMENTS 85

3.1 Introduction 85

3.2 Theory for a Rigid Rectangular Punch and an Elastic

Half Space 87

3.3 Introduction to Triangular and Rectangular Elements 91

3.4 Influence Coefficients 93

3.4.1 Influence coefficients for triangular elements 93

3.4.2 Influence coefficients for rectangular elements 94


3.6 Results of Comparing the Accuracy of Triangular and

Rectangular Elements 95

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Page

3.7 Theory for a Singularity Element 101

3.8 Influence Coefficients for Singularity Element 103

3.9 Setting Up the Matrix Involving the Singularity

Element Influence Coefficients 104

3.10 Results of the Accuracy of the Technique Employing

Singularity Element 106

3.11 Introduction to the Application of Three-Dimensional

Singularity Element in Stress Concentration Problems 106

3.12 Three-Dimensional Singularity Element 111

3.13 Influence Coefficients for Three-Dimensional Element ill

3.14 Setting Up the Matrix Involving the Three-Dimensional 1

Singularity Element Influence Coefficients 115

3.15 The Gap Shape Between a Sphere and a Cylinder Which is

Mounted on the Edge of a Cone 115

3.16 Results of Three-Dimensional Stress Concentration n g

3.17 Discussion 120

3.18 References 126

CHAPTER 4: OIL FILM THICKNESS AND PRESSURE DISTRIBUTION IN EHL

ELLIPTICAL CONTACTS 128

4.1 Introduction 128

4.2 Theory 129

4.2.1 Reynolds equation 129

4.2.2 Equations of state of the lubricant 130

4.2.3 Final form of Reynolds equation 131

4.2.4 Lubricant film thickness 133

4.2.5 Elastic film shape 136

4.2.6 Load 137

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Page


4.3.1 Mesh construction 137

4.3.2 Boundary conditions 138

4.3.3 Initial condition 138

4.3.4 The algorithm used to solve the equations 138

4.3.5 Convergence criteria and number of elements

used 142

4.3.6 Relaxation factors 142

4.4 Non-Dimensional Groups 143

4.5 Results 144

4.5.1 General 144

4.5.2 Effect of changing e 145

4.5.3 Squeeze effects 158

4.6 Regression Analysis 166

4.6.1 Final regression formula for minimum and

central film thicknesses 166

4.6.2 Speed and 9 166

4.6.3 Ellipticity ratio, normal speed, load and

material parameters 168

4.7 Testing the Regime of the Obtained Numerical Results 170

4.8 Comparison With Other Work 173

4.8.1 Relation between the groups used here and the

groups used in other work 173

4.8.2 Other formulae 174

4.8.3 Results of comparison 175

4.9 EHL Under Pure Spin 187

4.9.1 Introduction to spin 187

4.9.2 Theory of spin 188

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Page

4.9.3 Results of spin 190

4.10 Conclusions 191

4.11 References 199

CHAPTER 5: ELASTOHYDRODYNAMIC LUBRICATION OF FINITE LENGTH

CONTACTS 201

5.1 Introduction 201

5.2 Theory 202

5.2.1 Reynolds equation 202

5.2.2 Irregular mesh 203

5.2.3 Reynolds equation in central difference, form 206

5.2.4 Lubricant film thickness 208

5.2.5 Elastic film shape 210

5.2.6 Load 210


5.3.1 Mesh construction 210

5.3.2 Boundary conditions 211

5.3.3 Initial condition 211

5.3.4 The algorithm used to solve the equations 211

5.3.5 Convergence criteria and number of elements

used 211

5.3.6 Relaxation factors 211

5.4 Non-Dimensional Groups 211

5.5 End Closures 212

5.6 Oil Film Thickness and Pressure Distribution 213

5.7 Variation of Film and Pressure in the Direction of

Rolling Near the Roller Ends 230

5.8 Testing the Regime of the Obtained Numerical Results 230

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Page

5.9 Comparison With Other Work 232

5.9.1 Other formulae 232

5.9.2 Results of comparison 233

5.10 Discussion and Conclusions 236

5.11 References 237

CHAPTER 6: SUMMARY OF CONCLUSIONS AND SUGGESTIONS FOR FURTHER WORK 239

6.1 Summary of Conclusions 239

6.2 Suggestions for Further Work 240

6.2.1 Theoretical 240

6.2.2 Experimental Further Study 243

6.3 References 247

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LIST OF FIGURES

Figure Number

1 . 1

1.2

1.3

1.4

1.5

1.6-1.8

1.9

1.10

1.11

1.12-1.13

1.14

1.15

1.16

1.17

1.18

Title

Radial deflection on roller surface

Deflection components within roller

Alternative roller and hole loadings

Bodies before compression

Bodies after compression

Geometric representation

Pressure over the arc of contact

Elements of pressure distribution

Pressure distribution and distorted shapes inside and outside contact

Pressure distribution and distorted shapes inside contact

Variation of contact arc with load

Variation of total approach with load

Variation of deflection ratio with load

Effect on pressure distribution of truncated hole boundaries

Importance of nodal points density

Page

30

30

30

39

39

40-41

45

45

52

53-54

55

56

57

58

59

2.1

2.2

2.3

2.4

2.5

2.6

2.7-2.11

Simulation of conformal gear contact 65

Plan view of the half space 67

Diagram of pressure acting in rectangular boxes over each element 67

Arithmetic progression mesh 68

Condition of the bodies before and after compression 70

Diagram of sphere inside a cone before compression 73

Elastostatic isobar plot 78-82

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F igu re Number

3.1

3.2

3.3

3.4-3.5

3.6

3.7-3.9

3.10

3.11

3.12

3.13

3.14-3.15

T i t l e Page

Diagram of elastic half space 88

Rigid punch and elastic half space 89

Triangular and rectangular elements 92

Pressure distribution between the punch and

half space 97-100

Singularity element 102

Pressure distribution between the punch and

half space 107-109

Overriding gears 112

Three-dimensional singularity element 113

End view of the conical body showing a radial section of the cone and its relief 117 Mesh showing the top two rows being in the profiled regions 118

Elastostatic pressure profile 121-123

4.1

4.2

4.3

4.4

4.5

4.6

4.7-4.8

4.9

4.10-4.12

4.13

Film thickness components 134

Sphere inside a cone before compression 135

Rectangular grid over the computation zone 140-141

Figures (a) and (b): Computed isobars and contours when e = 0 Figure (c): Experimental photograph from reference (11) of Chapter 4 147-148

Figures (a) and (b): Computed and experimental (11) contours when 0 = 36° 149-150

Figures (a) and (b): Computed and experimental

(11) contours when 0 = 90° 151-152

Pressure distribution and film shape 153-154

Portrays of three-dimensional pressure distribution 155-157

Pressure distribution and film shape when there is a squeeze velocity 159-164

Contour plot with squeeze velocity present 165

- 13 -

F i gu re Number T i t l e Page

4.14-4.15 : Pressure distribution and film thickness graphs 171-172

4.16 : Variation of central and minimum film J thicknesses with speed (comparison with other

work) 176-178

4.17 : Variation of central and minimum film thicknesses with speed for different drive angles 179-180

4.18-4.19 : Variation of central and minimum film thicknesses with speed (comparison with reference (11)) 181-186

4.20 : Coordinates for spin speed 189

4.21-4.22 : Contours, isobars, experimental photograph (reference (l"l)) and three-dimensional portrays for spin 193-198

5.1 : Mesh construction 204-205

5.2 : Axial roller profiles 209

5.3 : Variation of minimum side constriction film thickness with the dub-off radius 215

5.4 : Variation of film thickness along the length of

the roller at outlet section 216

5.5 : Variation of film thickness with speed 217

5.6 . : Variation of the ratio with speed 218

5.7-5.8 : Contours, isobars and experimental photographs taken from reference (11) of Chapter 5 219-223

5.9 : Diagram of different sections to which the results will be referred 224

5.10 : Pressure distribution and film thickness along the central line of the roller 225

5.11 : Pressure distribution along the axial length of the roller 226-227

5.12 : Film thickness and pressure distribution along the restricted zone 228-229

5.13 : Film thickness chart from reference (6) of Chapter 5 231

5.14 : Variation of minimum exit and central film thicknesses with speed (comparison with other work) 234-235

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F i gu re Number T i t l e Page

6.1 : Dome ended roller with cross-section of its races 245

6.2 : Novikov simulation rig 246

- 15 -

Table Number

1.1

3.1

LIST OF TABLES

Title

Standard integrations from reference (8) of Chapter 1

Accuracy of the numerical method using different types of elements

4.1 The range of parameters, number of values taken and the accuracy of the regressed results

- 16 -

NOTATION FOR CHAPTER 1

: total approach of distant points in roller and body

= d/R2

radial deflection on roller surface due to a pressure triangle

: radial deflection on hole surface due to a pressure triangle

-

Young's moduli

line force per unit length

shear moduli of roller and body

VX

2

1/{4(X l i 2

+ Gh2)\

roller length

number of pressure elements

pressure

triangular element vertex pressure

maximum pressure in Hertz's theory

P.K> } P -K0 mg 2 mg <L

vector defining a point in the elastic body

vector defining a point in the roller

radii of roller and hole

R2/R

2

radial displacement at r,(f> in the elastic body

radial displacements at <j> on roller and hole surfaces

cartesian displacements in the roller

UYR2

- 17 -

" U2i/R2 : radial displacement at <j> on roller and hole surface

: tangential displacement at <{> on roller and hole surface

: load

= ( W K p / t L R ^ (materials the same)

: cartesian coordinates

: radial strain

: Lame's constant of roller and body

: Poisson's ratios

: stress in polar coordinates

: angular coordinates of displacement

: angular coordinates of pressure

: contact arc

: contact arc by Hertz's theory

: pressure triangle base arc

- 18 -


2a , 2b

d

2 d. .

d. .

d(x,y)

E E. '1,2

h

K2 n

P

P . 3

P. 3

P(x',y')

r1 , r

r2 ,R R,

R s

V. %

V.

V 1(0,0) ' v2(0,0) x , y

x' ,i/'

xb>Vb

dimensions of one element {2a,2b vary for elements)

total approach of the elastic bodies

d/Rx

influence coefficient for deflection (deflection at node i due to force at node j)

d. ./ (RJ?.) T>*3

1

3

deflection at a point x,y

E^l-v^/E^l-v*) Young's moduli

gap separation of the bodies at node i

W *1

number of elements

pressure

pressure at node j

• P / 2

: pressure at x',y'

: as shown in Figure 2.6

= r/flj

radius of one of the bodies (sphere in this case)

radius of sphere

total deflection at node i

• V/Ri

deflection of the bodies at node i

centre deflection of the bodies

cartesian coordinates of the mesh

coordinates for the elemental pressure

coordinates of the overestimated contact boundary

- 19 -

= xb/R

2 , y

b/R

2

e : half cone angle

: Poisson's ratios 1,2

: Poisson's ratios

<f> : skew angle

OJ , O) c s

: speed of cone and sphere

- 20 -


: dimensions of the punch

: dimensions of singularity element in three-dimensional case

= a t

/R1 , b '/R

2

: base length of singularity element in two-dimensional case

: base length of a rectangular element

: half base length of a triangular element

: influence coefficient in matrix form

: influence coefficient for deflection

: d(x,"^)/(aPj) for two-dimensional case, and d(x,y)/(R2P<c) for three-dimensional case

: Young's modulus

: gap separation for the cone and sphere

- h/R2

= (l-o2

)/irE

: pressure curve described by singularity element

: pressure

=

: as shown in Figure 2.6 =

: radius of relief cylinder

= Vs

:

: radius of sphere

: radius of one of the bodies (sphere in this case)

: deflection

= V/a

: load per unit length

: cartesian coordinates used

- 21 -

x , y : x/a,y/b for two-dimensional case, and x/R ,y/Rx for

three-dimensional case

x' ,yr : coordinates used for where the pressure acts

xf

,y' : x'/a,y'/b for two-dimensional case, and s'/f?^,

y three-dimensional case

C : coordinate used for singularity element

e : half cone angle

v : Poisson's ratio

- 22 -


a9b : the long and short axes of the ellipse

A 9B : the long and short axes of the ellipse (on the

graphs only)

CONVP : convergence required

d.. : influence coefficient for deflection dig>kl = dig,kl/Rl

= a/b E1 9

: Young's moduli of materials

E' : reduced Young's modulus

E ' = E^l-ofl/E^l-v*) ' f b J

3

: finite difference equation coefficients

G* = a/K'

h : film thickness

h = h/R1

hg : gap separation of the bodies

h 'g M YBi h : minimum film thickness m K - VRi h

Q : central film thickness

h* - V ^ I

Kf

=

n , .n 1

: number of rows and columns in the rectangular mesh VOW COL>

P : pressure

P = PK'

P^ : dimensionI ess EHL pressure in iteration procedure

p : dimensionless elastostatic pressure in iteration s

procedure

q : reduced pressure

- 23 -

q = qK>

R2 ,R : radii of curvature of the bodies

fl* = R / R2

u : speed component in x direction

us : spin component in x direction

U = /u2

+y2

U* = U^K'/Rj v : speed component in y direction

v : spin component in y direction

w(x,y) : deflection at x,y

w(x,y) - w(x,y)/R2

W : load

W ,u : dimensionless load, speed and material parameters

used by others

W* = WK'/Rj2

Wq : normal speed

w* = w /u s s

x , y : cartesian coordinates x = x/Rj , y/Rj x , v : coordinates with respect to the centre of contact a a o

= X

/R

l ^ c /R

l

X,Y : coordinates defining the geometrical gap

T,T = X/R2 ,1/R

2

a : pressure-viscosity coefficient of lubricant

8x , sy : element dimensions

, = 6 X / R2 , Sy/ff^

n : lubricant viscosity

n = n/n^

r\o : viscosity at inlet pressure and zero temperature

e = tan"1

v/u

- 24 -

X , X : r e l a x a t i o n f a c t o r s P 9

vj 2 :

Poisson's ratio of materials

p : lubricant density

P = P/P0

Q0 : lubricant density at atmospheric pressure

$ = q p /2

fi = ($2

fi = fi n

o fi

7 , fin : spin speeds of the bodies 1 and 2 (anti-clockwise)

- 25 -


The parameters which are defined in the notation for Chapter 4 and

have the same meaning in this chapter are not listed below, bearing in

mind that wherever i? is used in Chapter 4, R should be used in Chapter 5

instead, and wherever suffix 1 is used in Chapter 4, the parameters

corresponding to the ball should be used in Chapter 5.

A , D : as defined in Section 5.2.3

B : half elastostatic contact length in rolling direction (on the graphs only)

h : end closure minimum film thickness e

K = h /R

I : half axial length of the mesh m

L : length of the roller

Lp : flat length of the dub-off profiled roller

lf - y s

R : radius of the roller

: radius of the crown profile

• v *

: radius of the dub-off profile

=

V *

U* = VT)0K'/R

W : dimensionless load per unit length used by others

x : half flat length of the crowned radius profile roller

c R e

R

X c • V5

XR : coordinate used as shown in Figure 5.2

XR = V *

6y : dimension of the elements in y direction

- 26 -

<5y : by/R

bx. . : dimension of the element corresponding to the ith row and jth column in the x direction

bx. . - bx - ./R ^ 3

$ , $ : first and second derivative of the function $ with x

^ respect to x

- 27 -

CHAPTER 1

PRESSURE DISTRIBUTION BETWEEN CLOSELY

CONFORMAL CONTACTING SURFACES

1.1 INTRODUCTION

In engineering structures, there is often elastic contact between

closely conformal surfaces. Some examples of these are: complete and

partial journal bearings at start-up, columns on elastic foundations,

ball to race bearing contacts, turbine fir tree roots, artificial and

actual human hip joints, and Novikov gears. Problems related to the

latter example have prompted the study which is reported in this chapter.

The theory relating to a flat-ended or profiled rigid stamp

penetrating an elastic half space has been adequately covered in the

literature (1,2). A form of boundary integral equation is used to

calculate the resultant pressure distribution. Further examples of its

use relate to the contacts in ball and roller bearings and involute gears.

In these examples, the bodies are assumed to be elastic half spaces, with

the geometry in the vicinity of the contact be'ing counterformal or only

slightly conformal, making the planform (footprint) with at least one

dimension small compared to the local radii of curvature of the contiguous

surfaces (3). When, however, these surfaces are deeply curved and closely

conformal, having footprint dimensions nearly equal to their radii, the

equation used in the above examples is no longer accurate. Instead, a

solution to a more generalised form of boundary integral equation must be

sought. The model that is studied below is the two-dimensional contact

of a long cylindrical roller and a closely conformal circular hole in an

infinite medium. Some solutions to this problem can be found in

references (4) and (5). In reference (4), the hand computation resulted

in only a few points being taken, so that the accuracy of the result is

- 28 -

suspect. A comprehensive numerical technique with more realistic boundary

conditions is described.

1.2 THEORY

1.2.1 Radial Deflection of a Long Roller Due to Concentrated Line

Forces

Let the roller have two equal and opposite radial line forces,

F9 per unit length acting on its boundary (Figure 1.1). The resultant

radial displacement of the boundary at angle <j> to the line of action of the

forces is given by:

u

rl = C 0 S

^ * S 1

'n

*

Now, using Muskhelishvili's (2) complex mathematical functions for the

case of a circular disc under concentrated forces, applied to its

boundary, from reference (2) and Figure 1.2, for a point within the roller

at s, assuming a = 0 and plane strain (e = 0 ) , the deflection components z

parallel and normal to F are:

* - ih i- <2-Trf> ln ^ * 'cos 2e5 - cos 2V * jq f1-2) v = ttg L

(Th> + r s i n 2&i+ s 1 n 2

V

Relations (1.2) and (1.3) hold equally true on the boundary of the roller

and one can simplify the v component further by substituting

(e2 + e

2J = ir/2 for the points on the boundary of the roller.

For the boundary points, the following relations also hold:

COS 29 - COS 29- = 2 COS <j> (1.4)

- 29 -

and: sin + 2 e

2 ~ 5

sin cj> (1.5)

Let:

and:

K,

K.

\2 + 2Gj 2ir Gj (X2 +G2)

2(1-v*) IT F -

4(X1+G1)

(1+\>1)(1-2^1) 2E^

and note that: In r.

= In tan

Remembering also that for the points on the surface, (91'

h Q

2^ = 7 r / / 2

'

Therefore, from equations (1.1) to (1.5) for points on the roller boundary:

u

rl = F {- K

V (1 + cos tp In tan I ) + K

D sin |<j>|} (1.6)

1.2.2 Radial Deflection of a Circular Hole Due to Concentrated Line

Forces

From references (2) and (4), if a cylindrical hole is cut in

an infinite elastic body and two equal and opposite line forces, F , per

unit length act on the boundary (Figure 1.3(b)), then a point (i?2,<j>)

undergoes a radial displacement towards the hole centre of:

ur 2 = F {- K

f

2 COS <j> In tan + K ' sin |*|} (1.7)

where K^ and FJ are the corresponding elastic properties of the infinite

body. Such a loading of a disc and a hole was used in reference (4).

However, although each member is in equilibrium under these forces, they

cannot occur together when one is pressed against the other, so that a

more realistic hole loading is a single force, F, acting as in Figure

- 30 -

V

- 31 -

1.3(c). The double load on the disc can then be achieved by Figure 1.3(d),

although the hole is approximated to two semi-circular cuts on elastic

half spaces top and bottom. In reference (6), a stress function $ is

given for a line force acting on a hole in an infinite medium (or large

thin plate). This is given by (Figure 1.3(c)):

• = -i IT

7 7 r sin <j> - j (1 - W r In r cos <j> - j r <f> sin <f>

R 2

cos * 4 8

r

The direct stresses corresponding to <& in polar coordinates are:

Finally, the radial displacement at p is:

(1.8)

2 a

* r2

a*2

a , = (1.10) • ar

2

and the radial strain at P (plane strain) can be obtained from:

s = I {o (1-v2) - V aA (1+v)} (1.11) r E

1

r <{> J

= /e <fr (1.12)

In equations (1.9) to (1.11), the first and second derivatives of ip with

respect to v and 4 are needed. Therefore, from geometry referring to

Figure (1.3(c)):

f a n , _ r sin (j)

t a n

* - v COS - d/2

- 32 -

Therefore:

liL = r2

- r cos $ d/2 ^ (1.13)

W R

2 + D

L /4 -

R ^ COS (J)

32

\|> _ - sin $ r3

d/2 + v sin ^ ds

/8 ^ ^

3<|>2

(r2

+ d2

/4• - r d COS <j>J2

M = - sin 4> d/2 ^ (1.15) 3 r

r2

+ d2

/4 - r d cos . <j>

= sin ({> d/2 - sin $ cos $ d2

/2 }

(1.16)

Br2

(r2

+ dz

/4 - r d cos <f>;2

Using equations (1.8) and (1.13) to (1.16) in the relation for stresses

(equations (1.9) and (1.10)), the final expressions for or and <y , after

being written in an integratable form, are:

a _ _ F ^ r sin

2

(fr cos <f> d2

/2 sin2

<j> dz

/4

<f> * (r2 - r d cos cj> + d

2

/4)2

(r2

- r d cos <f> + d(2

/4)2

COS * - § - i - - g . <z-v) i - c o s (1.17) gZ. pi

a _ _ F | - sin

2

<f> £?/2 sin2

<f> r2

d/2 r 7 , 1

(r2

- r d cos <j> •/• (r2

- r d cos <j> + dz

/4)2

+ sin

2

<j) d3

/8 +

2v cos <{»

(r2

- v d cos (j) + dz

/4)2

(r2

- r d cos <j> + dz

/4)

c o s l £ ^ +

Lcos •

(r2

- r d cos * + c22

/^ 4

r2 4 r

+ & i - c o s <j> (1.18)

r3 *

Substituting (1.17) and (1.18) into (1.11), the final expression for the

radial strain is:

- 33 -

e . = F ^ - sin

2

* d/2 (1

.j2 , r

2

sin2

* d/2 ,

^ (r2

- rd cos * •/• d2

/4j (r2

- r d cos * + dz

/4) 2

sin2

* d3

/S

(r2

-rd cos <j> + dz

/4)2

2r cos *

Cr2

- r d cos $ + dz

/4) (l-v2)

cos2

* d

(r2

- r d c o s P+S-/4) (i-v2) cz+v; + 1 cos* r i - v

2

; ^ -

4 o r 2 s

d2

I , / 9 „ „, r sin2

* cos * dz

/ 2 , i + H_-±-COS * R - V

2

+ 2V + 3; = ^ Y

'

(r2

+ dz

/4 - r d cos *J2 16 3

(1.19)

Table 1.1 shows the standard integrations from reference (8).

Non-dimensionalising r with respect to r2 as r = r/R

2 and

substituting (1.19) into (1.12), the final expression for the radial

displacement after performing the integrations tabulated in Table 1.1 is:

*?2 =

+ "tan"

1 r

~ .c o s

• 1 ^ cos2

1) v - cos ji S 1 M

2 sin $ fr2

- 2r cos 1)

i _ tan

2 sin2

* sin * + N+1)2 r - cos *

r2

- 2r cos * + 1)

\ , tan- 1 r

~ ?o s

ft 2 sin * sin * J

+ 2 COS * (L-\>2

) 7 -=-ln I?

2

- 2r cos p +1

^ E S L l t a n "1 P

~ .C Q S

ft

sin * sin * J

- 2 cos2

<j> CZ-v2

J 1

ton-1 r

" .c o s

• sin* sin*

~(l+\>) +\r\r cos * — — c o s * (-V2

+ 2V + j; 2z» 2 4 ^

- COS * v(l+\>) r cos * - i s cos * t a n-i r - c o s *

L/-2 x , -r i sin * sin * J ' J

1 Hr^ - 2r cos * + 1 ) T

-1 + f

7(p)} (1.20)

To find the radial displacement at the boundaries of the hole, simply

substitute r = 1 into (1.20):

- 34 -

TABLE 1.1

Standard Integrations from Reference (8)

Let x =

r2

~ 2r cos $ + 1

( a ) j €

m l

t a n- i * - cos •

;

X s i n

<t> s i n

(b) /

(c) /

X

~p- dr

dr _ r - cos <(> +

1 2

2 sin2

<{» (r2

- 2r cos <f> + 1) 2 sin3

<p

tan -i r

" ,c o s

^ sin <p

(2 cos2

<p - 1) r - cos <f) J t a n - i ^ z c o s l

2 sin2

<f> (r2

- cos <f> + 1) 2 sin3

<J> *

(d) / = | In IF2

- 2r cos cj) -f- 2| + cos (j) fc-r-r tan -l r

- cos sin 4> sin <{>

(e) / r dr r cos <() - 1 +

cos 4> t a n

- i r - cos <j>

2 sin2

<j> CP- - 2? cos <j> + U 2 sin3

<j> s i n

*

- 35 -

u r2

= J L . (i - v 2

J fsinJii jJlL + 2 c

,os2

4> it En 2 _ 2

1

2 4 sin |cj>|

- COS <j> - 1 n - cos <j>;

4 sin2

|<i>|

2

< » 2 -2 ) sin

-

COS (f) {ln \2(1 - COS

+

cos <fr C2v„ - v 2 + 3) + 4 ( 1

- V l ( f i - v / j 2 2

V cos $ u

m

- V . tln'ui - ?>

+

/ j W (1.21)

or: u M - <H*>f2<*> (1.21a)

The integration constant, f2(<p)

9 is a rigid body term which must take the

form C cos <f>, so that when two equal and opposite vectors act with <f>

varying between 0 and TT, it cancels out. Below, it is shown that

equation (1.21) becomes equation (1.7) in this case.

If in equation (1.21) <j> is replaced by 6r-<|>J, the radial

displacement along the hole boundary is due to a single line force, F ,

acting upwards. By combining equation (1.21) with the new equation

containing the loading of a hole by two equal and opposite forces,

F , is obtained as in Figure 1.3(b). The resultant radial displacement at

the boundary is:

- 36 -

U

r2 ir E2 L"

(V2 + D2

F -sin

1*1 (1-v2

Z) (i-r-iiH + 4 sin

2

4

IT

4

cos 4 - v^2

; {In (l - cos

47 +

tan |<j>| (2 + cos

c o s 2

* ' W W V i •) - (l + v2) -

v2(l +\>2) cot 141 cos 4 j (1.22)

After much algebraic manipulation, equation (1.22) becomes equation (1.7).

Equation (1.21) is used in subsequent calculations with the

numerical value of C being found by an iteration procedure.

1.3 COMPRESSION OF TWO CLOSELY CQNFQRMAL CYLINDRICAL BODIES

Assume that A2 and A

2 are, respectively, two points on the boundaries

of the roller and hole, shown before compression in Figure 1.4 and after

compression in Figure 1.5. A is the point of initial contact and 4 is

the angle AOAR From Figure 1.4, let A^A^ and A^

2 be the elastic

displacements of the points A2 and A

2. It follows that the distance

between Aj and A£ will be the total approach of the bodies if they are to

be finally in contact. Also assume that the resultants of the

compression forces pass through the point A SO that a relative rotation of

the boundaries does not occur. Referring to Figure 1.4, if the radial

displacements u 2 and u

2 of points a

2 are AjA'j and A p p and d is the

total approach of the bodies (<i = A j A p , then if elastic displacements are

small, BC almost equals AJL'L, and, therefore:

- 37 -

u

rl + U

v2 = d C 0 S

$ ~ A

iB

(1

'2 3

)

Refer now to Figure 1.6, which shows also the body centres. As R2 -

O G B - 02B = 0

20

2 C O S (J)

or: r - (R2+A^) a (R2-R2) cos <f> (1.24)

Therefore, from equations (1.23) and (1.24):

u

rl + u

r2 ~ ^ C 0 S

^ " " C 0 S

^ (1.25)

Equation (1.25) applies only to the arc of contact. A more complete

discussion of the geometrical problem is given below.

Consider two closely conformal cylindrical bodies, and £ ,

initially touching at A . , having centres of and o and respective radii It l o

R2 and i? , r s r (Figure 1.7). Let Sj be rigid and S

2 be elastic.

After application of a vertical load on Sj, it penetrates S2 by u

Consequently, a typical point A ' on S moves to A0> being now on the O u Ci

common surface of contact at s2 with radius R

2. If U

2 and V

2 are the

radial and tangential components of displacement of A|, then, with

reference to Figures 1.7 and 1.8, we have:

i?7 cos ip + A = (R0 + U0) cos <j> + V0 sin <j> (1-26)

R 2 sin \|> = (r

2 + u

2) sin <p - v

2 cos <f> (1.27)

After squaring and adding (1.26) and (1.27):

- 38 -

(R2 + U

2)

2

+ V2

- R2

- A2

COS ip = 2A R

2

Therefore, eliminating * from equation (1.26) and letting:

E

= a

" U

o2 R

2 ~ Y

we have:

Uo2Z + 2UO2 2R2 U2 + U22 + V ~ R1 =

(UO2R

2 + U

o2 U

2 + e R

2 + £ U

2) C 0 S

* * 7

2 ^ 2 s i n

* + 7

2 e S l n

^

Remembering that R2

- i?^2

= ( R2~ R

2) ( R

2+ R

2) - 2ffe, then neglecting

second order quantities which include all tangential displacements of S2:

U

2 =

^ 2 C 0 S

ft ~ e (2 - COS <p) (1.28)

In a similar manner, if S2 is assumed rigid and S

2 is elastic, the

equation at the common surface of radius, R2, is:

U2 = U

o2 COS <f> + e (1 — cos <p) (1.29)

where, again, second order quantities are ignored. If both s2 and s

2 are

elastic, we can combine the two extreme cases described above by assuming

that the common contact surface is a circular cylinder of radius R as in

Figure 1.8 (similar to Figure 1.4). From equations (1.28) and (1.29), we

have :

U9 = V

N 9 cos <j>

9 - (R

9 - R)(L - cos <P

2) (1.30)

- 40 -

<h

FIGURE 1.6: GEOMETRICAL REPRESENTATION

- 41 -



- 42 -

and: U2 = U

2 cos 4

2 + (R - R^IL - cos 4 ^ (1.31)

as R2 « , 4

i

a

<J>2

s

4-

Substituting equations (1.31) from (1.30) and remembering that u2 and

U 2 are measured radially upwards:

U£ + U

2 = d cos 4 - (R

2 - R

2)(1 - cos 4; (1.32)

which is equation (1.25).

In deriving equation (1.32), relative rotations have to be shown to

be of second order. Furthermore, if the contiguous surfaces are smooth,

tangential stresses resulting from any relative rotation can be ignored.

A more complete analysis of the geometric problem is given in reference (5),

which also proves that the common contact region is a circular cylinder.

The radial displacements of the roller and hole boundaries due to any

radial pressure distribution between them can now be found. Taking

first the element KL on the roller (Figure 1.9), a radial force,

F = P(4') R2 <74', will act, together with (for equilibrium) an equal and

opposite diametral force. From equation (1.6), the sum of all these

forces cause a radial displacement of the roller at A' of:

4'-0

r j if d2 = / P < v ; R2 - K {1 + cos (4-4'; In tan ^ L i

K sin 14 - 4 ' | df (1.33)

Similarly, from equation (1.21a), the corresponding radial displacement of

the hole boundary is:

- 43 -

d 2 (1.34)

where * has been replaced by

1.4 METHOD OF SOLUTION

A numerical method is used to find the pressure distribution

resulting from the contact of the two bodies. The expected arc is

divided into a large number of pressure elements comprising overlapping

isosceles triangles. These form a piecewise linear distribution which

will approximate to the final pressure. Their application and accuracy

are fully discussed in Chapter 3. They are particularly useful when

pressures are expected to fall to zero from a maximum in the centre,

although their accuracy is suspect near pressure discontinuities.

However, for this method to be accurate, each base of a triangle has to be

a straight line, but the fact that each circular base is so small that it

approximates to a straight line removes the doubtfulness of using this

method.

An arc which exceeds that expected is divided into (M + l) elemental

arcs, each subtending an angle A single triangle of pressure will

rest on two adjacent arcs. Figure 1.10 shows two adjacent pressure

triangles with part of the total distribution joining their vertices.

Taking the pressure over a triangle to be:

Integrate equations (1.33) and (1.34) between ±<j> to give the radial

deflection due to that triangle only at all points relative to its base

centre. Therefore, using equations (1.33) and (1.35) and non-

0-35)

- 44 -

d imen s i o n a l i s i n g :

(d* J m K I A -

•m 171

) {- i - cos (6.-6*) In tan it

•

K sin |4>.-<f>'|} It

(1.36)

for the roller, where (d* .) = (d. J /(R^P'J, K = KJK-, P ' . = P

1.J m iy m 1 my 4 29 my my 2 Similarly, for the cylindrical hole, using equations (1.34) and (1.35):

(d.J„ 13 f

•a

/ "•a

(1 - m ) { f2

( d f (1.37)

where $ has been replaced by and where (%..)„ - (d..)J(RJR J , . X J " HI*}

and p . = p my my 2 Any singularities are avoided by making <j>' a little offset from zero

and at ±<j>. The solution of integrals (1.36) and (1.37) enables influence

coefficients of deflection due to any pressure distribution to be found.

By using the superposition theory of elasticity, the total radial

deflection at point i on the arc is, for the roller:

r v = i i v • j , = i ' n (1.38)

(n = number of elements), where (d^.)m = (d.J .)

9 and u

lr = u

lr /r

2>

iy ty my i i. For the cylindrical hole, it is:

r

V =

j2

{ r

V / V • J = }

'n (1.39)

where t / ^ = U ^ / R .

1 1 By using equations (1.38) and (1.39), together with equation (1.25),

and assuming corresponding i's on the two bodies come into contact

- 45 -

FIGURE 1.10: ELEMENTS OF PRESSURE DISTRIBUTION

- 46 -

(neglecting tangential movements), a set of n algebraic equations can be

written, which in matrix form become:

l(dYm +

" {1 C

°S

h + 1>(1

~ C0S

VI (K40

)

where d = d/R0 and *. is now the angle between the centre of the it\\ u 1>

element and the common axis of the loads. The angle p' 9 defining the

final contact arc, is unknown at first. It is found by assuming a value

of the constant C in equation (1.21) as well as expected arc size. Thus,

equation (1.40) can be solved. If there are any negative pressures, the

contact arc is gradually reduced until all pressures are positive over

2 f t I n addition over this arc, the contiguous surfaces should be

compatible to within a specified tolerance. If they are not, C is

altered and the procedure is repeated until both the compatibility

condition is reached and all the pressures are positive over the arc.

Having obtained the final pressure distribution, the respective deflected

shapes of the bodies outside the contact arc can easily be obtained from

equations (1.6) and (1.21). The correct value of c comes to 1.2 whatever

values are assigned to the radius ratio R or dimensionless approach d.

1.5 PRESSURE DISTRIBUTION FROM HERTZ'S THEORY

Having obtained by a numerical method the final pressure distribution,

the load per unit length is:

Y = / R0 P(p') COS dp' (1.41)

- f t '

To find the maximum pressure and the contact length for a given load from

Hertz's theory:

1 11

- 47 -

I f ma t e r i a l s are taken to be the same (6) :

P max

E W nk l2(l-v2) 7ri? Lj e

(1.43)

which, after non-dimensionalising and substituting for R from (1.42):

_ p P = I L Z M I (1.43a) max IT

v

'

W h e r e P

m ® =

W i ' a n d

^ =

Also, if in Hertz's theory the contact width is calculated along the

arc then if 2 4 ^ is the arc:

4 ' , - -JLj-w)* (1.44)

o h (1-RV Having found P

m a x and 4 ^ from equations (1.43a) and (1.44), Hertz's

pressure ellipse may be drawn from the expression:

P = ^ i1

- O -4 5

)

max 1

^ oh

1.6 TOTAL APPROACH OF THE BODIES FROM HERTZ'S THEORY

From reference (7), as the two materials are the same, the total

approach is:

W KL 2 2 D = — L N ( F J R }

L2

if L = L/Rji equation (1.46) can be written as:

- 48 -

Assuming L = R - 1:

d = 2.718 (1.47)

1.7 RESULTS

In all the results discussed below, both bodies are of the same

material. Figure 1.11 shows pressure distribution and distorted shapes,

both inside and outside the contact arc The shapes within the arc

are the same, with the hole surface having the greater deflections.

Outside the arc, the roller is seen to deflect radially outwards beyond

45°, whilst the hole surface has outward deflection up to ±90°. Even

outside the contact arc, the two deflected surfaces are still almost in

contact because R - 1.

Figure 1.12 shows pressure distribution and deflections within the

arc of contact for a greater approach, and hence a greater load. Both

the peak pressure and the arc are, therefore, greater than those of

Figure 1.11. Although the pressure distribution is approximately

elliptical, the numerical solution differs from that obtained by Hertz's

theory (6), which is shown dotted. The way in which it is solved is

given in Section 1.5. In both cases, the dimensionless load, w*, is the

same, although the areas under the respective pressure distributions

appear to be different because Hertz's theory does not assume a curved

base for its pressures but an elastic half space. Use of Hertz's theory

therefore underestimates the maximum pressure and overestimates the arc.

This is to be expected as his assumptions also include a footprint width

which must be small compared to the radii of curvature of the contacting

bodies. However, when both d and R are reduced considerably, causing <J>

to diminish from 54.7° to 1.92°, Hertz's theory becomes much more accurate,

- 49 -

as Figure 1.13 shows. This is again illustrated in Figure 1.14. For

R = 0.999 and low values of w*9 there is little difference between the

results. As p'Q approaches 90°, the curves diverge as it cannot exceed

90° in the present theory. When R = 0.99, Hertz's theory is again valid

because of the small contact arcs. Note that the arc is much reduced

when the surfaces are less conformal. Figure 1.15 shows total approach

against load. There is only a small slope difference for reduction of R

from 0.999 to 0.99, although approach increases for a smaller R.

Therefore, taking the average slope:

2 = 1.15 j/*o-937 (1.48)

But using Hertz's theory (Section 1.6):

2 = 2.718 J/*0

'91

Thus, Hertz's theory gives a reasonable prediction of approach, especially

for low loads.

A further comparison between the two theories can also be made. It

is based on the assumption by Hertz that both bodies can be considered as

elastic half spaces and, if they are of the same material, should suffer

the same deflections at corresponding contact points (6). With large

contact arcs, this is obviously not so, as Figure 1.12 shows, the maximum

deflection ratio being 1.84. From Figure 1.13 for a small arc, the

deflection ratio reduces to 1.17. Figure 1.16 shows the ratio of the

individual maximum deflections of both members plotted against total

approach. The ratio is seen to decrease as the total approach diminishes.

One would expect, however, that the two members would show nearly equal

deflections for such a small arc, because of the localised effect of the

- 50 -

stresses. That this is not so-might be because they are still closely

conformal, despite the small region of pressure. With counterformal

surfaces, the bodies are more alike and, therefore, the equal deflection

assumption appears to be more valid.

In the model in Figure 1.3(d), the extent of arc available for

pressure can be controlled. Thus, in Figure 1.17, the arc available in

the caps has been reduced. The effect on pressure distribution is to

cause a discontinuity at the contact edges. In the examples shown, only

when 4^ > 35° does the pressure fall to zero at the edges. This suggests

that careful profiling of the edges of the cylindrical cut can cause the

pressure distribution there to fall to zero, despite a limited available

arc. Such control over the pressure distribution can have important

implications in certain engineering applications, such as partial arc

bearings, prostheses in biomechanics, and the design of the edges of

Novikov gears.

1.8 IMPORTANCE OF NODAL POINT DENSITY

The number of elements on the accuracy of the results is important

and for the results presented, care was taken to prevent any errors from

being made. The criterion used to determine the optimum number of

elements was that when increasing the number of elements further, there

was not any noticeable change both on pressure distribution and contact

arc. Figure 1.18 shows a typical pressure distribution for varying the

number of elements (number of elements = 25, 33, 45). For the example

shown, the change on both pressure distribution and contact arc is very

little when increasing the number of elements from 25 to 45.

1.9 CONCLUSION

The pressure distribution, contact arc extent, and deflection are

- 51 -

determined for the elastic contact of a long roller and a cylindrical hole

in an infinite body, pressed against each other. It is shown that the

contact arc and approach of the bodies depend on the degree of conformity.

Although Hertz's theory determines arc and total approach reasonably

accurately, even when the assumption of small contact dimensions has been

violated, it considerably underestimates the peak pressure. When the

contact arc is small compared with their radii, as expected, the pressure

distribution from Hertz's theory becomes more valid.

For the case of Novikov gears, the conformity is never as severe as

that of the conformal surfaces discussed in this chapter (the ratio of

radii of curvature being - 0.85). Therefore, the contact arc for these

gears is small compared with their radii and hence it is quite justified

to use Hertz's theory for the case of conformal gears.

If the arc available is truncated too much, stress concentration can

result at its edges.

The results of the analysis have application in the design of some

engineering components, such as complete and partial arc bearings and

Novikov gears, where a knowledge of the magnitude and distribution of the

contact stress is important.

90.00

FIGURE 1.11: PRESSURE DISTRIBUTION AND DISTORTED SHAPES BOTH INSIDE AND OUTSIDE CONTACT

- 53 -

- - H E R T Z R =

TD) A P P R O A C H :

H A L F OF C O N T A C T :

OF E L E M E N T S

0 . 9 9 9

0 .0010 5 4 . 7 0 °

2 5

-90.00 -70.00 -50.00 •30.00 -10.90 10.00 RNGLECDEGREES)

90.00

FIGURE 1.12: PRESSURE DISTRIBUTION AND DISTORTED SHAPES INSIDE CONTACT

- 54 -

— H E R T Z

(D) A P P R O A C H

^ H A L F OF C O N T A C T

OF E L E M E N T S

- 4 . SO -3.50 - 2 . SO -'I .SO -O.St? O'.SO

ANGLE(DEGREES) l.so 2.50 3.50

0 . 9 9 0

0 . 0 0 0 0 2 5

1 . 9 2 °

7 5

4.50

FIGURE 1.13: PRESSURE DISTRIBUTION AND DISTORTED SHAPES INSIDE CONTACT

- 55 -

— HERTZ

VARIATION OF CONTACT ARC W I T H LOAD

FIGURE 1.14

- 56 -

VARIATION OF TOTAL APPROACH WITH LOAD

FIGURE 1.15

2 . 0

1 . 9

1. G

VARIATION OF DEFLECTION RATIO WITH A P P R O A C H

_L2 I r 1

1.6

1 . 5

1 . 4

1. 3 R = .999

1. 2

1. 1

1 . O

io7 • • « i «11 il 1 i M i nil 1 i " " " j ,

106

lo5

io'

FIGURE 1.16

e n

"Vj

' > • i m i l 1—l-L 10 10'

R= 0 - 9 9 9

FIGURE 1.17: EFFECT ON PRESSURE DISTRIBUTION OF TRUNCATED HOLE BOUNDARIES

R = 0 . 9 9 9

(d> RPPR0PICH= 0 . 0 0 0 5

I6.00 -b. dd 6.oo flNGLE(DEGREES)

ife.OO

FIGURE 1.18: IMPORTANCE OF NODAL POINTS DENSITY

- 60 -

1.10 REFERENCES

(1) LURE, A.I.

Three-Dimensional Problems of the Theory of Elasticity,

Wiley Interscience, New York (1964).

(2) MUSKHELISHVILI, N.I.

Some Basic Problems of the Mathematical Theory of Elasticity,

N.V. Noordhof (1953).

(3) HEYDARI, M., & GOHAR, R.

"The influence of axial profile on pressure distribution in radially

loaded rollers",

J. Medi. Eng. See., _21_ (1979) 381-388.

(4) SHETAYERMAN, Ya.

"Contact problem of the. theory of elasticity",

Koyvta.ktna.ja Zadatja TzanML tlpmgoAtlc (1949) Moscow.

(5) PERSSON, A.

"On the Stress Distribution of Cylindrical Elastic Bodies in Contact",

PhD Thesis, Chalmers University, Goteborg, Sweden (1964).

(6) TIMOSHENKO, S.P., & GOODIER, J.N.

Mathematical Theory of Elasticity,

McGraw-Hill, New York (1970) 3rd edition.

(7) GOHAR, R., & NIKPUR, K.

"Deflection of a roller compressed between plattens",

Talbology Int., 8 (1975) 1 - 8 .

(8) DWIGHT, H.B.

Tables of Integrals and Other Mathematical Data,

Macmillan, New York (1961) 4th edition.

- 61 -

CHAPTER 2

ELASTOSTATIC PRESSURE DISTRIBUTION

\

2.1 INTRODUCTION

Small variations in stress can produce large changes in the lives of

bearings and gears. Small changes in the profiles of the contacting

bodies can significantly influence the distribution of this contact stress.

Therefore, it is clear that the profiles of the two contacting bodies play

an important rOle in the life characteristics of bearings and gears.

Hertz (1) successfully analysed elastic body contact problems for

the case of contacting ellipsoids. However, for non-ellipsoidal contacts

in which the pressure distribution is non-symmetric or for cases where the

profiles of the contacting surfaces are not smooth and continuous where

discontinuities in profile are present, Hertz's theory does not apply.

Such examples occur in axially profiled rollers (dub-off radii at the

roller ends), helical Novikov gears (contact between a cone and a sphere),

a truncated contact ellipse (gear tooth tip contact), angular contact

bearings, dome-canted ended rollers touching the rib edge.

Harris (2) and Liu (3) have tried to extend Hertz's work to roller

shapes commonly found in anti-friction bearings. This was attempted by

dividing the roller into a series of discs, and then using Hertz's

results to predict surface stresses on individual slices. These results

provided an approximate means of analysing the stress state in rolling

element bearings. However, a major deficiency in this approach was that

the three-dimensional elasticity problem was not solved. Consequently,

an accurate appraisal of the surface stress could not be made.

Within recent years, researchers have attempted to solve contact

problems for a more extensive range of body shapes than the ellipsoids

considered by Hertz. Conry & Seireg (4) solved simple problems using a

- 62 -

numerical algorithm aimed at minimising the potential energy of the

contact. Oh & Trachman (5) extended Conry & Seireg's (4) method to the

contact of cylinders and were able to evaluate surface stresses for some

special cases of symmetry. Singh & Paul (6) employed the flexibility

method of structural analysis in conjunction with Boussinesq's relationship

for force and displacement on an elastic half space to derive a set of

equations in terms of surface pressures, but due to their system of linear

algebraic equations being ill-conditioned, their solution is limited only

to some special cases. Nayak & Johnson (7) presented a numerical method

to calculate the pressure distribution and the contact area for elastic

bodies having a slender contact area.

Gohar (8) described a general pressure distribution for obtaining the

deformed shape of the roll in the cold rolling process using overlapping

isosceles triangles of pressure of equal base length longitudinally and

elliptical transversely. The same method was used by Heydari & Gohar (9),

Johns & Gohar (10) and Rahnejat & Gohar (11) to calculate the pressure

distribution over a contact between either an aligned or misaligned roller

and an elastic half space, and for tapered roller bearings.

A numerical method is presented here for calculating the three-

dimensional pressure distribution and contact area shape between two

elastic bodies of arbitrary profile. This method has conjunctionally

been used by Hartnett (12), in which he analysed the dry contact pressure

distribution in rolling element bearings. The method utilises the force

and displacement boundary integral relationship, together with the

separation of the bodies and once the gap shape and either one of the

applied load or the penetration is known, the numerical scheme starts the

iteration on both the pressure distribution and the contact shape in

order to arrive at the final solution. The assumption embodied in the

method is that the contact can be flat in one plane and is therefore

- 63 -

suitable for concentrated contact problems in which the dimensions of the

contact are much smaller than the radii of curvature of the contacting

bodies.

The method is applied to the case of a sphere indenting inside an

elastic cone which is the simulation of helical Novikov gears used in

helicopter gear boxes. The influence of some parameters, such as

penetration, the angle of cone and of the conformity, on the pressure

distribution and the shape of the contact is examined.

One of the objectives of applying the numerical method to Novikov

gears is to analyse the shape of the contact and hence to check the degree

of unsymmetry of the contact ellipse in these gears and so to establish

that whether the assumption of treating the bent ellipse to a normal

ellipse holds true or not.

In later chapters, the same method is used to find the elastostatic

pressure distribution of a dub-off ended roller, a crown roller, and

varying elliptical contacts as an initial condition to the numerical

solution of the lubricated cases. Also in the next chapter, this method

will be utilised to find the pressure distribution in an overriding pair

of Novikov gears.

2.2 INTRODUCTION TO THE MODEL OF NOVIKOV GEARS

The conformal gear tooth system usually known as the Novikov gear

is used in some helicopter gear boxes. It has been shown to have a

superior load carrying capacity to the involute gear (13). However, its

failure characteristic suggests that the load capacity might be improved

even further if the lubrication of the contact area were better understood.

The gears themselves are not ideal for studying the mechanism of

lubrication but an arrangement of a sphere against a cone can closely

simulate all the essential motions and geometric characteristics of the

- 64 -

t oo th con ta c t (13) . Using such an arrangement, we can study the range

of variables which may influence the load carrying properties of the

contact. The model is shown in Figure 2.1 and, according to the figure,

adjustment of u> and u> can give variation of rolling speed and sliding c s

in the rolling direction. Adjustment of <p will change the transverse

sliding velocity, changes in R will alter the ellipse shape, and variation

in e will modify the rate of change of sliding in the rolling direction

across the contact area.

The shape of the contact area is usually described as elliptic and

for light loads, it would appear to be of this shape. However, as the

load is increased, it is apparent that this is not longer a simple

ellipse. The explanation is that the pinion tooth surface is wrapped

around an axis and is therefore curved in a direction normal to the load

vector at the point of contact. Thus, the contact area approximates to

an ellipse with its major axis bent around a circular arc or, in other

words, the contact area is an ellipse flattened on one side. Practical

measurements of the size of contact reported by West!and Helicopters have

confirmed that the theoretical Hertzian predictions are close to the actual

values.

2.3 THEORY

Consider a semi-infinite elastic plane with an arbitrary pressure

distribution P(x',y') applied over an area A of its surface. Then the

deflection at x,y is given by the force-displacement relationship (14,15):

d(x,y) - SlfgLj (2.1)

The geometry is described in the plan view of the elastic half space shown

in Figure 2.2, where Rr

= Ax-x')2

+ (y-y')2

.

- 65 -

LOAD

FIGURE 2.1

- 66 -

Letting the area A be rectangular in shape of dimensions 2b and 2a

in the x and y directions, respectively, and the pressure to be constant

over this area, then the double integration of the relationship (2.1) using

standard tables of integrals (16) can be performed analytically:

d(x3y) P (1-a2

) r / , 1n

7T E I ^

(y+a) In

(x-b) + Ay-a)2 + (x-b)2' Ax+b) + Ay-a)2 + (x+b)21

(x+b) In

(x-b) In

7x+b) + Ay+a)2 + (x+b)2'] Ax-b) + Ay+a)2

+ (x—b) (y+a) + Ay+a)2 + (x+b)2' Ay-a) + Ay-a)2

+ (x+b)21

"(y-a) + Ay-a)2 + (x-b)2] •(y+a) + / (y+a)2 + (x-b)21 (2.2)

The solution domain is divided into rectangular areas and the pressure is

taken to be uniform within each area (Figure 2.3). Since the problems

which will be considered here will usually have a sharp rate of change of

the pressure at the extreme ends of the zone, and much less change in the

central parts, the rectangular elements are constructed on an irregular

mesh shown in Figure 2.4, where the dimensions of the elements change as

an arithmetic progression, so that more elements are filled in the

sections where the changes of pressure are more critical.

Representing d . . as the influence coefficient defining the deflection

at element i due to the pressure acting at element j , by using the

superposition theory of elasticity, the total deflection at point i due to

the combined effects of all elements, using relation (2.2), can be written

in dimension!ess form as:

K- =

n

I d. • PJ i. = 1, n (2.3)

- 67 -

P L A N V I E W O F T H E HALF S P A C E

FIGURE 2.2

RECTANGULAR BOX OF

PRESSURE ACTS IN RECTANGULAR BOXES

OVER EACH ELEMENT

FIGURE 2.3

- 68 -

ARITHMETIC PR0&RESS10N MESH

FIGURE 2.4

- 69 -

(n = number o f e lements) where d. • = d. ./( i?7P J , P . = P-KL V. = 7 . / i ? 7 , "a v. 3 3 3 3 V I

and K'2 = ( I - v ^ A E ^ , or in matrix form:

I V - ^ l V • ) \ ] \ l (2.3a)

Feeding the deflections at the points of the mesh into the matrix relation

(2.3a) will lead to a set of linear algebraic equations in which their

solution will lead to the pressure distribution.

2.4 COMPRESSION OF TWO ELASTIC BODIES HAVING ARBITRARY PROFILES

Representing the gap separation of the bodies by h , Figure 2.5 shows

the state of the bodies before and after compression. For the sake of

clarity, one of the bodies is shown as flat. The top'part shows them

undeformed, whilst in the lower part both have been deflected by a load.

If d is the total approach or the maximum penetration of the bodies:

d m 7l(0,0)+72t0,0) (2"4>

where suffixes 1 and 2 stand for bodies 1 and 2, respectively. If

points Aj and A2 are representative of any pair of opposing surface points

which are to be brought into contact by compression, then:

71(030) + 72(0,0) - 71. + V2. + ' * = » (2-5>

(inside the region of contact only). In accordance with Hertz's theory

(17,1), assume that the ratio of the deflection differences suffered

respectively by the elastic bodies is inversely proportional to the ratio

of their Young's modulus, i.e.

- 70 -

BODY 2

C O N D I T I O N OF THE BODIES BEFORE AND

A F T E R COMPRESSION

FIGURE 2.5

- 71 -

V

1(0,0) ~ V

l. E0 (1-v

2

) % _ 2 1 _ T T • _ 1

= = E , ^ = I , n 72<0,0)~72. E1(1^2Z> ^

Therefore:

v ( 7

k o , O ) -v

i .i ( i

» £ = i

'n

ir "t» hi

and after dimensionalising with respect to Rj:

h - ^ v

l = v

l(o o) 1

» i = h n (2.6)

h U ° 3 0 ) (l + U/E))

Therefore, by keeping the approach of the bodies as an external variable

using relation (2.6), the corresponding deflections of the bodies can be

found for different elements in the contact zone.

It is assumed that the penetration is constant, so load is the

dependent parameter. The other practice can be to keep the load constant

by continuously adjusting the penetration. This condition imposes an

extra iterative loop with a corresponding increase in the computing time

and therefore, in order to produce some numerical results, it is not

justified to have the penetration as a dependent parameter.


The basic steps which must be executed in the numerical iteration

procedure are as follows:

(1) The approximate overestimated boundary of contact for a given

approach of the bodies is estimated from geometry considerations

alone. This is achieved by taking one of the members to be rigid

and indenting it into the other one.

- 72 -

(2) A rectangular grid with irregularly spaced elements is constructed

over the zone.

(3) The gap shape of the elastic bodies is calculated using parabolic

surface assumptions. Hence, from equation (2.6), according to the

amount of penetration, the elastic deflection for each element is

calculated.

(4) The influence coefficients for each element are calculated.

(5) The set of linear algebraic equations (2.3a) is solved for the

unknown pressure distribution.

(6) Are all the pressures positive?

Yes : Go to stage (9).

No : The elements which have negative pressure acting on them are

excluded from the mesh, since these elements are in tension

and therefore are outside the contact.

(7) The contribution of elements which correspond to negative pressures

are extracted from both the matrix of influence coefficients and from

the vector of deflections.

(8) Go to stage (5).

(9) The final pressure distribution is integrated over the contact area

to find the normal applied load.

(10) The final values of pressure distribution and the coordinates of the

edges of the up-to-date grid which represent the boundary of the

contact are printed out.

2.6 APPROXIMATION OF THE OVERESTIMATED CONTACT BOUNDARY SHAPE OF A SPHERE

INDENTING INSIDE A CONE

Figure 2.6 shows a sphere inside a cone before any compression takes

place. Assuming parabolic surfaces:

- 73 -

FIGURE 2.6

- 74 -

where x and y are the coordinates of the points which are to come into

contact projected onto a tangent plane touching the cone through the

initial point of contact.

Taking a typical point A on the sphere which is about to come into

contact with the cone:

v - t?2 + x sin e

Therefore: H = * ^ - -r->— ij^, ^ „ i cos 9 g 2Rj 2(x sin 9 + vn)

In order to obtain an overestimate of the contact area, suppose that the

sphere is rigid and after the bodies are compressed normal to the tangent

plane, the total approach must be equal to the separation of the bodies at

a distance beyond the exact contact boundary. Therefore, the following

expression holds true:

d = h at the overestimated contact boundary, where d - total

approach. Therefore:

+ yh2 yh2

d = 5 £ cos 9 (2.8) 2R2 2(xb sin 9 + v2)

where x^ and y b are the coordinates of the overestimated contact boundary.

After algebraic manipulations and non-dimensionalising with respect to i? ,

equation (2.8) will yield:

- 75 -

Assign values to ^ in small step intervals and find the corresponding

y^, and continue the procedure while the equation yields a real root for

The obtained x^ and y^ are the coordinates of the overestimated

contact boundary on the tangent plane.

2.7 RESULTS

The numerical technique developed can be applied to a wide range of

friction!ess contact problems. All the user has to supply is the gap

shape between the elastic bodies and either the penetration or the load.

To illustrate the method as well as to analyse the contact shape and

pressure distribution in conformal gears, the technique is applied to the

case of an elastic sphere indenting inside an elastic cone.

Figure 2.7 shows the contour plot of dimension!ess pressure

distribution for the half cone angle of 14° and the radius ratio of 1.166

and a dimensionless approach of 0.001. The degree of unsymmetry of the

contact shape about the long axis of the ellipse depends mainly on the

amount of penetration and partly on the cone angle. If the cone angle

increases, the unsymmetry of the contact shape about the long axis

increases and depends on the penetration, because if the load and

therefore the deflection are small, then the cone angle cannot really have

much effect on the unsymmetry of the ellipse. On the other hand, there

should be a limit on the penetration for the parabolic surfaces

assumption to be valid. In Figure 2.7, which is the simulation of

Novikov gears, although the load and therefore the approach have been

chosen as a larger value than that in practice, the unsymmetry of the

ellipse is very little. Therefore, the assumption of elliptical contact

for the gears under consideration is accurate. ec

Figure 2.8 shows the same contour plot for a stfp cone and a

different radius ratio. Figures 2.9, 2.10 and 2.11 show the same contour

- 76 -

plot with the same external conditions as those in Figure 2.8 but with an

increase in penetration on each plot. As illustrated, the unsymmetry of

both the pressure distribution and the ellipse becomes evident as the load

increases.

To ensure the validity of the parabolic surfaces assumption, a maximum

dimension of contact of about 16 percent of the radius of the sphere was

achieved with the greatest penetration, which makes the parabolic

assumption seem reasonable.

2.8 DISCUSSION

Once the profile of the contacting bodies is known, the numerical

technique developed here enables any frictionless concentrated contact

problems to be analysed. The assumption embodied in the method is that

the contact area must be in a flat plane. Using this technique, the

stress distribution due to a wide range of practical problems, such as

roller ends or misaligned rollers, or discontinuities on the surface, such

as dents, scratches or unsymmetrical contacts, can be analysed. It is

possible to use the computer program developed to apply this technique to

design roller shapes which minimise the contact stress. The designed

computer program was used by Karami (18) in order to study the effect of

roller misalignment and geometry on pressure distribution.

In later chapters, the technique is applied to roller bearings, and

the obtained elastostatic solution is taken to act as an initial condition

for the elastohydrodynamic case.

In the next chapter, the method is applied to overriding gears in

which singularities occur where the profile changes abruptly and the

assumption of pressure acting in the rectangular boxes over each element

averages the pressure over a rectangular segment. Therefore, the use of

the rectangular box elements where singularities arise is very approximate,

- 77 -

this being the reason why the aforementioned phenomenon has been left for

the next chapter.

/

- 78 -

FIGURE 2.7: ELASTOSTATIC ISOBAR PLOT

- 79 -

CONE APEX ANGLE=90°

< £ = • 0 0 1

R ' - J , 0 1

FIGURE 2 .8 : ELASTOSTATIC ISOBAR PLOT

- 80 -

FIGURE 2.11: ELASTOSTATIC ISOBAR PLOT

- 81 -

CONE APEX ANGLE

D = - 0 0 3

FIGURE 2 .11 : ELASTOSTATIC ISOBAR PLOT

- 82 -

CONE APEX ANGLE d=-004

R ^ L O L

FIGURE 2 .11: ELASTOSTATIC ISOBAR PLOT

- 83 -

2 .9 REFERENCES

(1) HERTZ, H.

Miscellaneous Papers,

Macmillan, New York (1896).

(2) HARRIS, T.A.

"The effect of misalignment on the fatigue life of cylindrical roller

bearings having crowned rolling members",

ASME, J. Lubn. Tack., 91_ (April 1969) 294-300.

(3) LIU, J.Y.

"The effect of misalignment on the life of high speed cylindrical

roller bearings",

ASME, J. Lubn. TecA., 93 (January 1971) 60-68.

(4) CONRY, T.F., & SEIREG, A.

"A mathematical programming method of design of elastic bodies in

contact",

ASME, J. Appl. Mecit., (June 1971).

(5) OH, K.P., & TRACHMAN, E.G.

"A numerical procedure for designing profiled rolling elements",

ASME, J. Lubn. Tack., 98 (October 1976).

(6) SINGH, K.P., & PAUL, B.

"Numerical solution of non-Hertzian elastic contact problems",

ASME, J. Appl. Mec/i., 41_ (June 1974).

(7) NAYAK, L., & JOHNSON, K.L.

"Pressure between elastic bodies having a slender area of contact and

arbitrary profiles",

Internal report, Cambridge University.

(8) GOHAR, R.

"A numerical method of obtaining the deformed shape of the roll in the

cold rolling process",

J. Medi. Eng. S<U., 16 (1974) 249-258.

- 84 -

(9) HEYDARI, M. , & GOHAR, R.

"Pressure distribution on radially loaded rollers",

J. Mec/i. Eng. Sex., 21_ (1979).

(10) JOHNS, P.M., & GOHAR, R.

"Roller bearings under radial and eccentric loads",

TtLbology Int., (June 1981) 131-136.

(11) RAHNEJAT, H., & GOHAR, R.

"Design of profiled taper roller bearings",

TtuLbolagy Int., (December 1979) 269-276.

(12) HARTNETT, M.J.

"The analysis of contact stresses in rolling element bearings",

ASME, J. Lubn. Tedi., 101 (January 1979) 105-109.

(13) SHOTTER, B.A.

"Evaluation of conformal gearing for use in helicopter applications",

Westland Helicopters Report, Research Paper 481 (November 1974).

(14) CAMERON, A.

Principles of Lubrication,

Macmillan, London (1966) 186.

(15) FORD, H., & ALEXANDER, J.M.

J. InU. MetcuU, 88 (1959) 193.

(16) DWIGHT, H.B.


Macmillan, New York (1961) 4th edition.

(17) TIMQSHENKQ, S.P., & G00DIER, J.N.

Theory of Elasticity,

McGraw-Hill, New York ( ) 3rd edition.

(18) KARAMI, G.

"Stress Distribution in Roller Bearing Rollers",

MSc Thesis, Imperial College, London University (198Q).

- 85 -

CHAPTER 3

SINGULARITY ELEMENTS

3.1 INTRODUCTION

Normally, stress concentrations appear at or near sharp edges,

corners, cracks or slope discontinuities. Material failure is often

initiated in these regions. Therefore, it is a matter of some importance

to develop accurate theories which explain these kinds of problems.

Unfortunately, for these regions, an exact theory is not yet available

and a numerical method must be applied. Now, the more accurate the

numerical method, the more realistic will be the picture of occurrences in

these regions. However, there is always an error involved in numerical

methods which magnifies where singularities might occur. The

singularities can be categorised as due to geometry and due to load.

In plane strain contacts of two-dimensional bodies, a discontinuity

of profile slope within the contact leads to a 'logarithmic singularity'

in the contact pressure (12). Thus, the pressure at a small distance x

from the discontinuity varies as In Singularities which arise at

the edges of' contact have been studied by Dundurs (1), who has shown that

in friction!ess contacts, a 'power singularity' would be expected,

i.e. the contact pressure at a small distance x from the edge would vary

as x71, where m varies from 0.5 to 1.0, depending on the angle of

discontinuity in the profiles and the reduced elastic constant of the two

bodies.

The singularity which is considered here is due to a sharp edge.

This is a 'power singularity' and is catagorised as of the geometrical

type. The order of singularity is very important for following the

discontinuity. A singularity element is devised here in which it

minimises the error in the numerical scheme of finding the stress

- 86 -

distribution where sharp edges or discontinuities, such as change of slope,

appear in the surface. The idea was first suggested by Johnson through

a private communication and it is put into mathematics and computation in

this chapter. The idea is to use numerical methods to represent the

pressure distribution by triangular or rectangular elements, but to change

these elements in the vicinity of discontinuities to a new type of element,

referred to here as a singularity element and which describes the correct

order of singularity there.

Numerical methods are only useful when the classical solution to a

problem is not known. Therefore, the element is most useful when the

order of singularity can be forseen in problems with no classical solution.

The way that the order can be estimated is to compare the problem with a

similar case with a known solution.

To apply the method to a physical problem, a rigid punch indenting

an elastic half space is analysed. The case of an elastic punch acting

on an elastic foundation has been analysed by Fredriksson (2) using finite

element methods. Assuming plane stress and no friction, Fredriksson (2)

studied the effect of the number of contact nodes on pressure distribution

using different finite element models and compared the solution with Okubu

(3). Using the finest mesh, the discrepancy between (3) and (2) was 3%

at the centre and increasing towards the singularity zone. The reason

for this discrepancy, which was explained in.(2), is that the finite

element model is not able to follow the singularity. Nayak & Johnson (4)

noticed that where there is a small discontinuity in the slope of the

profile, the logarithmic singularity appears to be extremely local and

their numerical method was considered to give good results at all points,

except very close to the singularity.

The problem of the contact of a rigid stamp on an elastic half-plane

has been analysed in (5) and (6), in which they give an expression for

- 87 -

pressure distribution in terms of load, which increases without limit with

the approach to boundaries of contact. In reality, the real profile of

an elastic body will never have corner points, so that the right-angled

stamp used here is an abstraction, which leads to a solution of the

contact problem with an unreal distribution of pressure in the region of

contact. Shetayerman (5) has examined the case when the profile of the

stamp has a continuously revolving tangent. However, since the objective

of the work presented in this chapter is to demonstrate the singularity

element and to check its accuracy against other types of element, it is

justified to analyse the problem of a sharp cornered punch which suffers

from a discontinuity at its edges.

Biswas & Snidle (7) have also represented the pressure distribution

with paraboidal surfaces in general contact problems to calculate the

surface deformations and found accuracies of about one percent in

calculating the surface deformations.

3.2 THEORY FOR A RIGID RECTANGULAR PUNCH AND AN ELASTIC HALF SPACE:

RELATION BETWEEN PRESSURE DISTRIBUTION AND CENTRE DEFLECTION OF

THE ELASTIC HALF SPACE

To find a relation between pressure distribution and centre deflection

of the half space, the force-displacement integral equation is solved

simultaneously with the pressure-load relation derived in (5) and (6) for

a rigid long stamp indenting an elastic half space. Consider a semi-

infinite elastic plane (Figure 3.1) with an arbitrary pressure

distribution q(x',y') applied to its surface. The deflected shape of the

plane, together with the punch, are shown in Figure 3.2. The force-

displacement integral relationship is:

v(x,y) = (1"g2; J f qte'tV1) dv' 77 E -b -a SCx-x1)2 + (y-y')z

(3.1)

- 88 -

ELASTIC HALF SPACE

FIGURE 3.1

- 89 -

RIGID PUNCH AND ELASTIC HALF SPACE

FIGURE 3.2

- 90 -

where x',y' are the coordinates at which pressure acts, and x,y are the

coordinates where the deflection is required.

The deflection at the central point is:

mto) - n f / 0.2) -b -a vx'

2

+ y'2

where K'2 = (l-a

2

)/-aE. Since an infinitely long stamp is considered,

q(x',y') is independent of y'. Therefore, integration with respect to

y' can be done in the following manner:

J dy' = ln

j b + A'2

+ b2

| = 2 ln

jb + Jx'2

+ £2

j

-b Jx'2

+ y '2

- b + A'2

+ b2 X

'

Therefore:

7(0,0) = 2q f q(x') In ib +

^ + h

-\ dx' -a

Since x2 « £>, hence x'

2

« b2

. Therefore:

a 7(0,0) = 2K'

2 j q(x') In {p-} dx' -Ia

a a = 2K'

2 [In (2b) f q(x') dx' - f q(x') In \x'\ dx'] (3.3)

-a -a

but from (5) and (6):

q(x') = JL— (3.4) IT / A ^ - x

where W = load/unit length.

Substituting (3.4) into (3.3):

- 91 -

V(0,0) = 2K> W a 1 „ ( 2 B

) / / 1N M D X ' -a /a2 - x'

2

-a /a2 - x'2

•

= a ' w [in <2b) - 1 / 1 n

l«'l d c

'" 2

L " L j„z _ * tz. -a va - x (3.5)

From Dwight (8):

a In la?'I dx'

-a Ja2

- x'2

= IN (A) IT - 2A (3.6)

where:

I X 1 1 X 3 1 X 3 X 5 A = 1

2 x 3 x 3 2 x 4 x 6 x 5 2 x 4 x 6 x 7 x 7 = 1.0769331


v(o,o) = 2K' w {in r—; + — } 2

1

a IT

J

Therefore: r, V(030) • 2K' {In (2b/a) + (2A/-U)}

But: q(x') = W

IT /a2

- x'2

i.e. q/a?U = Sir /a2 - a;'2 {In (2b/a) + (2A/v)}

(3.7)

3.3 INTRODUCTION TO TRIANGULAR AND RECTANGULAR ELEMENTS

It is shown in (9) and (10) that a large number of elementary

overlapping isosceles triangles replaces a given pressure profile. As

illustrated in the left hand side of Figure 3.3, the contact area of

length 2a is divided into 2n strips, each of width cT. A triangular unit

- 92 -

FIGURE 3.3

- 93 -

of pressure is taken to act on each pair of adjacent strips having a

central ordinate P. and falling to zero at a distance ±Cm on either side.

Since the pressure at the ends of the contact is not zero, a half unit

triangle of pressure is added on either side. Each unit of pressure acts

on a rectangular base of width 2C^ and breadth 2b. Superposition of

2n-l complete overlapping triangles and two half triangle units results in

a piecewise linear distribution of pressure.

An arbitrary pressure distribution can also be represented by a large

number of rectangular elements of either varying or equal base length, as

shown in the right hand side of Figure 3.3. In this method, the contact

area is divided into m strips and a rectangular unit of pressure with

ordinate p. is taken to act on each strip. Abutment of these elements

will again yield a linear distribution of pressure between element centres.

3.4 INFLUENCE COEFFICIENTS

3.4.1 Influence Coefficients for Triangular Elements

Once again, using the force-displacement boundary integral

equation, the deflection at a point x,y due to a triangular pressure

actin on x' = ±CT and y

r

= ±b can be written as:

:jy) = (I-*

1

) f JT

P(ss') dx' dy' d(x,y) = i i - q p - / / v (3.8) 7 7

* -b -CT Ax-x')

2

+ (y-y')2

where P(x2) = p.(l - \x'\/C

T), P. being the central ordinate of each

tri ang1e. Non-di men s i ona1i sing:

x a ' a

x' - — , x = I , yr B

u± , ZG,y) = i

a P. 3

where P. = P.K' Therefore: 3 3 2

- 94 -

P(x2) = P. (1 - \x'\) (3.9)

since deflections at the mid plane are required, therefore y = 0.


1 C

T/ A

, •

= / i ( 1 r ( a / 0 ) | g y | ; d x t d y '

-1 -Cy/a Ax-2')2 (a/b)2 + y'2

One of the double integrations using (8) can be performed analytically:

} ciyj = ln ^CZ-2')2 (a/b)2 + 1 + 1^ -1 Ax-2')2 (a/b)2 + y'2 A2-2')2 (a/b)2 + 1 - 1

Therefore:

V A 5

lU,y) = / (1 - <a/o)\3'\) In + ( 3 1 Q ) -Cy/a Ax-x')2(a/b)2 + 1-1

3.4.2 Influence Coefficients for Rectangular Elements

Following the same procedure as for triangular elements:

Ae , (l-o

2

) ? ^2

P(x') dx' dy' d(x,y) = — r - p — J J , * -

"b "C/2 S(x-x')2 + (y-y')2

For the rectangular elements, p(x') = P..

After non-dimensionalising in the same way as for triangular

elements and setting y to zero and carrying out the integration with

respect to y analytically, the final expression for the influence

coefficient is:

CJ2a 2(2,y) = / In ^

(a/h)Z + 1 + (3.11) -C]/2a Ax-x')2 (a/b)2 + 1 - 1

- 95 -


Representing d.. as the influence coefficient defining the deflection

at element % due to the pressure acting at element j, by using the

superposition theory of elasticity and setting a/b to a large number, say

100, in expressions (3.10) and (3.H), the total deflection at point i due

to combined effects of all triangles is:

_ ^ f 1 "3 p £ - 1, 2n+2 for triangles i ~ -_7 1*3 3 9 -i = 1, m for rectangles

where V. - V./a, or in matrix form: % %

ITr I - a i ip I i = 1, 2n+l, j = 1, 2n+l for triangles n { v ' 3 3 s i = , j = 1, m for rectangles

Any singularities due to loading in expressions (3.10) and (3.11) are

avoided by making x' a little offset from zero and ±Cr]/2a in (3.10) and

from zero in (3.11). For the half unit triangles, the influence

coefficients change accordingly due to the change in limits of the

integration in (3.10). Inputting the deflection vector of the elastic

half space as a constant value beneath the punch, (3.12) yields a linear

matrix equation which can be solved by straightforward inversion to find

P.. 3

3.6 RESULTS OF COMPARING THE ACCURACY OF TRIANGULAR AND RECTANGULAR

ELEMENTS

For a certain deflection of the half space, expression (3.7) will

represent the pressure distribution for the classical solution.- The same

pressure distribution is calculated by the numerical method using the same

number of both triangular and rectangular elements and is then compared

with the exact solution.

- 96 -

It was found that the rectangular elements give a closer solution to

the exact one and for both cases the pressure distribution converges to

the classical case with increasing the number of elements, the rate of

convergence being higher for rectangular elements. Figures 3.4a and b

show the comparison between the exact theory and rectangular elements case,

for different numbers of elements, and Figures 3.5a and b illustrate the

same situation for triangular elements, using half triangle units at the

ends. For both types of element, the effect of singularity is very local

and it affects the pressure profile in the vicinity of discontinuity most

severely and the effect diminishes further away. The local effect of

singularities is more severe with the triangular elements and their

accuracy in this region is suspect.

Table 3.1 shows the percentage difference between the numerical

methods and the exact solution at the central section of the stamp. The

table shows the obvious inaccuracy in the results caused by using complete

triangles at the ends instead of half unit triangles.

TABLE 3.1

Accuracy of the Numerical Method Using Different Types of Elements

Types of Elements Number of Elements % Difference

Rectangular 17 45

1.5 . 0.75

Triangular with two half units at the ends

19 47

2.25 0.75

Triangular with complete units at the ends

17 45

6.0 2.25

00

FIGURE 3.4b: PRESSURE DISTRIBUTION BETWEEN THE PUNCH AND HALF SPACE

FIGURE 3 .5a : PRESSURE DISTRIBUTION BETWEEN THE PUNCH AND HALF SPACE

M-

FIGURE 3.5b: PRESSURE DISTRIBUTION BETWEEN THE PUNCH AND HALF SPACE

en

- 101 -

3.7 THEORY FOR A SINGULARITY ELEMENT

The pressure profile in the vicinity of a discontinuity is represented

by an element which describes the right order of singularity in that

region. To illustrate the element, the same problem of an elastic punch

indenting an elastic half space which suffers from a sharp edge singularity

is again studied. By using expression (3.4) to find the order of

singularity at the sharp edge concerned, the order is:

(3.13) /a2 - x f 2

where x' is the coordinate measured from the centre of the punch. Let

C = a - x ' , i.e.

x' - a - c (3.14)

Substitute (3.14) into (3.13):

2 order of singularity =

/ 2 A C + V but since c2 << 2ac, i.e.

7 order of singularity =

/ 2 A C

where c is the coordinate measured from the edge of the punch.

The pressure curve which describes this singularity is as shown in

Figure 3.6a:

P(d = constant { 3 } s ) /2az

where the constant must be found from boundary conditions.

An element with the profile of the order of singularity is constructed

- 102 -

F T C I I D R -5 a

- 103 -

at the edges of the punch, as shown in Figure 3.6b, and just adjacent to

this element, a half unit triangle is constructed, the pressure being

represented by triangular elements elsewhere.

To find the constant in the pressure curve of equation (3.15),

according to Figure 3.6, at c = C:

P(V = P2

where P2 is the maximum ordinate of the half unit triangle, and C is the

length of the singularity element. Therefore, using (3.15):

constant = P^ v2aC

i.e. P(z) = P2 JcK (3.16)

3.8 INFLUENCE COEFFICIENTS FOR SINGULARITY ELEMENT

Let the edge of the singularity element be at a minute distance of,

say, 0.001. c from the sharp edge. Following the same procedure as that

for triangular and rectangular elements by assuming an infinitely long

stamp, and therefore pressure to be independent of y, and setting y to

zero since the deflections at the mid-plane are required, the force-

displacement relationship can be written as:

d(x,y) TR E _ b

P(x') dx' dy' ^ ]

(1 - a2

) -ib 0 y/Cx-x')2

+ y'2

where x' is the coordinate measured from the edge of the singularity

element. Remembering that c is the coordinate measured from the sharp

edge itself:

- 104 -

P(x') = P2 Jc72 = P

2 JC/(x' + 0.001C) (3.18)


d(x3y) IT E = P /c j* j'"

9 C

dx' dy' (1-a

2

) 1

-bO /x'+0.001C Ax-x')2 +y'

2

After non-dimensionalising the coordinates in the same way as for

triangular elements:

1 0.999C/a j-,

- I I >- ** $ - , , = < 3' 1 9 )

-1 0 Vx' (a/C) + 0.001 Ax-x')2(a/b)2 + y'2

where d(x3y) - d(x

3~y)/aP

2, and where P

2 =

Therefore, the influence coefficients for the singularity element is

in terms of the pressure at the half unit triangle adjacent to it.

After performing the integration with respect to y' analytically in

expression (3.19), the final relation for the influence coefficient is:

0.999C/a - rA2-^')2(a/h)2 + l+l, -d(x3y) = /

1 In { ZZ (a/b) +1 + 1) dx' (3.20)

0 Ax'(a/C) +0.001) Ax-x')2 (a/b)

2

+1-1

3.9 SETTING UP THE MATRIX INVOLVING THE SINGULARITY ELEMENT INFLUENCE

COEFFICIENTS

Let 2n+l be the number of triangles, including the half unit ones,

and let suffixes sl,sr represent the parameters associated with the

singularity element at the left and right hand side of the punch,

respectively.

Because the pressures at singularity elements are prescribed in

terms of the pressure of the neighbouring half triangles. There will be

- 105 -

a mat r i x of 2n+3 equat ions and on ly 2n+l unknowns. Once the mat r i x i s

solved and hence the pressures at the half unit triangles known, the

pressure at the edges of the singularity element is found from relation

(3.18) by setting x' to zero.

Once again representing the deflection at node % due to pressure at

node y by d. - and V. as the total deflection at node the following v, y it matrix is constructed in dimensionless form using the superposition

principal, numbering the triangles from left to right consecutively:

Vsl = P1 (dsl,sl+dsl,l} + P2 1sl,2 + * P2n+1 idsl,2n+l+ dsl,sr}

V1 = F1 (dl,sl+dl,l} + F2 \2 + — * P2n+1 (11,2n+l+dl,sr)

V2n+1 " F1 (d2n+l,sl^2n+l,l) * P2^2n+1, 2 + " ' + P2n+1 (d2n+l, 2n+l+d2n+l,sr)

Vsr a P1 (dsr,sl + 1sr,l} + F2^sr,2 + + F2n+l (1sr,2n+l+ dsr,srJ

Taking the 2n+l equations from V2 to V

2n+V m a t r i x c a n s°lved by

straightforward inversion to find {p} for a given deflection vector.

Once the pressures P^ to P 2 n + 1 a r e the following relation is

used by direct substitution to find the pressure at the edges of the

singularity element:

*e t = d = = . P

s r = T = (

3

"2 1

> s i

/OoT s r

SoTooT

- 106 -

3.10 RESULTS OF THE ACCURACY OF THE TECHNIQUE, EMPLOYING THE SINGULARITY

ELEMENT

The accuracy of the method is analysed by comparison with the

classical solution. Repeating the same procedure as in Section 3.6, the

accuracy is illustrated in graphs of Figures 3.7, 3.8 and 3.9 for

different numbers of elements. For scaling purposes, the point

corresponding to the singularity element is only shown in Figure 3.8 and

is excluded from the other two graphs. As we see, the agreement between

the numerical method and the classical solution is excellent and, with an

increasing number of elements, the error is almost zero at the central

point and increases to a negligible value in the vicinity of singularity.

It is therefore concluded that using the newly constructed singularity

elements, the inaccuracy presented in the numerical methods discussed

above due to geometrical discontinuities is minimised over the whole

solution domain, especially in the vicinity of the discontinuity.

3.11 INTRODUCTION TO THE APPLICATION OF THREE-DIMENSIONAL SINGULARITY

ELEMENT IN STRESS CONCENTRATION PROBLEMS

Many practical problems arise, in which there are real or effective

discontinuities in profile, which lead to stress concentrations having

implications for plastic flow and fatigue failures. Some of these

problems are discussed here for introductory purposes.

Since cpnformal gears function by movement of the contact area

across the face width, the continuity of action from one tooth to the next

occurs by transfer of load from one end of the face width to the other.

Thus, deflection under load can cause errors in the uniformity of tooth

motion. Such problems exist with involute spur gears, where tip and root

relief are used to assist the transfer of load from one tooth to the next.

The equivalent in conformal gearing is end relief, this being introduced

FIGURE 3 .7 : PRESSURE DISTRIBUTION BETWEEN THE PUNCH AND HALF SPACE USING SINGULARITY ELEMENT


+ + + 2 5 T R I F L N G U L R R E L E M E N T S O

A N D 2 S I N G U L A R I T Y E L E M E N T S

E X A C T T H E O R Y

T = 0 - 0 0 0 5 O

•

R-

O

•

X C M G

U J

O C

Z D

U J «_ Q C O Q _

I

M O

O '

1 1

I 1 R 4 1 T 4 4 >•• I • 1 1 H 1 1 1 + 1 ' ^

1 . 0 0 - B . 7 8 - 0 . 5 6 - B . 3 3 - B - 1 ? 0*. 1 1 O ' . 3 3 O ' . 5 6 O ' . 7 8 L'.

X ( M E T E R S )


<N

- no -

in the pinions in the earliest design, to ensure that the end of tooth

stresses were less than in the central region. Although no failures have

been found to originate at tooth ends by introducing the end relief into

conformal gears, it has been reported in (11) that the tooth contact

marking still extends across the full face width. Figures 3.10a and b

illustrate this phenomenon, in which the ellipse of the contact overrides

the edge of the available area, leading to a truncated contact ellipse.

As the lubricant film is generated by the motion of the contact

area across the face width, the initial contact occurs at a tooth end.

If the tooth ends are left sharp, or a relatively crude chamfer is applied,

as is the practice with some involute gears, then there is a tendency for

a line of scuffing to be produced up and down the tooth height. However,

careful profiling at the edges seems to alleviate the problem.

Similar effects have been noted with the tips of involute gear teeth,

which for that system is again the start of motion over the tooth surface.

In spiral bevel gears, for example, it has been the practice to radius

and buff the tooth edges.

Certainly, whenever the tooth ends are left sharp and the contact

area extends over the tooth tip, there is a greatly increased tendency

towards scuffing.

The other common problem, as discussed in (4), is when the ratio of

the thrust load to radial load in an angular contact bearing becomes

excessive, the contact angle increases sufficiently for the elliptical

contact area of the ball to override the edge of the race.

These common practical problems have prompted the study of analysing

the elastostatic pressure distribution in truncated elliptical contacts.

The method developed in the previous chapter can very well be applied to

this case. In accordance with the aforementioned method, it was thought

that devising a three-dimensional singularity element, instead of

- m -

rectangular boxes of pressure, and constructing it over the areas that

stress singularities are to be expected, will lead to a much more accurate

solution.

3.12 THREE-DIMENSIONAL SINGULARITY ELEMENT

This section is concerned with replacing the rectangular boxes of

constant pressure in the vicinity of the discontinuity zone by boxes of

varying pressure, in which their heights describe the order of

singularity in the direction approaching the discontinuity. Sufficiently

close to the truncated region, Johnson (4) suggests that the stress

distribution might be expected to approach the two-dimensional situation.

It is reasonable, therefore, to expect similar singularities in these

situations. The gradient of the pressure curve in the direction of

approaching the discontinuity is so steep that the assumption of zero

pressure gradient for each element along the discontinuity zone seems to

be reasonably accurate.

Figure 3.11 illustrates the three-dimensional singularity element,

together with its neighbouring rectangular box of constant pressure.

3.13 INFLUENCE COEFFICIENTS FOR THREE-DIMENSIONAL SINGULARITY ELEMENT

The procedure for obtaining the influence coefficients for three-

dimensional singularity element is the same as for the two-dimensional

case discussed in Section 3.7 and will not be repeated here. The order

of singularity is taken to be the same as that in the two-dimensional case

for sharp edges and this is justified since near the truncated zone, the

stress distribution for each element might be expected to approach the

two-dimensional situation (4).

Now, depending on whether the truncated ellipse is like that in

Figures 3.10a or b, the direction of the order of singularity and therefore

- 112 -

TRUNCATED CONTACT

E L L I P S E

(a)

(b) TRUNCATED ELLIPSE OF CONTACT

OVER RIDING GEARS

FIGURE 3.10

- 113 -

EDGE OF DISCONTINUITY

3 D SINGULARITY ELEMENT

FIGURE 3.11

- 114 -

the coordinates chosen, and hence the limits of the double integration

of force-displacement relationship will change accordingly.

If the direction of the singularity order is in the x direction:

PSN ) = P

i S P A (3.22)

where P is the height of the pressure curve described by the singularity s

element at a distance x' from the edge of the element, and c is the

coordinate measured in the x direction from the edge of discontinuity, and

P. is the pressure at the neighbouring rectangular box.

If the direction of the singularity order is in the y direction:

P. /2aJ P = (3.23)

where Po is now measured at a distance y' from the edge of the element, s

and e is the coordinate measured in the y direction from the edge of

discontinuity.

If the direction of the singularity order is in the x direction:

d(2,y) = I'"91"' 1 In {(^)+A^)2

+ (x^J)2

} £,

( 3 a 2 4 )

0 Six'/b'+0.001) (y-a') + S(y-a ')2 + (x-x')2

and if the direction of the singularity order is in the y direction:

M Y ) - yd 9 9 ( 2 a , )

_ _ i ^

0 Ay '/2a'+0.001) (x-5*/2) +Ax-b '/2)2+ (y-y ')2

where all the lengthwise dimensions are non-dimensionalised with respect to

Rj, and l(x3y) = d(x

3y)/Rp^.

- 115 -

3.14 SETTING UP THE MATRIX INVOLVING THE THREE-DIMENSIONAL SINGULARITY

ELEMENT INFLUENCE COEFFICIENTS

The singularity elements are taken to act in one column or row

(depending on the singularity direction) adjacent to the discontinuity

zone. Let:

mn = total number of elements

m = number of elements in one column

n = number of elements in one row

The method of setting up the matrix and finding the pressures at the

discontinuity zone is the same as in the two-dimensional case discussed in

Section 3.9 and therefore will not be repeated here. The only difference

is that now, since pressures at singularity elements are prescribed in

terms of the neighbouring blocks, there will be mn equations and only mn-m

or mn-n (depending on the singularity direction) unknowns. Choosing the

equations which correspond to the deflections under the rectangular boxes

of pressure and solving them for a given deflection vector will lead to

the pressure distribution. The pressures corresponding to the singularity

elements are then found by straight substitution into the relations (3.22)

or (3.23) (depending on the singularity direction) by setting c to

0.0012?' or to 0.001(2ar), again depending on the singularity direction.

3.15 THE GAP SHAPE BETWEEN A SPHERE AND A CYLINDER WHICH 15 MOUNTED ON THE

EDGE OF A CONE

In order to be able to study the effect of profiling the edges of

conformal gears (end reliefs) on the pressure distribution, a model has to

be made. The model used here is the sphere inside the cone, discussed in

the previous chapter.

- 116 -

In order to reduce the available area and to model a sharp edge

situation, the conical member is cut somewhere in the elliptical zone in a

direction parallel to either the major or minor axis of the ellipse. In

order to study the effect of profiling, a cylindrical member is mounted on

the edge of the cut conical member. Now, instead of a sudden change in

profile, there is some material beyond the cut conical section, which will

give rise to the gradual change of profile and hence to relaxing the

pressure distribution. For the relieved part, the equation for the

separation profile of the bodies has to be modified. Figure 3.12

illustrates the geometry, and the following relations which refer to

Figure 3.12 are purely geometrical. r, r 2 and e are illustrated in

Figure 2.6 of the previous chapter, and R is the radius of the relieved

cylinder. Referring to Figures 3.12 and 3.13:

3 = sin"1 ( aA ; (3.26) r2 + x sin 9

y-*T ABi AB - AC - BC , AC = , BD =

cos 8 r BC = BD tan 8 , AB = - -75 - tan 8

Y

"A

T AB2

~ O D

cos 8 r Therefore:

- 1 ± JL + (2/R ) tan 8 CY-df)./cos 8 AB =

tan 8 / RR

(only the positive root is relevant). Therefore:

R + tan 8 (y-aj/cos 8 - R„ /I + (2/R ) tan 8 (y-aJ/cos BD = —

tan2 8

Now DE = CE- CD = (Y-CLY) tan &-BD/cos 8, therefore:

- 117 -

END VIEW OF THE CONICAL BODY SHOWING A RADIAL SECTION OF THE CONE AND ITS RELIEF

FIGURE 3.12

- 118 -

JL Y

• i k- . • • • • • •

» 0

1 t-

T t. - • • • • •

• •

1

a T

» • • • • • •

• • f

• •

• m • • • • • «

• « • 0 » • • •

• • • V « # •

THE: TOP TWO ROWS OF THE MESH ARE IN THE PROFILED REGIONS

FIGURE 3.13

- 119 -

DE = (y-aJ tan 3 -R + tan 3 (y-a

T)/cos $-R Jl + (2/R ) tan 3 (y-a

T)/cos 3

(3. T cos 3 tan2 3

The perpendicular distance from D to the tangent plane equals:

cos Q + DE cos e 2r

where e is the cone angle.

Non-dimensionalising all the parameters with respect to the radius

of sphere, the final expression for the undistorted separation profile is:

where DE and 3 are given by equations (3.26) and (3.27).

Figure 3.13 illustrates the solution domain with regular elements in

which rectangular boxes of pressure are to act over each rectangular area.

For illustration purposes only, if the top two rows of the mesh are to be

in the profiled region, then aT and (y-a

T) are shown in the contact on the

figure.

3.16 RESULTS OF THREE-DIMENSIONAL STRESS CONCENTRATION

The results presented here are for pictorial purposes only and

portray the three-dimensional pressure distribution. In all the plots

presented in this section, x and y coordinates are scaled differently

and therefore the pictures are not the true representation of the shape of

the contact.

Figure 3.14 shows the elastostatic pressure distribution in an

elliptical contact identical to that of conformal gears when the two mating

gears override and hence lead to a truncated ellipse. The developed

X2 + Y

2

cos e - DE cos e (3.28) 2(r- + x sin 6)

- 120 -

singularity element is used to represent the pressure curve in the numerical

methods accurately at the sharp edges. The amount of the stress

concentration in comparison with the central pressures is very well

illustrated, which is due to sharp edge effects.

Figures 3.15a and b, which portray identical results from two

different viewing angles, illustrate pressure distribution in an override

case for conformal gears. When the ends are profiled, the pressure

distribution is relaxed at the ends and behaves much more gently there.

The profile used in the relief region is part of a cylindrical body mounted

near the edge of a cut conical member.

By comparing Figures 3.14 and 3.15, the importance and merits of

profiling stress concentration problems become evident.

3.17 DISCUSSION

In performing the analysis, two classes of problem must be

distinguished. In the first class, the profiles are smooth and

continuous at the ends of the contact, so that the mid-point of the contact

is located at an unknown distance from some arbitrary chosen origin.

The dimensions of the contact is one of the unknowns of the problem.

This is self-determined in the solution because wherever the pressures

become negative, the material is in tension and therefore the nodes there

are outside the contact. In the second class, the length and position of

the contact area is uniquely defined by discontinuities in the profile.

In the first case, where the profiles of the contacting bodies are smooth

and continuous up to and beyond the ends of the contact pressure falls to

zero at the ends of the contact area and in the cases where a discontinuity

of profile prescribes the contact length, the pressures at the ends of the

contact are no longer zero.

Since the triangular elements represent a pressure distribution

FIGURE 3 .14: ELASTOSTATIC PRESSURE PROFILE

SHARP EDGEVINQ"^IGUl/A&lTY ELEMENT

DIMEN5IONLE 5S A P P R O A C H = 0 0 1

PROFILED END

DIMENSIONLESS APPROAC H=001

RO ro

R k = 1 0 0

FIGURE 3.15a: ELASTOSTATIC PRESSURE PROFILE

PROFIED END

DIMENSIONLESS APPROAC H=z-001

R r = 1 0 0 FIGURE 3.15b: ELASTOSTATIC PRESSURE PROFILE

- 124 -

which rises smoothly from zero at the ends of the contact, this kind of

element is preferable to use in the first class of problems discussed.

However, if discontinuities in the profiles are present and stress

singularities are to be expected, then it has been shown that if

singularity elements are not used, even with adding two half triangles at

the ends of the contact, the rectangular elements would give a more

accurate solution. The other advantage of the rectangular elements is

their flexibility in varying their base length. Hence, more pressure

elements can be constructed at the places where the slope of the pressure

curve is sharp.

Using the singularity element developed in this chapter and placing

it where sharp edges exist makes the numerical solution very accurate in

the whole solution domain, especially in the vicinity of discontinuities.

The other merit of the singularity element is saving on the computing side

of the problem where an accurate solution can be obtained with a small

number of elements.

A less severe case of a singularity can be due to a slight

discontinuity in the slope of the profiles within the contact zone,

i.e. contact is maintained on either side of the discontinuity. A

logarithmic type of singularity is the mathematical representation of

these kinds of discontinuities and the validity of the present method

applied to these types of problems can very well be tested by using the

model of a sharp wedge indenting into an elastic half space, in which the

exact theory for it is given by Sneddon (12). In most practical cases,

the slope discontinuity is rounded off by finishing procedures, whereupon

its effect becomes negligible. End effects are not negligible, however.

A sharp end leads to a power singularity and even well radiused ends give

rise to a stress concentration which is revealed in the numerical method.

Singularity elements can be useful in three-dimensional problems, in

- 125 -

which their use is expected to lead to a more accurate solution. Close

to the discontinuity region (Johnson (4)), the stress distribution for

each element approaches the two-dimensional case. In order to be able

to estimate order for the singularity of a three-dimensional case, where

an exact theory is not known, the three-dimensional case can be compared

with a similar type of two-dimensional problem with a known order. Using

that order of singularity in the three-dimensional case seems to be quite

justified.

In order to test the order of stress concentrations, this method is

very useful and if the end profiles are used, the method of using ordinary

rectangular boxes of pressure is very useful to design the optimum shape

of the profiles in order to lead to a maximum relaxation in the stress

concentration regions.

- 126 -

3 .18 REFERENCES

(1) DUNDURS, J.

J. Elasticity, 2 (1972) 109.

(2) FREDRIKSSON, B.

"On elastostatic contact problems with friction",

Linkoping Studies in Science & Technology Dissertations, No. 6.

(3) OKUBO, H.

"On the two-dimensional problems of a semi-infinite elastic body

compressed by an elastic plane",

Jap. Soc. Medi. Eng. T/ians., 65 (1952) 58-62.

(4) NAYAK, L., & JOHNSON, K.L.

"Pressure between elastic bodies having a slender area of contact and

arbitrary profiles",

Internal report, Cambridge University.

(5) SHETAYERMAN, Ya.

"Contact problem of the theory of elasticity",

Kontaktnaja Zodatja Taciti Lipnagostic (1949) Moscow.

(6) TIMOSHENKO, S.P., & 600DIER, J.N.


McGraw-Hill, New York (1970) 3rd edition,

(7) BISWAS, S., & SNIDLE, R.W.

"Calculation of surface deformation in point contact EHD",

ASME, J. Lubn. Tcch., (July 1977) 313-317.

(8) DWIGHT, H.B.


Macmi11 an, New York (1961) 4th edition.



loaded rollers",

J. Mcch. Eng. Set., 21_ (1979) 381-388.

- 127 -

BENTALL, R.H., & JOHNSON; K.L.

Int. J. Mack. Set., 9 (1967) 389.

SHOTTER, B.A.

"Evaluation of conformal gearing for use in helicopter applications

West!and Helicopters Report, Research Paper 481 (November 1974).

SNEDDON, I.M.

Fourier Transforms,

McGraw-Hill (1951).

- 128 -

CHAPTER 4

OIL FILM THICKNESS AND PRESSURE DISTRIBUTION IN

ELASTOHYDRODYNAMIC ELLIPTICAL CONTACTS

4.1 INTRODUCTION

It is now well-established, both theoretically and experimentally,

that machine elements having concentrated, contact, and in relative motion,

can be separated by a coherent liquid lubricant film. If the load is

high enough to distort locally their surfaces and increase lubricant

viscosity in the gap between them, elastohydrodynamic lubrication (EHL)

results (1,2,3,4,5). This condition can occur in engineering

applications such as between rolling elements and their races in ball and

rolling bearings and between pairs of teeth in spur gears. Practical

evidence that EHL does indeed occur is shown in the complete absence of

wear in precision ball and roller bearings or by the cessation of wear,

after an initial period, in more commercial applications of such bearings.

The formulae obtained from numerical solutions to the EHL problem are

therefore a useful design tool for forecasting minimum lubricant film

thickness.

Apart from numerical solutions to "infinitely long" line contacts

(2) and Grubin type solutions for the entry film thickness (1), numerical

solutions to date for point contacts have only succeeded for moderate

loads and a relatively low material parameter such as would result from

the EHL between glass and steel (3,4). However, interferons try

experiments using sapphire, diamond and tungsten carbide as the contacting

bodies (6) show that the regression formulae obtained from the numerical

methods can be extrapolated with reasonable confidence to the high load,

steel on steel region. Apparently, the work is in progress elsewhere to

overcome the convergence problems associated with these numerical methods.

- 129 -

This chapter presents more numerical results for point contacts,

again under moderate loads, using a relatively low value material

parameter. However, some additional features are included in the

solutions. These include a simulation of the EHL between dome ended

rollers and the outer race ribs of a roller bearing when subjected to

combined radial and axial loads, and the lubrication of helical Novikov

(circular arc) gears when the direction of velocity vector is either along

the long axis of static contact ellipse or skewed to it. More results

include the effect of squeeze velocity on EHL, an important factor when

considering transient loads on rolling element bearings, or even the

effect of the transition of the elements from the loaded to the unloaded

zones. In addition, the EHL of a point contact under pure spin is

discussed.

4.2 THEORY

The general approach to the numerical solution of the EHL problem

for point contacts is similar to the method used in references (3) and (4).

The following governing equations are employed.

4.2.1 Reynolds Equation^

_ 3 _

3a: h

3

3 P r\ 3a: 31/

p h3

3P

9J/. N = 12 3 (ph) ^ , 3 (ph)

TT u " t o * * p ^ (4.1)

If:

x x • 57 ' * = JL R

1 ' P =

t ' *

h_ R4

P = PK'

* See Notation

- 130 -

U = /u2

+ v1

, W

e = tan-1 (-) , w* = — u s u

equation (4.1) becomes:

J L [ l i d + ±

BX L 71 AP AP"

p /z3

3P

- n

By-

12 U K' R,

„„„ . afp/J ^ . - B(ph) . — rrjt cos 9 — + sin 9 — + p w*

Bx By S

-(4.2)

4.2.2 Equations of State of the Lubricant

As the subsequent gap shape is determined in the inlet

conjunction between the contiguous bodies, pressures there are not high,

so the single formula (1):

n = * CtP

can apply. Furthermore, a reduced pressure

1 M

- a P , a - — (1 - e ) ^ A

is used to prevent excessive peak pressures from appearing during the

numerical procedure (1,4). Let G* = ct/X', the materials group.

Therefore:

T) = e (4.3)

and: q = i d - f 0**) (4.4)

Also, the relation between lubricant density and pressure is taken from

- 131 -

r e f e rence (2) to be:

= 2 ^ 0.6P __

( 4 > g )

K' + 1.7P

4.2.3 Final Form of Reynolds Equation

Letting u* = (ut\oK')/R2, the dimensionless rolling speed,

equation (4.2) can now be written as:

_3_ V F ^ J TP & dx ^ dX- 3j/J

12 U* cos sin s

dX (4.6)

As in reference (3), by means of the substitution $ = qJp/ 2, terms

containing products of the derivatives of h and pressures are eliminated,

thereby considerably reducing computation time. Equation (4.6) becomes:

S* - L (7 X + 4 : ( P %

L

dx dx dy dy J

= 1-2U*

2 L

3a: 3a: dy dy

cos e ^ P h L + Sin e i<JkL + p w* dx dy

s

•

(4.7)

Equation (4.7) can now be written in finite difference form as

i*+

l*3 1**3 1>-1*3 i*3 1**3+1

S. . , - AP. . - SO. . = 0 1*3 1**3-1 1**3 1**3 1**3

(4.8)

where:

- 132 -

E. . = 1,3

W. . = 1>,3

N. . = 1,3

S. . -

4 P . . i,3

t\ .

4 6 X2 ^ - 1 + 1 * 3

+ P

J + 4 Q

T , 3 'M

4$y2 L ^ < 7 + 7 " *

R _ •

4i-p. I "i.3+1 +

P

i,j-1 +

J K . 3 " . ( H

I + I . I " V J . ,7J

, L

^ 4SP

7 T _ < K+ 1

I - K - I I >2

2 1 , 5 " T - , 5

O . . S T . 1 +

7 T.,J 6 a '

P . . (

\ I + I - \ I - I> Z

+

2 T , 5 ^

1,3 1,3 8 Y

4

SO. . 1,3

4 - rrk 4 -p . . 7z. . + p . . 7r. .

S A F Z B Y2

= 12U*

2<SIC

cos B in. * — • — + p

2Sy

h. . . siri B i n . * — — « — + p

26y

P V • 1,3 S . .1

1 > , 3J

- 133 -

E. ., W. ., N. . and S. . refer to the coefficients applied to the nodes ti 3 1*3 3 1*3 3 3 3

east, west, north and south of the central node. AP. .is the coefficient i»3

applied to the central node. SO. . is the source term and, finally, 6x 1*3 3

and 6y are the element dimensions (6a = 6«/r2> &y = Sy/R^).

Equation (4.7) is highly non-linear and the most important

factor of all which has prompted the study reported here is the

distribution of film thickness which itself is dependent on the pressure

distribution.

4.2.4 Lubricant Film Thickness

The EHL film thickness, 7z, can be written as:

h(x,y) = h (x,y) (x3(0,0) - w(x,y)) _ g

+ h* (4.9) o x

where, from Figure 4.1, h^ is the separation of the bodies in their

undeformed state, $w(0,0) -fw(x,y) + hg(x,y)] is the elastic film shape, and

h* = h(0,0) (suffix (0,0) corresponds to the central point). H is

determined from the chosen geometry. For example, the simulation of a

pair of Novikov helical circular arc gear teeth is by a sphere contacting

the inside of a cone having an apex half angle 3 (about 14°).

Therefore, from Figure 4.2, which is similar to Figure 2.6 of

Chapter 2, assuming that the dimensions of the footprint will be small

compared with the local radii of curvature of the contacting bodies (7),

let their shape be parabolic there before deformation, making:

h _ X

2

+ I2

_ J 2 cos 3 g 2R

1 2(R

2 cos 3 + X sin 3;

where x and 1 are the coordinates of the projections of points on the

bodies onto a tangent plane situated at their touch point, R^ is the

- 134 -

FILM THICKNESS COMPONENTS

FIGURE 4.1

- 135 -

SPHERE INSIDE A CONE BEFORE COMPRESSION

FIGURE 4.2

- 136 -

sphere radius, and R is the cone local radius of curvature at the u

contact point, as shown in Figure 4.2. If R* = R^RY

h = X2

* *2

(4.10) 9 2

2(R* cos $ + X sin B)

where X = x/R2, and I = Y/R

2 .

As the examples discussed below involve elliptical footprints,

let the contact be a sphere radius R2 touching the inside of a cylinder

radius thus, making 3 = 0:

- (4.1D

4.2.5 Elastic Film Shape

Assuming that the contacting bodies behave as elastic half

spaces, the solution zone of a finite difference mesh is divided into

equal rectangular elements with the pressure co^tajSJt over each element

(3). Using the integral force-displacement relationship:

7 P('x .,y .) dx . dy .

d(x.,y.) = Kr

(l+±) J 3 3

(4.12) * * E A Ax.-x.)

2

+ (y.-y.)z

V 3 I> 3

where A is the area over which the pressure P acts. Equation (4.12) is

solved analytically for a constant pressure acting over an element as in

Section 2.3 of Chapter 2. Hence, a set of influence coefficients is

obtained, thereby allowing the deflections to be calculated by using the

superposition principle. Thus, in matrix form:

n n * row col __

- 137 -

^ = l j n

row9 a n d J

' =

^ n

ool)3 W h e r e d ( x

i3 y

i) =

d(xi3yJ/R

r

4.2.6 Load

The load w is found from the integrated pressures as:

W = / P dx dy A

In dimensionless form, it is:

w* = ^JLL = f p & (4.14)

R2

A


To solve the non-linear Reynolds equation simultaneously with the

equations for elasticity, film thickness and lubricant state, the

following procedure is adopted.

4.3.1 Mesh Construction

For a chosen load and geometry, the elastostatic footprint

dimensions are calculated from the numerical method discussed in

Chapter 2. The method is very useful if the shape is unusual, such as

that given in the example above (a deformed ellipse). If the shape is

symmetrical, such as that given by equation (4.11), then the known

solutions are found in reference (8).

Assuming the velocity vector u* is along the long axis, then

referring to Figure 4.3a, a rectangular grid is constructed over the

computation zone, its size being such that it is about 3.5 to 4.0 times

the elastostatic footprint half length in the inlet to the expected EHD

pressure distribution and at outlet to about 1.25 to 1.5 times that

- 138 -

factor. Transversely, it is about 1.5 times the footprint half width.

These dimensions correspond approximately to 'flooded' lubrication

conditions (3). If u* is at some angle e to the long axis in Figure

4.3b, then the domain is adjusted to ensure that inlet distances are

sufficient to preserve flooded conditions.

4.3.2 Boundary Conditions

(1) Take $ as zero along the edges of the computation zone.

(2) Avoid tensile pressures in the film by applying the Reynolds

condition $ = 3$/3aT = 3<&/3y = 0, whenever-they go negative

during the iteration procedure. This condition will prevail

by immediately setting any value of <& to zero whenever they

happen to be negative. This will establish the outlet

boundary.

4.3.3 Initial Condition

The pressure distribution is initialised to be the elasto-

static distribution as that discussed in Chapter 2 over the dry contact

footprint and zero outside it-.

4.3.4 The Algorithms Used to Solve the Equations

The following is the algorithm used to solve the equations.

Choose materials and geometry, and then:

(1) Choose load w* and calculate hg(x3y).

(2) Pressure distribution p is elastostatic. ' s

(3) Guess a value for the central film thickness h*. x ' o

(4) Calculate film thickness distribution by:

(a) Using known P in equation (4.13) to calculate the

- 139 -

deflections.

(b) Find film thickness by using equation (4.9).

The EHD pressure distribution is initialised as P^ (zero in

the first iteration) and hence reduced pressures q are

calculated.

<& is initialised as

Using P^, calculate the density distribution from equation

(4.5).

Calculate coefficients of finite difference equation.

Solve Reynolds equation (4.8) for $ by a line iterative

method (tridiagonal matrix algorithm).

The up-to-date values of q are obtained from $ and are

checked for convergence with previous values.

(a) If not converged, modified values of q are calculated by

under-relaxation, producing new values of P^ from

equation (4.4). Then the algorithm returns to stage 6.

(b) If converged, the algorithm moves to next stage.

New values of p^ are calculated from q.

From equation (4.14), P^ is integrated to produce a load w*.

Is w* within a specified tolerance of the required initial

load?

(a) No: adjust the film thickness h* in equation (4.9) and

return to stage 5.

(b) Yes: move to next stage.

The distribution of P and p, are checked for convergence. s rt

(a) If not converged, new values of Pq are calculated from P.

with heavy under-relaxation and the algorithm returns to

stage 4.

(b) If converged, the solution is obtained.

RECTANGULAR GRID OVER THE COMPUTATION ZONE

FIGURE 4.3a

RECTANGULAR GRID OVER THE COMPUTATION ZONE

FIGURE 4.3b

- 142 -

4.3.5 Convergence Criteria and Number of Elements Used

In order to stop the most outer loop iteration, the following

convergence criteria (as used in references (3) and (4)) is employed to

lead the final solution:

row 9 = 1

>n

cot

I lp

/z. . -p

s. •

4 C 0 N V P

1*3 3 1*3 3

where CONVP is varied between 0.02 and 0.03, depending on the required

accuracy of the solution.

Similarly, for reduced pressures in the most inner loop:

J = '"col

I k - q « 0 -0 1

' L l

new ^oldx

new

The convergence requirement for load in the inside loop is taken to be:

IW* - W* 'i\ ' requ-ired obtained

1

< q gg

W* . , requ^red

The number of elements used in the x and y directions depends on the

direction of the velocity vector. If the velocity vector is in the x

direction, then 57 and 15 elements are used in the x and y directions,

respectively (Figure 4.3a). If the vector is skewed at, say, 45° to the

x axis, then a 29 by 29 nodal points mesh is used (Figure 4.3b). If the

vector is along the y axis, then a 15 by 57 nodal points in the x and y

directions, respectively, are used.

4.3.6 Relaxation Factors

The under-relaxation used for the pressures in the most outer

loop is of the form:

- 143 -

New P s

The value of \p is adjusted to give the required convergence within the

least number of iterations. The optimum value of \p is found to vary

from 0.05 to 0.02 as the number of iterations increases.

The under-relaxation factor for reduced pressures in the most

inner loop is of the form:

where x^ is kept at a constant value of 0.1 throughout the iteration.

4.4 NON-DIMENSIONAL GROUPS

Dimensionless central film thickness, H* = H/R« o o i

Dimensionless minimum film thickness, h* -

The independent variable parameters which influence the film thickness

can be put into the following dimensionless groups:

Dimensionless film thickness h = h/Rj

Y M I - f' (**>

- 144 -

8 = tan"1

(-) u

W* = JL W _£

u

The dimension!ess central and minimum film thicknesses can be written as:

h* = fjCU^W^G^e^BtV*)

ho = f2(U*'W*>G*'ep>Q>Ws}

For design purposes, it is usually desirable to predict the value of

the central and minimum film thicknesses and one of the most important

aspects of the theory developed in this chapter is the effect of changing

the angle e on these two dependent variables.

4.5 RESULTS

4.5.1 General

For design purposes, the most important parameters are the

minimum film thickness, h*, and the maximum pressure. The central film m

thickness is also of interest for comparing with earlier approximate

solutions or with experimental results. From the governing equations:

h* = f2(U*,W*,G*,R*,B,W*

3) (4.15)

All the results presented below are for geometries giving an elastostatic

elliptic footprint. This is defined by the ellipticity ratio e = a/b,

where a/b = f'(R*). Using e i n s t e a d of R* as one of the controlling

groups immediately gives one an idea of the contact shape. Thus, an e

- 145 -

of one is for a ball on a plane, whilst an e o f 20 would approximate a

crowned roller. However, a line contact represented by, for example, a

cylindrical roller and its race cannot be simulated by & . The line P

contact is finite and normally has dub-off or crown radii at each end to

minimise stress concentrations there. Two separate geometry groups are

therefore required as controlling variables.

4.5.2 Effect of Changing 9

The first results show the effect of varying the direction of

the rolling velocity vector u*. Thus, Figures 4.4a and 4.4b show isobars

and contours for u* directed along the long axis of the ellipse (e = 0).

In these and all contour and isobar plots to be represented, the + symbol

indicates the centre of the elastostatic contact. Figure 4.4c shows the

experimental results, taken from reference (11), showing interference

fringe contours under the same external conditions. Figures 4.5a and

4.5b again show theoretical and experimental contours when e = 36°, whilst

Figures 4.6a and 4.6b are corresponding results for e = 90°. For each

angle e chosen, the external parameters were varied to comply with the

experimental conditions, so a quantitative comparison between the above

figures cannot be made. However, the contour shape in Figures 4.4a and

4.4b shows that the oil is entrained through a narrow U shaped gap with

minimum thickness at side lobes just past the elliptical footprint centre.

The outlet boundary is shown by a dashed line on contour plots. The

Reynolds exit boundary condition follows the Hertzian ellipse boundary

there in Figures 4.4b, 4.5a and 4.6a. The boundaries diverge further out

in Figures 4.4c, 4.5b and 4.6b because, in the experimental case, different

boundary conditions (like Swift Steiber) probably obtain there. The

isobars in Figure 4.4a show a main peak on the x axis with a subsidiary

bump towards the exit. This is in keeping with other experimental and

- 146 -

theoretical findings (2,3,4). For the angle drive in Figure 4.5, the

contours distort to create a minimum gap at the through point exit. In

Figure 4.6, when e = 90°, we have the typical ball bearing contours (3,

11). The slowly varying gap in the y direction has resulted in gentle

contours with the two side lobes having moved from the footprint major

axis to a single minimum on the minor axis. Actual pressure distribution

and gap shapes are shown in Figures 4.7 and 4.8 for a different set of

operating conditions, with e = 0° and 30°, respectively. Despite the low

value of w*9 the minimum film thickness on the footprint major axis,

associated with a subsidiary pressure peak, is evident in Figure 4.7b.

There is a slight side closure on the minor axis (Figure 4.7a). Making

8 = 30° in Figure 4.8 removes the subsidiary pressure peak and reduces the

elastic distortion of the surfaces, the pressure becoming more like that

of an isoviscous distribution. As the velocity vector is not along an

axis of symmetry, the surfaces distort accordingly (Figure 4.8b).

Therefore, from comparison of Figures 4.7 and 4.8, increasing the value of

e leads to higher films being formed and the conditions become less

elastohydrodynamic. Under the same external conditions, highest films

would be formed if the direction of the velocity vector is along the

short axis of the ellipse.

Figures 4.9a, 4.9b and 4.9c are for pictorial purposes only

and portray the three-dimensional pressure distribution for some given

external conditions when 0 = 0° , 30° and 60°, respectively. In order to

keep all the represented portrayals well into the EHL regime, the load is

increased at each case as 0 increases. Note that these figures are a

matrix map and therefore the spacings of the grid points in the a and y

directions are artificially scaled to be equal and therefore the figures

are not a true representation. Nevertheless, the plots give one an idea

of how the pressure distributions vary .in the computation zone. As is

FIGURE 4.4a

- 6

-11 DIMENSIONLESS L0AD=0 .240 «io DIMENSIONLESS SPEED=1.170 *io CONTOUR PLOT OF DIMENSIONLESS PRESSURE (pxio )

G r l 072o TETA=0°

FIGURE 4.4c: EXPERIMENTAL PHOTOGRAPH FROM REFERENCE (11) (EXTERNAL PARAMETERS NEARLY THE SAME AS IN FIGURE 4.4b)

FIGURE 4.4b: COMPUTED CONTOURS

CAVITATION LINES

-6

11 D I M E N S I O N L E S S L O R R r r O . 24 0 xio

• ! • I M E N S I O N L E S S S P E E D = 1 . 1 7 0 * i o

• C O N T O U R P L O T O F D I M E N S I O N L E S S F I L M T H I C K N E S S ( h / R S ) *io

G = 1 0 7 2 o

T E T R = 0°

-p. cc

FIGURE 4.5a: COMPUTED CONTOURS

- 150 -

FIGURE 4.5b: EXPERIMENTAL PHOTOGRAPH FROM REFERENCE (11) (EXTERNAL PARAMETERS ARE NEARLY THE SAME AS IN FIGURE 4,5a)

FIGURE 4.6a: COMPUTED CONTOURS

- - --"

+

------CAVlT,'\ TlON LINES

-6 OIMENSIONLESS LOnO=O.230xll 0=12150

. -II 0

DIMENSIONLESS SPEED=O.283 KJO TETR=90 J

CONTOUR PLOT OF 0 I MENS I ONLESS FILM TI-I [CI~NESS ( h/RS ) KIO

- 152 -

FIGURE 4.6b: EXPERIMENTAL PHOTOGRAPH FROM REFERENCE (11) (EXTERNAL PARAMETERS ARE NEARLY THE SAME AS IN FIGURE 4.6a)

- 153 -

33

0 IMENSIQNLESS L0A0=0 .057 «io G=7L46 0 IMENS IONLESS. SPEED= 1 .055* io TETfl=0

FIGURE 4.7: PRESSURE DISTRIBUTION AND FILM SHAPE

- 154 -

D 1 M E N S I Q N L E S S L G A D = 0 . 0 6 0 *io G = 7 1 4 S .

0 I M E N S I O N L E S S S P E E D = ' 1 . O S S *1 0

T E T A = 3 0

FIGURE 4.8: PRESSURE DISTRIBUTION AND FILM SHAPE

PORTRAY OF A TYPICAL 3D P R E S S U R E DISTRIBUTION

WHEN 0 = 0

FIGURE 4.9a

PORTRAY OF A TYPICAL 3D PRESSURE DISTRIBUTION

WHEN 0=30°

PORTRAY OF A TYPICAL 3D PRESSURE DISTRIBUTION

W H E N 0 - 6 0 °

CJl -M

I

- 158 -

shown, the pressure spike has emerged along the long axis when 9 = 0 ° and

as the direction of velocity vector changes, the spike locations change

accordingly.

4.5.3 Squeeze Effects

There are occasions, in EHL, when squeeze effects may become

significant. These may occur in radially loaded ball bearings as the

elements roll from the loaded to the unloaded zones and can become more

important if, in addition, there are shock or periodic loads. The effect

of such a. downward squeeze velocity plus a rolling velocity on pressure

distribution and film thickness in an EHL elliptical contact, with e = 90°,

is shown in Figure 4.10, together with an insignificant squeeze velocity

condition for comparison. We see that a pressure distribution, which was

due solely to U*9 is altered considerably when a large w* is superimposed

s

(Figures 4.10a and 4.10b). Its effect on the film shape is to dimple it

upwards near the peak pressure (Figure 4.1Qc and 4.10d). Note that as

the resultant integrated pressure f/* has been kept the same, the central

film thickness h* has actually increased when W* is added. Of course, in o s

an actual situation, W* would become time-dependent and be increasing,

making h* diminish, although only slightly because of the extreme

stiffness of EHL films. More significant is the rise of maximum

pressure with squeeze which may cause eventual failure of the elements,

especially if the loading is periodic.

Figures 4.11 and 4.12 show the pressure distribution and the

film thickness along the two axes of the ellipse at some intermediate

values of the squeeze velocity examined, and Figure 4.13 is the contour

plot for the highest value of the examined squeeze velocity.

Y/B ( A L O N G MINOR AXIS)

FIGURE 4.10a: PRESSURE DISTRIBUTION

X / A ( A L O N G MAJOR AXIS)

FIGURE 4.10b: PRESSURE DISTRIBUTION

Y/B (A L O N G M I N O R A X I S )

FIGURE 4.10c: FILM THICKNESS

FIGURE 4.lOd: FILM THICKNESS

in" hi z *c A 5

28 too z tu -e—

-b.M 0-17 0.80 X/n IRL0N0 MAJOR AXIS)

TTri -b.B« 0-83 1.17

-b.60 -U-lT 0.17 0 80 X/n (ALONO MAJOR AXIS)

DIMENSIONLESS L0RD=0 .175 «io 0=7146

DIMENSIONLESS SPEED= 0 .317**" TETR=90'

IGURE 4.11: PRESSURE DISTRIBUTION AND FILM THICKNESS

CTT C O

SQUEEZERRTIO =-0.0010

- • « , X/" IALON MRJOB AXIS]

-TTio -i.ii

OtMENSlONLESS L0R0=0.178-to OlMENSIONLESS SPEEO= 0 .317»«°

0=7146 TETR= 90*

-Flo -In -I-it -i,n -fc.n -*.n d o'.oo ?15 Sli T-m Y/b IALOHO MIHOR AXIS I

-i.« -i,. U'.ALoU'n.Ho^^al"

cn

SQUEEZE RRTl0=-0.0050

FIGURE 4.12: PRESSURE DISTRIBUTION AND FILM THICKNESS

FIGURE 4.13: CONTOUR PLOT WITH SQUEEZE VELOCITY PRESENT

-6 DIMENSIONLESS LOAO=O.175 K IO

-u o I MENS IONLESS SPEEO=O. 317 "10

CONTOUR PLOT OF DIMENSIONLESS

G=7146 o

TETR=90 SQUEEZE RATIO::: -0 .oto FILM THICKNESS(h/RS}.J

- 166 -

4.6 REGRESSION ANALYSIS

4.6.1 Final Regression Formulae for Minimum and Central Film

Thickness

Central and minimum film thickness formulae are obtained by

independently varying each dimensionless group in equation (4.15), whilst

holding the others constant.

The range of parameters, number of values taken and the

accuracy of the obtained regression formulae in comparison with the

numerical results are shown in Table 4.1. The regression formulae

obtained by least squares come to:

i.* N NNTT*-0*045 TJ

J0.S62+Q.04B) RIK

0.46Z 1.S71B ~1 2 Z W

*S °'°

? 6 e

p H* = 0.112W* U* G* E E E * m

(4.16)

h* = 0.07ZW* o

0.004 A

(0.484+0.068Q) Q

^0.4Z9 &

2.1ZZB ~1 2 8 W

*S 0.02SE^

(4.17)

where e is exponential, e is in radians, and w* is negative. s

4.6.2 Speed and 8

The exponent for u* is found by holding the material, squeeze

and ellipticity, independent groups constant (G* = 7146, w* = 0, e =.3.56) s p

and assuming that h* = j / * ^6

^ . For each of four different values of e

(0, TT/6, ir/3, TT/2), W* was kept constant at 0.95 x l O "7

, 0.17x10-6,

0.27xlO"6

and 0 . 1 7 x l O "6

, thus enabling an index to be found. A second

fit was then made between e and the index to give a f(Q) of the form

(a'+b'Q).

The results are:

TABLE 4.1

The Range of Parameters, Number of Values Taken and the Accuracy of the Regressed Results

Group Range Number of Values Maximum Difference Between Numerical and

Regressed Results as a Percentage Difference

U* 0.31 x 10"

1 1

+ 0.44x 10"1 0

for 0 = 0 , TT/6, IT/3, TT/2

10 for 0 = IT/2 and 6 for each

of e = 0, TT/6, IT/3

3.5% for h* m

5.8% for h* o

w* 0.12 x 10"7

•»• 0.5x lO"6

8 3.6% for h*

m

5.9% for h* o

G* 0.57x10^ + 0.12x105 5 3.1% for h*

m

5.3% for h* o

0 0 + IT/ 2 11 16.0% for h*

m

7.6% for h* o

1.2 5.34 6 11.1% for h*

m

9.8% for

W* s

-10"4

-10"2

11 22.6% for h*

m

20.1% for h*

- 168 -

H* « V

*(0.562+0.04Q)

m

ht a

^(0.484+0.0686) o

where e is in radians.

The exponential form for e is chosen so that when it goes to

zero, the exponential becomes unity. The effect on the film thickness

of 0 alone varying is found by holding the other groups constant

(J/* = 0.62x10-7, u* = 0.105x10-10 q* = 7i4

6 w*

= 0,

e = 3.56), and

s p

varying the index of U* according to the above law.

The results are:

t.* 1.5716 h* <* e tn

•l* 2.1336 n* e o

where e is in radians.

4.6.3 Ellipticity Ratio, Normal Speed, Load and Material Parameters

Exponential forms for e a n d y* are chosen so that when any

were zero, the exponential becomes unity.

By changing the radius ratio of the contacting bodies and

hence altering the ellipticity ratio, while keeping the other independent

groups constant (j/* = 0 . 1 8 x l 0 "6

, u* = 0 . 3 5 x l O ~1 0

, G* = 7146, e = TT/2,

J7J = 0), the following relation in the exponential form is obtained by a

least square fit:

- 169 -

0.076E FL* oc 0 P m

0.02Se h* « e P

Figure 4.14 shows the pressure distribution and film thickness along the

two axes of the ellipse for the case when e = 1.606. P

In pursuing the regression analysis for W*, the other s

independent groups are held constant at:

V* = 0 . 1 8 x 1 0 - 6 , U* = 0 . 3 1 6 x 1 0 "1 1

, G* = 7 1 4 6 , 8 = tt/2, e = 3 . 5 6 P

The regressed exponential form is found to be:

-123W*

h* « e 6

m

-128W* h* * e s

where W* is negative. s

Load is found to have the least effect on the film thickness

and is the most important parameter in determining the profile. To

pursue the regression analysis for the case of varying load, after assuming

a power-law form, the other independent parameters are held fixed

(U* = 0.105 x l O "1 0

, G* = 7146, E = TT/2, W* = 0, e = 3.56) and the results s p

are:

h* « w*-°-0 4 5

m

h* cc &P-004 o

- 170 -

Figure 4.15 shows the pressure distribution and the film thickness along

the two axes of the ellipse for.a particular value of W* used.

After assuming a power-law form for material parameters and

holding the other independent groups fixed at:

W* = 0.18 x 10"6

, U* = 0.105 x 1 0 ~1 0

, 0 = tt/2, W* = 0, e = 3.56 ' s p

The following regressed law is obtained by varying G*:

ht « G* m 0.463

h* « G* o

0.439

The multiplying constants for the regression equations is

found by computing the film thickness from chosen values of all the groups,

then raising them to their appropriate indices, or putting them in

exponential form where appropriate, and then regressing the product

against the film thickness.

4.7 TESTING THE REGIME OF THE OBTAINED NUMERICAL RESULTS

The transition from an undeformed film shape to the heavily loaded

situation has been categorised in reference (16) as a regime in which the

elastic deformation of the surfaces is of about the same magnitude as the

film thickness. As the condition of full EHL approaches, the ratio of

elastic distortion to the film thickness increases.

In most of the results presented, care was taken for the values of

deflections in the central parts of the contact to be reasonably larger

than the film thickness (by a factor varying approximately between 2 and 6)

to position the results in the elastic range.

-

LZL -

- ZZL -

- 173 -

However, in the numerical results used in performing the regression

analysis, very occasionally the values of elastic deflections and film

thickness came to be almost equal. Remembering that the more it is EHL,

the more expensive it is to achieve the convergence for the solution.

4.8 COMPARISON WITH OTHER WORK

4.8.1 Relation Between the Groups Used Here and the Groups Used in

Other Work

Er

= 2 (4.18)

((l-vfi/Ej) + ((1-v2

)/E2.)

where E' is the reduced Young's modulus. Let:

Ejl-v2

)

E = — (4.19)

V W c i - v

2

; K> = — (4.20)

TT E2 Substitute (4.20) and (4.19) into (4,18):

E ' = h r - 4 (4.21)

it (1 + (1/E)) K

If the materials are the same, then E = 1 and (4.21) becomes:

= ^ r (4.21.)

Let the symbol with the heading = on top represent the groups used by

others, i.e.

u* = u ( £ ; (4.22)

it (1 + (1/E))

- 174 -

If materials are the same:

U* = - (4.22a) TT

W* = W ( ) (4.23) ir (1 + (1/E))

If materials are the same:

W* = £ (4.23a)

G* = G/( - ) (4.24) 7T (l + Cl/E,'))

and, finally, is the materials are the same

G* = • G IT (4.24a)

4.8.2 Other Formulae

Converting the Archard & Cowking (12) formula for the central

film thickness to the symbols used here, making materials the same and

substituting the geometry of the cone and sphere {R2 = 0.046 m,

R0 = 0.053 m , & - 3.56), in which many numerical data are available here

a p

for.

If the velocity vector is along the long axis of the ellipse:

h* = 0.468U*0

'7 4

G*°-7 4

U*-°-0 7 4

and if along the short axis:

- 175 -

h* = 1.67V0

'7 4

G*0

-7 4

W*-°-0 7 4

o

Similarly, Cheng's formula (13) for the central film thickness

in terms of the symbols used here, making the materials the same and the

geometry for a long ellipse with ellipticity ratio of nearly 4 (11).

If the velocity vector is along the long axis of the ellipse:

h* = 0.447V*0

-6 3 9

G*°-6 3 9

W*'0

-0 2 8

o

and if along the short axis:

h* = 1.33V*0

'7 3 8

a*0

'7 3 9

W*-°'°7

o

Converting Hamrock & Dowson's formula (3) for the central and

minimum film thicknesses to the symbols used here, making the materials

the same.

If the velocity vector is along the short axis of the ellipse

h* - M M * * )0

' " (G*/*)0

'4 9

( W H f0

'0 7 3

(l-e'0

'6 8

*?)

h* = 2.69(U*ir)°'6 7

(G*/*)0

'5 3

(W*irf0

'0 6 7

(1 -0.61e

4.8.3 Results of Comparison

Figure 4.16a shows the variation of central film thickness

with speed for e = 90°. Some results using approximate methods are shown

for comparison (3,12,13). Figure 4.16b shows the comparison with

references (12) and (13) for when 9 = 0° .

-3 1 0

- 176 -

Numerical Solution Hamrockand Dowson Cheng Archard and Cowking

o A

-5

-12 10

VARIATION OF CENTRAL FILM TH1CK NE SS WIT H SPEED FOR 0-90°

FIGURE 4.16a

- 177 -

N u m e r i c a l S o l u t i o n

Cheng Archard and Cowking

-11

U

VARIATION OF CENTRAL FILM THICKNESS WITH

1C

SPEED FOR 6 = 0

FIGURE 4.16a

- 178 -

U VARIATION OF MINIMUM FILM THtCKNESS WITH SPEED

FIGURE 4.16c

- 179 -

U Variation of minimum fi lm thickness with speed for

different angles* of ve loc i ty vector

FIGURE 4.16a

- 180 -

. U

Variation of central f i lm thickness with speed for

d i f ferent angles of velocity vector

FIGURE 4.16a

- 181 --3

- 3

- 1 8 2 -

FIGURE 4.18: COMPARISON WITH EXPERIMENTAL RESULTS OF REFERENCE (11)

- 183 -


- 184 -

- 186 -

—3

(e)


- 187 -

Hamrock & Dowson's numerical method graph for e = 90° is

closest to the numerical results obtained in this chapter, although, in

common with the other results, it has a greater slope. The differences

in h* are, however, not large. There is a similarity with Cheng's

results in that in both, graphs reduce in slope between e = 90° and

9 = 03

. Figure 4.16c shows the variation of minimum film thickness with

•speed. Hamrock & Dowson's graph (3) for when e = 90° is shown for

comparison and the agreement is quite good.

The more important minimum film thickness results are given in

Figure 4.17a for 0 = 09

to 9 = 90°. Note that there is a progressive

fall in film thickness at any speed as the velocity vector direction

changes from 9 = 90° to 9 = 0°. Figure 4.17b shows the same phenomenon

for the central film thickness. One conclusion we can therefore make is

that minimum films between pairs of circular arc gear teeth or between

rollers and the outer rib of a roller bearing (0 = 03

) are generally lower

than those in, for example, a ball to race contact or at the inner race

rib of a roller bearing, where 9 = 90°.

Figures 4.18a,b,c,d,e,f and 4.19a,b,c,d,e show the comparisons-

of minimum and central film thicknesses against speed with Thorp's

experiments (11) (fully flooded) for various angles of velocity vector.

The agreement is quite good and sometimes it is excellent.

4.9 EHL UNDER PURE SPIN

4.9.1 Introduction to Spin

The lubrication mechanism when one of the bodies is under pure

spin only is mainly of academic interest. In angular contact or thrust

ball bearings, there may be a spin component about the z axis, in addition

to rotation about the y axis due to gyroscopic effects, whilst the elements

roll about the x axis (the notation used here).

- 188 -

Snidle & Archard (14) have shown that a coherent oil film is

possible in a contact between a spinning ball and a straight circular

groove with no distortion anywhere. They also suggest'that even under

EHL conditions, an oil film is possible with the load supported by the

wedge forming areas, these being hydrodynamically improved by elastic

distortion.

Gohar & Thorp (15) showed by experiment that a film due to

pure spin can indeed form under light loads and moderate rotational speeds,

and with some elast'ic distortion present.

4.9.2 Theory of Spin

Equation (4.1) can be easily modified to account for spin in

addition to rolling. A point on one of the bodies at a distance r from

the dry contact centre will have a tangential velocity due to spin of rn^

and the corresponding point on the other body velocity ru0. If u and 2 S

v are the components of velocity in Reynolds equation due to spin only, s

then from Figure 4.20 (positive quadrant):

u = - ft y s **o

where: n = — -

Hence, neglecting the squeeze term, the right hand side of equation (4.1)

can be written as:

COORDINATES FOR SPIN SPEED

FIGURE 4.3a

- 190 -

12 '(u-Qy ) Si&L + (v + Six ) ^ h i 9a: a By

12 ~u *

v

8

W 9a: By

+ 12 '.ay m i + a x m l a

o 9a: o By

where the signs of x and y must be taken into account, depending on the

quadrant.

After non-dimensionalising, the right hand side of equation

(4.7) can be written as:

12U* cos 6 ^ h L +

sin e

9a: By -1

+ 12a ' - y M + x i < M C

Bx ° ty

(4.25)

where: a = a K* r\

Reynolds equation now has an extra variable term accounting

for the spin which creates variable velocity components. There was no

problem in solving the Reynolds and elasticity equations simultaneously at

light loads, but as load and hence deformation increased, no compatible

solution between the elasticity and Reynolds equations could be found.

The reason for this is that there are convergence problems at high loads,

or low speeds, or both. Under pure spin, the speed is very low indeed

because it is proportional to radius vector within the region of pressure,

Indeed, there can be no contribution to pressure at the centre of contact.

The only way a solution could be found was to make the spin speed

unnaturally high.

4.9.3 Results of Spin

Figures 4.21a and 4.21b show isobars and contours for a ball

spinning at a high speed and rolling at a very low speed in a straight

- 191 -

stationary circular groove. The geometry chosen is the same as in

reference (15). The contours for pure spin found by experiment

(Figure 4.21c) are shown for comparison with Figure 4.21b. Although the

experimental spin speed is much lower, the similarity between Figures

4.21b and 4.21c is shown by the dashed lines, showing cavitation

commencement (p = zp/%x = 3P/3y = 0), Note that there is very little

change in distortion from the elastostatic footprint. The regions of

pressure are clearly shown in Figure 4.21a with the spin velocity anti-

clockwise, there are pressure nests in the second and fourth quadrants,

with the zero pressure lines in the first and third quadrants (Figure

4.21b). A similar pattern for the cavitation zones occurs in Figure

4.21c. Observe in Figure 4.21a that the pressure forms a saddle around

the centre of contact, where surface speeds are negligible.

Figure 4.21d is for pictorial purposes only and portrays the

three-dimensional pressure distribution corresponding to the isobar plot

of Figure 4.21a. Again, as Figure 4.9, the plot is a matrix map and

therefore the spacings of the grid points in the x and y directions are

scaled to be equal. The two mountains of pressure can clearly be

observed.

Figures 4.22a and 4.22b illustrate the isobars and contours

for a higher spin speed. The dashed lines corresponding to cavitation

commencement are again marked on the contour plot.

4.10 CONCLUSIONS

A converged general solution to the EHL point contact problem has

been obtained with moderate load and material parameters. The values of

oil film thickness are similar to those obtained by other workers when the

rolling velocity vector is along the short axis of the static contact

ellipse. When the velocity vector is along the long axis, there is a

- 192 -

considerable reduction in minimum film thickness. If a downward squeeze

velocity is present in addition, the pressures are much higher than those

resulting from pure rolling, and an upward dent in the film shape is quite

evident under the peak pressure. When spin velocities are dominant,

there are two distinct pressure regions, with the minimum oil film

thickness occurring near the contact centre, it being an order of magnitude

less than when under a pure rolling velocity of similar magnitude.

FIGURE 4.21a: ISOBAR PLOTS WITH SPIN PRESENT

DIMENSIONLESS LOflD^O .056xio G=1072o -10

DIMENSIONLESS SPEED=2-638*io CONTOUR PLOT OF DINENSIONLESS PRESSURE^**))

FIGURE 4.21b: CONTOUR PLOTS AND LINES OF CAVITATION WITH SPIN PRESENT

- 195 -

FIGURE 4.21c: EXPERIMENTAL PHOTOGRAPH FOR THE CONTOURS OF PURE SPIN FROM REFERENCE (11)

PORTRAY OFA TYPICAL 3D PRESSURE DISTRIBUTION FOR THE CASE OFA VERY HIGH SPIN SPEED AND A VERY LOW ROLLING SPEED

FIGURE 4.21d

FIGURE 4.22a: ISOBAR PLOTS WITH SPIN PRESENT

DIMENSIONLESS LORD^O.057*1° G=1072o -10

DIMENSIONLESS SPEED=4 . 396«» CONTOUR PLOT OF DIMENSIONLESS PRESSURE (P-io)

FIGURE^4.22b: CONTOUR PLOTS AND LINES OF CAVITATION WITH SPIN PRESENT

CAVITATION LINES

D I M E N S I O N L E S S L 0 A D = 0 " . 0 5 7 x i o G = 1 0 7 2 o

D I M E N S I O N L E S S S P E E D = 4 . 3 9 6 x 10

C O N T O U R P L O T O F D I M E N S I O N L E S S F I L M T H I C K N E S S ( h / R S )i

oc

- 199 -

4.11 REFERENCES

(1) GRUBIN, A.N.

Investigation of Scientific and Industrial Research,

Book 30, Central Scientific Research Institute for Technology &

Mechanical Engineering, Moscow (1949) 115-116.

(2) DOWSON, D., & HIGGINSON, G.R.

Elastohydrodynamic Lubrication,

Pergamon Press, SI edition (1977).

(3) HAMROCK, B.J., & DOWSON, D.

"Isothermal EHL lubrication of point contacts",

Part 1: ASME, J. Lubn. Tack., Series F, 98 (April 1976) 223-229;

Part 2: ASME, J. Lubn. Tech., Series F, 98 (July 1976) 375-383;

Part 3: ASME, J. Lubn. Tech., Series F, 99 (April 1977) 264-274.

(4) RANGER, A.P., & ETTLES, C.M.M., & CAMERON, A.

"The solution of point contact EHL problem",

Vtwc. R . S e e . LandA346 (1975) 227-244.

(5) HOOKE, C.J.

"The EHL lubrication of heavily loaded point contacts",

J. MecA. Eng. See., 22 (1980) 183-187.

(6) GOHAR, R.

"Oil film thickness and rolling friction in EHL point contact",

ASME, J. Lubn. To.dk., Series F, 93 (1971) 371-382.

(7) MOSTOFI, A., & GOHAR, R.

"Pressure distribution between closely contacting surfaces",

J. Meo/i. Eng. S<U., 22 (1980) 251-259.

(8) TIMOSHENKO, S.P., & GOODIER, J.N.


McGraw-Hill, New York, 3rd edition (1970).

- 200 -

(9) HARTNETT, M.J.

"The analysis of contact stresses in rolling element bearings",

ASME, J. Lubn. Tech., 101 (January 1979) 105-109.

(10) RAHNEJAT, H., & GOHAR, R.

"Design of profiled taper roller bearings",

Tnlbology Int., (December 1979) 269-276.

(11) THORP, N.

"Oil Film Thickness and Friction in Elliptical Contacts",

PhD Thesis, Imperial College, London University (1972).

(12) ARCHARD, J.F., & COWKING, E.W.

"EHL lubrication at point contacts",

PA.oc. I. MejcJh. E., 180 (1965-66) 47-56.

(13) CHENG, H.S.

"A numerical solution to the EHL film thickness in an elliptical

contact",

ASME, J. Lubn. Tack., (January 1970) 155-162.

(14) SNIDLE, R.W., & ARCHARD, J.F.

"Theory of hydrodynamic lubrication for a spinning sphere",

Vfuod. I. Me.dk. E., 184 (1969-70) 839-848.

(15) THORP, N., & GOHAR, R.

"Oil film thickness and shape for a ball sliding in a grooved raceway",

Paper No. 71-lub-18, ASLE Lubrication Conf., 5-7 October 1971,

Pittsburgh, Pennsylvania, USA.

(16) BISWAS, S., & SNIDLE, R.W.

"EHL of spherical surfaces of low elastic modulus",

ASME, J. Lubn. To.dk., Series F, 98 (1976) 524-529.

- 201 -

CHAPTER 5

ELASTOHYDRODYNAMIC LUBRICATION OF FINITE LENGTH CONTACTS

5.1 INTRODUCTION

The EHL of line contacts has a wide engineering application.

Nominal line contacts occur between the races and the rollers in a

cylindrical roller bearing, between a pair of involute gear teeth, and in

the contact between a disc cam and its roller follower. All the elements

in these examples have finite length and are normally radiused slightly

towards their ends in order to overcome the problems of edge stress

concentrations caused by their finite length, and by misalignment. The

external publications presented by the major bearing manufacturers have

traditionally used elastostatic infinitely long line contact Hertzian

theory to predict pressures and stresses in the rolling elements (1).

More recently, with improvement in numerical methods, finite elastostatic

line contacts have been studied; in particular, the edge stress

concentrations and their modification by suitable axial profiling of the

rollers (2,3,4). With a lubricant present, and smooth contiguous

surfaces, EHL occurs under engineering loads. Its presence has been

established theoretically, general refined solutions having been

produced (5,6). These all assume an infinitely long line contact between

the two rolling elements, thus making it a one-dimensional problem

involving the elasticity and Reynolds equations.

Hooke (7) and Gohar & Bahadoran (8) have attempted to study finite

line contacts using modifications to the infinite solution. Dowson &

Hamroch^(9) have recently solved the point contact EHL problem in two

dimensions for moderate material parameters and loads. They obtained

regression formulae for film thickness for a wide range of ellipticity

ratios for the static contact ellipse. They approximate an EHL point

- 202 -

contact to a line contact by increasing the ellipticity ratio to 8. Such

a result would apply to a crcwned roller in a spherical roller bearing,

but would not necessarily simulate a cylindrical roller because of its

end profiling. Experimental evidence of finite EHL contacts is given in

references (8) and (10), where the interference fringe pictures show

clearly that the minimum oil film can occur near the roller ends and not

at the trailing edge of the contact centre. The end closure is shown to

be critically dependent on the local geometry there, it becoming more

severe the less the profiling. In fact, in reference (8), the edge film

is shown to be theoretically zero for the EHL of a finite right cylinder

rolling over an infinite plane. The results presented in this chapter

give a numerical solution to the two-dimensional EHL problem for finite

line contacts.

5.2 THEORY

5.2.1 Reynolds Equation

For zero velocity terms in the x and z directions, Reynolds

equation (from expression (4.1) of Chapter 4) can be written as:

_3_ P h3

BP + JL P h3

BP' 3a; TI 3a;_ + 9 2/ N By_

= 12v 3fp h)

92/ (5.1)

Following exactly the same procedure as in Section 4.2 of Chapter 4, the

final form of the Reynolds equation in the $ form is:

3 3$ I , 3 /— 3$» — (p — ) + — (p — ) 2 L

3a; 3a; 3 y 3 y

= 12U*

4 - rp

•Bx Bx By By •

3(p h) L

By J

(5.2)

where x = x/R, y - y/R, h - h/R, and u* = (vr\ K')/R9 where R is the

- 203 -

radius of the roller. All other parameters have the same meaning as in

the previous chapter.

5.2.2 Irregular Mesh

Since the pressure distribution and the film thickness change

most sharply at the ends of the roller, situated in the profiled regions,

the rectangular elements are constructed in an irregular way over the

computation zone. So that the dimension of the elements changes as an

arithmetic progression in the axial direction of the roller (x direction),

decreasing in magnitude towards the ends, and equally spaced elements in

the direction of rolling (y direction) (Figures 5.1a and 5.1b). Thus,

more elements are filled in the regions where the dependent parameters

change most rapidly.

In order to find the central difference formula for first and

second derivatives of an irregularly spaced function in the x direction:

_ Ax2. . = $ . . - Aa. - . • + $ (5.3a)

x 2 xx v ' _ 'Ax1. .

$ . , . = $ . . + Ax. .<& + — ^ <$ (5.3b) 1+1*3 x ^ xx v ' where $ and $ are the first and second derivatives of the function $ at

X XX

the point respectively, and:

_ Ax . Ax. i . = x. . - x. ~ . , Ax. T •. = —^

_ Ax. \ Ax. . = a?..- x. . , Ax. . = —

Multiplying equations (5.3a) and (5.3b) by Ax1. . and AX2. 7 respectively,

and subtracting the obtained expressions will lead to a relation for the

- 204 -

ROLLER AXIAL PROFILE

[-•—EXTENSION OF THE MESHH INTO THE PROFILED REGtONS ,

U

I m

half ax iat mesh j length '

l f 25 IRREGULAR ELEMENTS

i

CENTRE OF CONTACT - f

• - »

I»'J+1 * • »

• •

R M I ' J T+1U R •

• • • / - •

CO I— Z lii

LlI

•J

CR

1 I

u

DIRECTION OF VELOCITY VECTOR

MESH CONSTRUCTION

FIGURE 5.14b

T

n

-j-

y 1 I Q5 b

\ 7 /

\ i t \ /

V 1 } / 1

} 4 | I / \ w / \

t V

C O M P U T A T I O K I

B O U M D A C V

i u FIGURE 5.1b: MESH CONSTRUCTION

- 206 -

first derivative of

4> .. - . AX2

. 7

. - $ . - . Ax2

. 7

. - AP. .J $ = l*+l*3 1*-1*3 1-1*3 1*3 1**3 i*-l*3 1**3 (5.4) X

Ax. . Ax. * . (Ax. 7

. + Ax. J

1**3 1-1*3 i*-l*3 1**3 Multiplying equations (5.3a) and (5.3b) by Ax. . and Ax.

1 respectively,

1**3 i*—-i*3 and adding the obtained expressions will lead to a relation for the second

derivative of

2(<&.j7 . AX. * . + .Ax. (Ax.

7 . + Ax. .)) $ = 1-1,3 1-1,3 1**3 1**3 i*-l*3 1**3 (55)

^ Ax. . Ax. * . (Ax. * . + Ax. .)

v,3 1-1,3 i*-l*3 1**3 5.2.3 Reynolds Equation in Central Difference Form

Using relations (5.4) and (5.5), equation (5.2) can be written

in finite difference form as:

E. . $ . . + J7. . $ . 7 . + N. . $ . .

7 +

1**3 l*+l*3 1**3 l*—l*3 1**3 1**3+1 S. . $ . . 7 - AP. . <b. . - SO. • = 0 1**3 1**3-1 1**3 1**3 i*,3 (5.6)

where: E . . 1**3 = 7?. . [to2 7 . — + 2p . .

1**3 I i*-l*3 Dz 1**3

Ax. 7

.-

i*-l*3 D Wi*3

N . . 1**3

S. . 1**3

^ V a Aa?« .1 S * . - . S.

+ 2J. . -2*1 1.3 L I'll DZ 1.3 D J

(p. - p . . 7 + 4p . .) 1**3+1 1**3-1 1**3

* • C- p . .

7 + p . . 7 + 4p . .) 1**3+1 1**3-1 1**3

- 207 -

tf. • 1*3 p ., - . Ax? - . - p . - . Ax

2

. . - p . . ( Ax? - . - Ax? J Z+1,3 1-1,3 1-1*3 I*3 1*3 , C

D

h. . Ax1

. , . - h. , . Ax? . - h. . (Ax2

. - . - Ax? J

j 'L—lj 3 1,3 ^-^J + D

i u . Ax1

. * .-h. * . Ax2

. . - H . . (Ax2

. - . - Ax2

. J)2

1

T 1-1*3 1-1*3 1>,3 1,3 I - L ^ , 3 ~O AZ • « P • » 2 I,3 ^,3

DZ

, 2(hj.« -Ax. , ,+h. ~ .AX. .-h. . ( A x . - . + Ax. .)) JFC 1-1*3 I~L*3 1*3 1*3 1-1*3 I>»3

+

I*3 I*3 D

K 3 4&Y

2

(h. - ft. . J2

IhT^.j. . - J ^ t l — +

. h. • .«+ h. . - - 2Ti. . - JFE 1,3+1 1*3-1 1*3 _

<>,3 I>3

4 T?. . (- A^? 7 . + ASS J — +

3 I,3 D

2

— —

, / Ax. - . + Ax. - J . 7z. n ±J& T 1+1,3 . 4 -

•sr AZ . « p » « 1 1

' TT P . . — — — <s y4

2<5y 26y J

Ax. . Ax. - . (Ax. - . + Ax. J FC^J 3

= P . Ax? - . - p . - . Ax? . - p . . (Ax? - . - Ax? J . , - . ox. - . - p . - . AX. . - p . . ox. - . I+L J 3 1-1*3 L-L J J 1>*3 1*3 ^-1*3 1,3

- 208 -

5.2.4 Lubricant Film Thickness

Relation (4.9) of the previous chapter now holds (the

parameters being normalised with respect to the radius of the roller in

this chapter). However, the new profile of the bodies (h ) has to be

substituted for. The variation of h depends on the axial profiling used.

Normally, the roller bearing manufacturers use two main types, these being

dub-off and crown profiling. The aim is to relieve slightly the

geometry towards the roller ends by an amount of the same order as the

elastic deflection.

In the case of the dub-off, a small radius is struck off

the roller cylindrical surface near each end so as to occupy 10% of its

total length (Figure 5.2a). With a crown profile, a circular arc,

having its centre on the z axis, intersects the roller surface near each

end (Figure 5.2b). Normally, this radius is greater than ten times that

of the roller. In both designs, the region of pressure terminates very

close to the commencement of the profiling and, hopefully, does not rise

sharply prior to this.

Thus, for the cylindrical region in both designs (Figure 5.2):

y2 _ y2 h = -sr , or in dimensionless form h = (5.7) g2R g 2 v '

where I = Y/R, and the parabolic approximation is justified because of the

long slender aspect ratio of the elastostatic footprint.

In the profiled region, we have for the dub-off profile

(Figure 5.2a): _

y2 Y g d 2R

e where R is the dimensionless dub-off radius, and x is as shown in

Q tt Figure 5.2a.

- 209 -

AXIAL ROLLER PROFILES

FIGURE 5.2

- 210 -

For the crown profile, we have (Figure 5.2b):

F - 12

h = L . + — 2 - -(5.9)

where R is the dimensionless crown radius, and x is the half flat length o c

as shown in Figure 5.2b.

5.2.5 Elastic Film Shape

The procedure for finding the elastic film shape is exactly

the same as that in Section 4.2.5 of the previous chapter and will not be

repeated here.

5.2.6 Load

The same as Section 4.2.6 of the previous chapter.


To solve the non-linear Reynolds equation simultaneously with the

equations for elasticity, film thickness and lubricant state, the same

procedure as in Section 4.3 of the previous chapter is adopted.

5.3.1 Mesh Construction

For a chosen load and geometry, the elastostatic footprint

dimensions are calculated from the numerical method discussed in Chapter 2.

If the velocity vector u* is along the y axis, then, referring to Figure

5.1b, a rectangular irregular grid, as mentioned in Section 5.2.2, is

constructed over the computation zone, its size being such that it is

about four times the elastostatic footprint half length in the inlet to

the expected EHD pressure distribution and at the outlet to about 1.5 times

that factor. Along the axial length of the roller {x axis) at both sides,

- 211 -

it extends to a fraction of the roller length into the profiled region.

This fraction is varied to obtain the best position of the start of the

pressure curve. These dimensions correspond approximately to 'flooded'

lubrication conditions.

5.3.2 Boundary Conditions

These are the same as Section 4.3.2 of the previous chapter.

5.3.3 Initial Condition

This is the same as Section 4.3.3 of the previous chapter.

5.3.4 The Algorithm Used to Solve the Equations

The procedure of solving the equations is the same as in

Section 4.3.4 of the previous chapter and will not be repeated here.

5.3.5 Convergence Criterion and Number of Elements Used

The convergence criterion employed is the same as in Section

4.3.5 of the previous chapter.

25 irregular elements in the axial direction (x) and 29

regularly spaced elements in the direction of rolling (y) are constructed

over the computation zone.

5.3.6 Relaxation Factors

These are the same as Section 4.3.6 of the previous chapter.

5.4 NON-DIMENSIONAL. GROUPS

Dimensionless film thickness, H = H/R

Dimensionless central film thickness, h* = h ^ R

- 212 -

Dimensionless minimum film thickness in the central section, h* = h^R

Dimensionless end closure film thickness, h* = h^/R

The independent variable parameters which influence the film

thickness can be put into the following dimensionless groups:

u* =

w* =

R W K

r

R2

G* = 1L u

Kr

The geometry groups are:

_ R __ X R = — , x — , for the crown profiled roller ° R Q R

and:

R -

R for the dub-off profiled roller R f R

5.5 END CLOSURE

The well known feature of end closures was quite evident 1n all the

computational results; the rollers with small blended radii showed more

severe end closures than those with large blended radii, an observation

also made by Wymer & Cameron (10). Figure 5.3 illustrates this

phenomenon which shows the variation of side constriction film thickness

with the degree of blending for a dub-off profiled roller. It was also

observed from the results that different end blendings did not have a

- 213 -

significant influence on the central and exit film thicknesses. Where

the end constriction is the most severe, the reduction in film thickness

occurs over a very small area at the extreme ends of the contact.

Adjacent to this area along the length of the roller, the film thickness

achieves its constant central value. Figure 5.4 shows the plot of

dimensionless film thickness along the length of the roller at the outlet

section for a dub-off profiled (R = 1 . 5 ) and a crown profiled (R = 1 0 0 )

roller subject to the same external conditions. It is clear that the end

closure is more severe for the dub-off design.

Figure 5.5 shows a logarithmic plot of the film thickness against

speed for a blended dub-off profiled roller (R = 1.5). On the figure

is plotted central, minimum exit and side constriction film thicknesses.

The severity of end closures is evident. The behaviour of the side

constriction in comparison with the exit constriction 1s illustrated in

Figure 5.6, which shows the plot of their ratios for the blended dub-off

profiled roller. Since decreasing the speed makes the conditions more

EHL, one can judge from Figure 5.6 that the thinning of the film at the

sides depends upon the severity of the EHL conditions.

5.6 OIL FILM THICKNESS AND PRESSURE DISTRIBUTION

Figures 5.7a and 5.7b give contours and isobars for a crown blended

profiled roller ( r = 100, x = 0 . 7 ) , subject to a given external

o o condition. In this and all contour plots to be presented, the + symbol

indicates the centre of the roller and because of the symmetry, only half

of the plot is shown. Also, in order to make the plots presentable, the

distances along the length of the roller have been divided by a factor of

10. The well known dog-bone feature of shape reported in experimental

investigations (shown in Figure 5.7c), which are the experimental

photographs taken from Wymer's thesis (11), is clearly observed with

- 214 -

islands of film closures being formed at the ends. Figures 5.8a and

5.8b show the contour plots and isobars for a slightly lower material

group. It can be seen that there is hardly any change in the shape of

the plots.

For the sake of clarity, Figure 5.9 shows different sections of the

computation zone in which the results of pressure distribution and film

thickness will in future be referred to. The other existing theoretical

results, which are for an infinitely long roller, are given by section

1-1' (a middle cut). The end closures in the axial direction are shown

by sections 2-2' at the inlet, 3-3' on the central section, and 4-4' on

the section through the minimum exit film. 5-5' is a section through the

maximum closure in the direction of the velocity vector.

Pressure distribution and film thickness along the middle cut

(section 1-1') are shown in Figure 5.10 for a crown profiled roller.

Figures 5.11a and 5.11b illustrate the pressure distribution along the

sections 3-3' and 4-4', respectively. On the same figures, the elasto-

static pressure distribution is shown by a broken line. At inlet, except

for the end pressure spikes, the axial gradient of pressure is negligible

near the central section; this gradient increases, however, at outlet

sections. Figures 5.11 illustrate this clearly, the reason being that

near section 3-3' the film is parallel axially because there is not much

diffusion flow there in comparison with the velocity induced flow. As

the oil advances through the contact into section 4-4', the axial film is

no longer parallel. As the ratio of end constriction to the central

section film thickness drops, the oil in the central regions starts to

feel the extreme end blocked areas, caused by the magnitude of diffusion

side flows increasing so that these are not negligible any more in

•comparison with the velocity induced flows. Thus, the pressure gradient

will build up in the axial direction towards outlet sections. The other

- 215 -

•3 k

.25 b

h -2

y / o

•15 h-

•1 L

.075

variation of minimum side constr ic t ion film thickness

WITH THE DUB OFF RADfUS

(W=-56x l0 5 ; U=-96xlD * G = 10720 , L f /R=1.6)

FIGURE 5 .3

- 216 -

Dub of f ro l le r

T f =1.6 Re = 1-5

. 2 5 -

.211

cn -171 ' S

x

1-c -13-1

.091

•054

•01

Crowned roller

Xc =-7 Rc = 1 0 0

.95 —74 —53 --32 --11 . - j i -32 -53 -74

X / lm

VARIATION OF FILM THICKNESS ALONG THE

LENGTH OF THE ROLLER AT OUTLET SECTION * - 5 * - 1 1 Mr

(W=>5 8x10 9 U =.63x10 , G =• 965 0 )

95

FIGURE 5.4

- 217 -

Minimum exit film (near B fig 5«9) n i

M r A f. ~ rr h m Central film (at A f ig 5-9)

Minimum side constriction film(near C fig 5«9)

h

-A 10

-5 10

10 10 TR

variation of film thickness with speed — C _

(Wd56x10 9 G - 1 0 7 2 0 ; Re=l-5)

10

FIGURE 5.14b

FIGURE 5.6

i i l l i I I I I I 1 1 1 1 — i — i — L

-11 10

" - 5 m — variation of the rati6 with speed ( W ^ 5 6 xiO , G = 1 0 7 2 0 y L

f/ R

=1 6 iRe-1-5)

I ro UD i

D I M E N S I O N L E S S L 0 R D = 5 . 6 4 7 * i o G = 9 6 5 o

D I h E N S I O N L E S S S P E E D = 0 . 6 3 5

C O N T O U R P L O T O F D I M E N S I O N L E S S F I L M T H T C K N F S S f ^ / R * 10

FIGURE 5.7a

DIMENSIONLESS L0flDi=5 .647 *io G=965o DIMENSIONLESS SPEEDS.635*1° CONTOUR PLOT OF DIMENSIONLESS PRESSURE^ \

FIGURE 5.7b

- 221 -

FIGURE 5.7c: EXPERIMENTAL PHOTOGRAPHS FROM REFERENCES (10) AND (11)

X/10

DIMENSIONLESS L0RD=5-537*io G=858o DIMENSIONLESS SPEED=0 .635*iol JSO CONTOUR PLOT OF DIMENSIONLESS FILM THICKNESS(h/Rxjo )

FIGURE 5.8a

ro ro CO

DIMENSIONLESS L0RD=5 . 537 *1° G=858o DIMENSIONLESS SPEED=0 -635*i° feioo CONTOUR PLOT OF DIMENSIONLESS PRESSURE^

FIGURE 6.2: NOVIKOV SIMULATION RIG

- 224 -

5 -

N L E T

S S I O C L O S U I 2 E

OUTL.ET

EXIT CO*4$TlZlCTIOKJ

CENTRAL. E-SGiIOM

2 3 4-

DIFEERENT SECTIONS WHICH THE RESULTS WILL BE REFERRED TO

i FIGURE 5.9

FILM THICKNESS AND DISTRIBUTION IN THE SECTION 1-1 " I N THE OF ROLLING

FIGURE 5.10

PRESSURE CENTRAL DIRECTION

316-267-217-167 -118 -68 -19 -31 -9l 1-3 Y/B

* -316 -267 -217-167-118-68 -19 31 -81 13

- 5 * -11 * Y/B _

ro ro tn

(W=-56xlO > U=-63xlO } G=9650)Xc=-7 , Rc=100)

SECTION ALONG 3-3' EHL -6 ELASTOSTATIC

X/Xc (a long long axis) — 5 —11 5k

(W=S6xlO* U—63xl0 , G-965 0 > Xc/R-.7;jRc=100)

FIGURE 5.11a: PRESSURE DISTRIBUTION ALONG THE AXIAL LENGTH OF THE ROLLER

FIGURE 5.11b: PRESSURE DISTRIBUTION ALONG THE AXIAL LENGTH OF THE ROLLER

PRESSURE

FILM THICKNESS

ALONG SECTION 5-5 \

1 I I I T

-3-66 -316 -2-67-2-17 -1-67 -1-18-68 - 49 - c * -11 * Y/R

(W=-56x 10 U=-63x l0" G=9650jXc/R=-7/ p

Rc =100)

ro ro oo

FIGURE 5.12a: FILM THICKNESS AND PRESSURE DISTRIBUTION IN THE DIRECTION OF ROLLING NEAR TW^ ROLLER ENDS \»

PRESSURE

FILM THICKNESS

ALONG A SECTION

PARALLEL AND

ADJACENT TO

SECTION 5 - 5 /

\ \ \ \ \ \ \ \ \

\

? o - 1 2 <M X

•8-

— 11 tH X

•7- - 9

• 6 -~8 A ny

A- - 6

•3- - 5 \

• 2 - - 4 '

0. .2 i i i i r 1 r

^ -3-66 -316-2-67 -217 -167 -118 - 6 8 -19 -31 - 5 * -41 * v /R

( W = 56 xlO» U=63 x lG , G- 9 65 0 5XC/^ - . 7 , R c =10 0 )

FIGURE 5.12b: FILM THICKNESS AND PRESSURE DISTRIBUTION IN THE DIRECTION OF ROLLING NEAR THE ROLLER ENDS

- 230 -

observation is that the local minimum exit film occurs first in the

central regions and further out in the y direction in the end regions.

Therefore, any section like 4-4' in Figure 5.11b will show the pressure

has already dropped considerably near the roller centre, but is climbing

to a maximum near the roller ends as the side constriction is approached.

Figures 5.11a and 5.11b also show the difference between lubricated

and elastostatic pressure distributions at different sections. The two

pressure curves are quite similar in the central section, but 1n outlet

sections the elastostatic pressure distribution behaves more gently,

having a higher value along the length and much lower spikes at the ends.

This observation is significant in the design of roller bearings which

traditionally have used elastostatics or infinitely long rollers in EHL.

5.7 VARIATION OF FILM AND PRESSURE IN THE DIRECTION OF ROLLING. NEAR THE

ROLLER ENDS

Figure 5.12a shows the plot of dimensionless film thickness and

pressure distribution along the section 5-5' for the crown profiled roller

example. Figure 5.12b illustrates this variation at an inboard section

parallel and adjacent to section 5-5'. Although the pressure spike is not

so significant along the central section, it is quite evident along the

section 5-5'. From analysis of the pressure spikes for different

external conditions, it is found here that the magnitude of the maximum

spike varies between 1.5 to 2.0 times the central point pressure.

The film is found to be flat along this section, dropping to a minimum

at the point of maximum closure just before the surfaces tend to relax

towards their undeformed shapes.

5.8 TESTING THE REGIME OF THE OBTAINED NUMERICAL RESULTS

For the contours, isobars and two-dimensional plots of pressure and

- 232 -

film thickness represented, care was taken to ensure that the values of

deflections in the central parts were larger than the film thickness (by a

factor varying approximately between 2.5 and 4.0) so as to position the

results in the EHL domain. Unfortunately, when the conditions approach

the severe EHL regime, there will be large pressure spikes at the roller

ends making convergence of the numerical method difficult.

The other check on the operating conditions regime is from Figure

5.13 obtained from reference (6) for line contact, which defines different

regimes of lubrication for different loads and speeds when the materials

parameter has a specific value. Again, as in Section 4.8 of the previous

chapter, let the symbols with the heading = on top represent the groups

used by others. If the materials are the same, use relations (4.22a) and

(4.24a) to convert the speed and material properties to the symbols used

here. However, the load used in Figure 5.13 is load per unit length,

i.e. if the materials are the same:

W* = 2 1 (5.10)

IT R x ' Although the value of G* used here is a little more than two thirds of the

value used by (6) in their contour plots (Figure 5.13), the check is still

valid. For the operating conditions which are used here to represent the

results, a cross is shown on Figure 5.13. As illustrated, this is in

the elastic regime. However, the operating conditions concerning the end

of the curves of Figures 5.5 and 5.6 move into the intermediate region, as

shown by a box in Figure 5.13 for the furthest move into that zone.

5.9 COMPARISON WITH OTHER WORK

5.9.1 Other Formulae

The Dowson & Higginson (6) formula for minimum film thickness

- 233 -

is:

H* = L.SU0

-7

G0

-6

W-°-1 3

m In terms of the variables used here:

H* = 1.69V*0

'7

G*0

'6

W*~0

'1 2 m

Wymer & Cameron's experimentally based formulae in terms of the symbols

used here are:

H* = 0.44V* G* W* 0

H* = 1.72V*0

-7 1

G*°'3 7

W*~°-1 3 m

5.9.2 Results of Comparison

Figure 5.14a shows the variation of exit minimum fiTni thickness

with speed from different approximate methods and experiments (6,9,10).

In order to make the Hamrock & Dowson (9) formula applicable to the line

contact, an ellipticity ratio of 8 is substituted in their expression as

they suggested. Figure 5.14b shows the variation of central film

thickness with speed compared with experimental formula (10).

As shown for the exit minimum film, the agreement is closest

to the Dowson & Higginson (6) formula. Although Wymer & Cameron's (10)

experimental formula predicts a low value for exit film in comparison with

other works, which may be due to the tolerance put on their indices, the

agreement is very good with their central film thickness relation.

- 234 -

-3 10

— NUMERICAL SOLUTION — Dowson and Higginson

— Hamrock and Dowson (EP=8) — Wymer and Cameron

*

-5 1 0 I I I I ' ' ' ' » i i i i i i i i

-12 - 1 1 - 1 0

1 0 * 1 0 1 0

U Variation of minimum exit film thickness with speed

fc — 5 ^ (W=-56xl0 > G =10720)

FIGURE 5.14a

- 235 -

- 3 10

h o

- 4 1 0

-t5 10

—

— - NUMERICAL SOLUTION

—

WYMER AND CAMERON

y yy yy

— yy

—

—

1 1 1 1 1 1 1 1 I I I I I I I I -12 10 *

U

-11. 10

-1C 10

Varia-tion of central f i lm thickness with speed

(W=.56xl0 3 G= 1072 0)

FIGURE 5.14b

- 236 -

5.10 DISCUSSION AND CONCLUSIONS

The theoretical studies of line contact presented here have enabled

a number of features to be found that were either not suspected or not

shown in theory. The other existing theoretical solutions give profiles

and pressure distributions for an infinitely long roller on the middle

section only and their main aim has been the determination of minimum and

central film thicknesses along that section only. They have ignored both

the end closure effects and the axial pressure gradient at the outlet

sections and, indeed, their assumption is totally justified, since it is

found that these phenomena have virtually no effect on the film thickness

along the central part. However, the designer of roller bearings is

concerned with the maximum stresses which occur with the severe closures

at the ends and the other concern is the value of minimum film thickness

in the whole of the zone. Thus, the most practical aspect of the theory

developed here is to show how the film and pressure behave in the

restricted end closure zones.

In the computed results, pressure spikes are very moderate and often

not noticeable on the central section. However, the maximum EHL pressures

occur near the start of profiling and their magnitude is found to be

nearly 1.5 times the central maximum pressure.

One of the findings of the work presented here is to show the build-

up of axial pressure gradient which must be due to the development of

diffusion side flows as the oil advances through the contact, and the

other finding is to show how external parameters such as degree of blending

and speed affect the degree of severity of end closures.

Finally, computed contour and isobar plots are reported for the

first time. The predictions of film thickness compare favourably with

experiments which use the optical interference method, as well as with

other theoretical results for an infinite line contact or an ellipse with

a slender aspect ratio.

- 237 -

5.11 REFERENCES

(1) HARRIS, . .

Roller Bearing Analysis,

Wiley (1966).



loaded rollers",

J. Mec/t. Eng. SCA.. , 21_ (1979) 381-388.

(3) HARTNETT, M.J.

"The analysis of contact stress in rolling element bearings",

ASME, J. Lubn. Tech., 101 (January 1979) 105-109.

(4) KANNEL, J.W.

"Comparison between predicted and measured axial pressure distribution

between cylinders",

T/Luyu. ASME, (July 1974) 508-514.

(5) GRUBIN, A.N.

Investigation of Scientific and Industrial Research,

Book 30, Central Scientific Research Institute for Technology &

Mechanical Engineering, Moscow (1949) 115-116.

(6) DOWSON, D., & HIGGINSQN, G.R.

Elastohydrodynami c Lubri cati on,

Pergamon Press, SI edition (1977).

(7) H00KE, C.J.

"The EHL of heavily loaded contacts",

J. MecA. Eng. SU., (1977) 149-156.

(8) BAHAD0RAN, H., & GOHAR, R.

"End closure in EHL line contact",

J. Medi. Eng. ScZ., 16 (1974)

- 238 -

HAMROCK, B.J., & DOWSON, D.

"Isothermal EHL lubrication of point contacts",

Part 1: ASME, J. Lubn. Tec/1., Series F, 98 (April 1976) 223-229;

Part 2: ASME, J. Lubn. Tech., Series F, 98 (July 1976) 375-383;

Part 3: ASME, J. Lubn. Tech., Series F, 99 (April 1977) 264-274.

WYMER, D.G., & CAMERON, A.

"EHL lubrication of a line contact. Part 1: Optical analysis of

roller bearing",

VHJOc. I. Mec/i. E., 188 (1973-74) Paper No. 18.

WYMER, D.G.

"EHL of a Rolling Line Contact",

PhD Thesis, Imperial College, London University (1972).

- 239 -

CHAPTER 6

SUMMARY OF CONCLUSIONS AND

SUGGESTIONS FOR FURTHER WORK

6.1 SUMMARY OF CONCLUSIONS

For the elastic contact of a long roller and cylindrical hole in an

infinite body which are highly conformal, the pressure distribution,

contact arc and deflections are determined. It is shown that the contact

arc and approach of the bodies depend on the degree of conformity.

Although Hertz's theory determines arc and total approach reasonably

accurately, even when the assumption of small contact dimensions has been

violated, it considerably underestimates the peak pressure. The case of

circular arc gears prompted the study outlined in Chapter 1, that whether

for the degree of conformity used in these gears, it is justified to use

Hertz's theory, it is shown that Hertz's theory for these gears is quite

accurate.

A method for finding the three-dimensional elastostatic pressure

distribution and the contact area shape in frictionless concentrated

contact problems between two elastic bodies of any arbitrary profiles is

developed, which is very useful when dealing with the EHL case.

When using singularity elements developed in Chapter 3, which utilise

the order of singularity in describing the pressure curve, very accurate

solutions of elastostatic problems, when discontinuities are present, can

be obtained.

A converged solution to the EHL point contact problem has been

obtained in Chapter 4 with moderate load and material parameters which has

lead to the following formulae for minimum and central film thicknesses:

z,* N -Jtor7*'0'045 TJ

JO.562+0.04Q) -+0.463 1.571Q ~1 2 3 W

S °-0 7 6 E

P N* = 0.112W* U* G * E E E 777

- 240 -

F\ MITT*0

-0 0 4

„JO.484+0.0686) N

*0.439 2.1336 ~1 2 8 W

*S 0.02SE

H* = 0.073W* U* G* E E E C o

where e is exponential, e is in radians, and w* is negative. When the s rolling velocity vector is along the long axis of the static contact

ellipse, there is a considerable reduction in minimum film thickness.

If a downward, squeeze velocity w* is present in addition, the pressures

are much higher than those resulting from pure rolling, and an upward dent

in the film shape is quite evident under the peak pressure. When spin

velocities are dominant, there are two distinct pressure regions, with the

minimum oil film thickness occurring near the contact centre, it being an

order of magnitude less than when under a pure rolling velocity of similar

magnitude.

The EHL of finite length rollers under flooded conditions has shown

that the minimum oil film thickness and maximum pressure occur near the

roller ends. The film thickness there can be influenced by the choice

of roller end blend geometry, whilst the maximum pressures can exceed

those estimated using elastostatic theory only.

6.2 SUGGESTIONS FOR FURTHER WORK

6.2.1 Theoretical

Chapter 1: The analysis dealt with in Chapter 1 can also be

used to find the pressure distribution and contact shape for a sphere

indenting inside a conformal spherical cavity. Once this is done, the

influence coefficients for deflection can be calculated. The elasticity

part can then be solved simultaneously with Reynolds equation in polar

coordinates to examine the effect of squeeze on film thickness and pressure

distribution. The application of this work will be in biomechanics

dealing with artificial and actual human hip joints.

- 241 -

Also, the stress equations derived in Chapter 1 for the

circular hole and the stress equations in reference (1) for the roller can

be used to find the maximum shear stresses in the components corresponding

to the pressure distribution under conformal contacts. Isobars of

principal stress differences can then be plotted in the two components and

compared with the isobars derived from the Hertzian distribution of

pressure.

Chapter 2: The program developed in Chapter 2 is very general

and knowing the profiles of the contacting bodies, the elastostatic

pressure distribution can then be obtained. Thus, the program can be

applied to rough surfaces and misalignments if desired. It can also be

used to design the optimum profile shapes to minimise the stress

concentrations.

Chapter 3: The accuracy of the singularity element, introduced

in Chapter 3 for the case of the logarithmic type, can be tested by

applying it to the model of a sharp angled wedge indenting an elastic

half space, in which the exact theory for it is given in reference (2).

Chapter 4: The program developed here can be used to analyse

the EHL of rough surfaces by superimposing the shape of the roughness on

the undeformed smooth profile.

The regressed formula for minimum film thickness, which takes

the normal speed of approach into account, can be used to study the

dynamic behaviour of the oil when subjected to a periodic forcing function.

Although some interferons try experiments (3) show that the

regression formula obtained from numerical methods can be extrapolated

With reasonable confidence to the high load, steel and steel region, it is

- 242 -

still desirable to find a mathematical solution for the heavily loaded

cases. Dowson & Higginson (4) made their classic contribution to the

theory of heavily loaded line contacts by the inverse solution of the

Reynolds equation. Whereas the forward iteration method of solution

fails at relatively high loads, the use of the inverse method might enable

very heavily loaded cases to be solved. Apparently, the application of

the inverse solution to the point contact problems is in progress at

Cardiff University, which overcomes the convergence problems associated

with numerical methods.

Alternatively, the work of Rohde & Dh (5), who reported a

novel method of solving the EHL equations by using a higher order element,

together with Newton's method, can be extended to point, contacts. This

method has already been used by Ruskell (6) for the case of rectangular

rubber seals.

Chapter 5: By using the relations for the flow in the x and

y directions, one can compute the angle of flow for each element in the

mesh. Then, for the case of finite length contacts, 1t would be possible

to compare the axial with side flow at different sections of the EHL zone.

One of the parameters which has an influence upon the side

constriction minimum film thickness is the curvature of the profiled

region. By defining a mathematical expression which shows the curvature,

one can find a relation between the end closure film and this curvature by

simply substituting different profiles into the program and regressing the

end closure film thickness against this curvature in an appropriate form.

Repeating the same analysis for speed, load, material properties and

squeeze velocity, a general formula can then be obtained for side

constriction minimum film thickness which will be very useful for design

purposes.

- 243 -

6.2.2 Experimental Further Study

The theory developed in this thesis also applies to the contact

between dome ended rollers and their ribs in roller bearings, as well as

between a pair of circular arc (Novikov) gears. With dome ended rollers,

the elastostatic contact with the conical race is an ellipse. If oil is

present and EHL conditions obtain, oil flow is along the ellipse long axis

on the outer race and along its short axis on the inner race. A dome

ended roller, with cross-section of its race, is shown in Figure 6.1. Oil

flow at about 15° from the long axis also occurs between the contact of a

pair of helical circular arc (Novikov) gears.

The main objective of the experimental research proposal is to

study experimentally the EHL mechanism in an elliptical contact with the

main oil flow along the long axis.

In addition to the film thickness, a further experimentally

unknown factor is the exact distribution of both axial load on the rollers

under combined loading and the pressure in the circular arc gears.

Experiments carried out by Korren (7) and at SKF (8) suggest that the

rollers in the radically loaded zone are subjected to the highest axial

loads. This is shown by the recorded temperature rise. Direct

measurement of the load distribution can be made by employing Manganin

gauges developed at Imperial College by Professor O.C. Anderson. In

addition, these gauges, when specially adapted, can measure discrete

pressure, temperature and film thickness in an EHL contact. They can be

placed at chosen points on one of the conical surfaces.

For further experimental confirmation of the theory developed

in this thesis, an experimental rig has been designed by Westland

Helicopters and later on redesigned by Mr P.M. Saunders of the Mechanical

Engineering Department at Imperial College (Figure 6.2). It basically

comprises a steel sphere contacting the inside of a glass cone or a

- 244 -

sapphire window fitted into a steel cone (which, as described in Section

2.2 of Chapter 2 and shown in Figure 2.1, simulates the Novikov gears and

the dome ended rollers). Both members are independently rotated in the

presence of a lubricant, their speeds being adjusted to fix the slide/roll

ratio. The angle between their respective spin axes can also be changed

in order to simulate a Novikov gear or slewed roller. Direct comparison

between the experimental and theoretical predictions of oil film thickness

and pressure distribution for the oil flow along or skewed to the long

axis of an EHL contact can then be made, using optical interference

techniques and Manganin gauges.

- 245 -

DOME ENDED ROLLER WITH CROSS SECTIONS OF ITS RACES

FIGURE 6.1

FIGURE 6 .2 : NOVIKOV SIMULATION RIG

- 247 -

6.3 REFERENCES

(1) MUSKHELISHVILI, N.I.

Some Basic Problems of the Mathematical Theory of Elasticity,

N.V. Noordhof (1953).

(2) SNEDDON, I.N.

Fourier Transforms,

McGraw-Hill (1951).

(3) GOHAR, R.

"Oil film thickness and rolling friction in EHL point contact",

ASHE, J. Lubn. Tack., Series F , 93 (1971) 371-382.

(4) DOWSON, D., & HIGGINSON, G.R.

"A numerical solution to the EHL problem",

J. Hadi. Eng. Scl., 1_ (1959) 6-15.

(5) ROHDE, S.M., & OH, K.P.

"A unified treatment of thick and thin film EHL problems by using

higher order element methods",

Vhjod. R. Sac. Lond., A343 (1975) 315.

(6) RUSKELL, L.E.C.

"A rapidly converging theoretical solution of the EHL problem for

rectangular rubber seals",

J . Hock. Eng. SU., 22 (1980) 9-16.

(7) KORREN, H.

ASME Paper 69-Lubn., 5-9 June 1969.

(8) IKQ, 0., & ORTE, S.

"Axial load carrying capacity of cylindrical roller bearings",

Bali BacvUng Journal, (1967) No.-l, 13-20 and 21-26.

oil film thickness and pressur distributioe n in elastohydrodynamic elliptica contactl s · 2016....

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