Isothermal elastohydrodynamic lubrication of point contacts I ...

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Bernard J. Hamrock, Lewis Research Center; and. E-8295. Duncan Dowson .... The reason why a complete, isothermal, point-contact EHL solution has not yet.
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g^S g^Y KIRTLAND AFB ^g S

0

Z

^

>~~

1^1

0^^ ^^B JP

1

(X

Inside the Hertzian contact ellipse the pressure is initially assumed to be Hertzian; that

is,

P

Jl

3F

(X

m)2

(Y

I)2

(64)



2!rabE’ or

0

3FH3

2

2!rabE’

^/l

(X

m)2

(Y

when

(X

16

m)2 + (Y I)2 < 1

I)2

(65)

Relaxation Method

If subscript n is the iterant and relaxation method known as the

is the particular solution to be found, i \

the

be expressed as

Ai’](?>i+l^n ^A 1-1^-1 ^j^i-lj, n+1 Pj.j^i.j-.l. n

L^.

^

(66) \

^^)

where X is an overrelaxation factor which is initially set to 1. 4. Therefore, equation (66) is used starting with node (2, 2) then continuing with (3, 2), until (M, 2); followed by (2, 3), (3, 3), (M, 3); and ending with (2, N), (3, N), (M, N), where

(n + m)d

M N

2Zc

1 1

The relaxation procedure described by equation (66) is continued until

N

M V^ \

Z

/

j=2, 3, where

^

i=2, 3,

l0l^x.j. n+l d>. ^i. j. nl d>.

^

L n+l

^

1

0. 1, initially.

Dimensionless Pressure Loop The relaxation method provides values of cf>_ for every point within the mesh. Having determined cf), we can write the dimensionless pressure as

^J ^iJ^J^2

PH / where P is dimensionless pressure and H is dimensionless film thickness, was introduced in order to help the relaxation process. The analysis is applicable to the complete range of elliptical parameters and is valid for any combination of rolling and sliding within a point contact. Lewis Research Center, National Aeronautics and Space Administration,

Cleveland, Ohio, May 28, 1975, 505-04.

21

APPENDIX constants defined in eq. (34)

A, B, C A, B, C, D, L, M a

SYMBOLS

^

J

relaxation coefficients

semimajor axis of contact ellipse

a

a/2c

b

semiminor axis of contact ellipse

b

b/2d

c

number of equal divisions of semimajor axis

D

defined by eq.

d

number of equal divisions of semiminor axis

E

modulus of elasticity

E’

(30)

2

.idh ^-i / \

^ ^

S

elliptical integral of second kind

F

normal applied load

F

integrated normal applied load

y

elliptical integral of first kind

G

dimensionless material parameter

H

dimensionless film thickness,

H

dimensionless minimum film thickness, h

H

dimensionless central film thickness, hYR

h

film thickness

hmm

minimum film thickness

h

central film thickness

J

function of k

k

elliptical eccentricity parameter,

P

dimensionless pressure, p/E’

22

h/R

(eq. (9))

a/b

/R

p

pressure

Q

dimensionless mass flow rate,

q

mass flow rate

R

effective radius

r

radius of curvature

U

dimensionless speed parameter,

u

surface velocity in X direction

V

Vu2 + v2

v

surface velocity in Y direction

W

dimensionless load parameter,

w

total elastic deformation

W

total elastic deformation at center of contact

x X X

pT^p E’R

V^VRE’

F/E’R R

’r^ \ y, Y,

coordinate systems defined in report

z

viscosity pressure index, a dimensionless constant

r^

constants defining fluid used (eq. (23)) curvature difference

77

lubricant viscosity

T]

dimensionless viscosity, rj/ri

rj

atmospheric viscosity

6

angle of resultant velocity vectors (eq. (18))

^.

overrelaxation factor

^ v

P/r?

p

lubricant density

p

dimensionless density, p/p

(p

defined by eq. (75)



(i+ i,j)

/

h

0(i,j- l)

\ Figure 5.

Figure 4. Components of film thickness for ellipsoidal solid near plane.

Mesh used in numerical analysis.

0

^1,)

^i-l,j

---------^ --------Y^^^ _________________!________

xi-l,j

1/d

Figure 6.

Y

xl,j

xi+l,j

I/d

Parabola and corresponding points representing ill as a function of X.

29

&0

o

-J^- _^_--I p65

^’’"’

Figure 7.

by using eq, (28)

\},n-^ /-l-\ (RETURNJ

0,

\

(^sroTV-^-results Write ’rlte resl"ts J

0^

by using eq. (30)

,-----’------’

by using eq. (69)

Calculate Hj

^^>--

PIJ Inltially" ’(64)___

using eg,

/ No

_______I

^n^U.n+l ---r^

^L ^SrS’t^K/ ~^by

WCMe

__________[__________

i--i--i Calculate Q and