Surface roughness effects on externally pressurized bearing performance

Wear, 103

(1985)

SURFACE BEARING

(Received

1

- 10

ROUGHNESS EFFECTS PERFORMANCE

M. F. KHALIL Mechanical Alexandria

1

ON EXTERNALLY

PRESSURIZED

and K. A. EL-SHORBAGY

Engineering (Egypt) February

Department,

6,1984;

accepted

Faculty March

of

Engineering,

Alexandria

University,

8,1985)

Summary The effect of surface roughness on the performance characteristics of externally pressurized bearings is analysed. The pad surface is assumed to have irregularities of sine wave shape. It is shown that surface roughness has a pronounced effect on the operating characteristics of bearings, especially at lower values of the lubricant film thickness and higher values of the wavenumber. The surface roughness always increases frictional power and decreases lubricant flow rate while its effect on bearing load is dependent on the ratio of recess radius to outer radius.

1. Introduction Externally pressurized bearings are finding increasing use in a wide variety of applications, particularly for machine tools, aerospace industries and large pumps used for pumped storage projects [ 11. Externally pressurized bearings have numerous advantages over conventional hydrodynamic and rolling bearings. These include a high load-carrying capacity, high stiffness with low power loss and good reliability [ 1, 21. Most treatments of externally pressurized bearings until now have been limited to the case of perfectly plane smooth surfaces [2 - 51. Owing to machining limitations it is not possible to manufacture bearings with perfectly smooth surfaces. It has long been recognized, however, that surface roughness has a major effect on the characteristics of hydrodynamic bearings [6 - lo]. To the present authors’ knowledge, the only paper which deals with the surface roughness effects in externally pressurized bearings is by El-Kadeem et al. [ 111. In this analysis the surface roughness of a stationary rectangular bearing was represented by a series of rectangular waves. It was concluded that the surface finish has no effect on bearing load although it increases the lubricant flow rate and bearing stiffness. 0043-1648/85/$3.30

@ Elsevier Sequoia/Printed

in The Netherlands

2

In the present paper, a theoretical investigation of the behaviour of externally pressurized bearings is given, with account being taken of the rotational inertia and surface roughness effects.

2. Analysis Figure 1 shows a rotating circular bearing. The coordinate system is fixed in space with the origin located on the lower fixed plate. The equations expressing the conservation of mass and momentum within the lubricating film approximation are as follows: for the conservation of mass,

&r)=O

(1)

for the conservation V2

dp

r -p-=-Z

of momentum a2U (2)

+/-l--a22

and +!$

(3)

The boundary

conditions

2 = 0:

u=v=o

z=h:

lL=O

are

u=CJJl-

The solution of eqns. (2) and (3) with the boundary conditions expressions for the radial and tangential velocities. Thus,

’ dp =-z/Q+

u=--(z

2~ dr

f$(zh3-z4)

(4) (4) yields

(5)

and v = w-z/h

Fig.

1. Externally

(6)

pressurized

bearing.

3

The lubricant flow rate is given by dz

q = ,/kru

which with the use of eqn. (5) becomes ‘=

nrh3 0.3p02r6~ i

dp s

(7) i The film thickness h at any distance r from the origin (Fig. 1) is given

by + fr sin 0)

h = h,(l

(8)

Since the film thickness h is a function of the radial coordinate r, the pressure distribution cannot be obtained by integrating eqn. (7). Since the lubricant flow rate is constant along the flow, i.e.

aq

-t ar

0

eqn. (7) gives

where P = PIP,

R = r/r2

H = h/h2

S = 0.3p02r,fPo

The boundary conditions for pressure are R = 1:

P=O

and

(10) P=l

R=R,:

Solution of the differential equation (9) with the boundary conditions (10) yields the pressure distribution in the lubricating film. Equation (9) is written in finite difference form as -Pi

%, lpi-1

+ ai,2Pi+l= Ui,3

where 1 %l=;

-

+ 3 dH HdR

-

2

1.5R

aj,3=s

i

1+H

dH

do

!

tARI

(11)

4

i = 1 - m is an index defining the value of R in the bearing mesh and m is the number of mesh subdivisions in the r direction. Equation (11) is applied to all points in the pressure field except at the recess edge since the pressure function is not continuous. Applying eqn. (7) before and after the recess edge and keeping in mind that the pressure is single valued at the recess edge, we then obtain Pk-l - (1 + x3)Ph + x~P~+~ = 2SR(x3 - 1) AR

(12)

where X=

h,(l p +

+ eRI sin p) + ER, sin 0)

h,(l

and h is an index defining the value of R,. The pressure distribution is obtained numerically, for a given film thickness shape, by solving eqns. (11) and (12) with the boundary conditions (16). 2.1. Bearing performance characteristics After determining the pressure distribution bearing, the bearing characteristics are obtained. 2.2. Load-carrying capacity The load-carrying capacity through the integration

is obtained

with the lubricant

numerically

film in a

using Simpson’s rule

(13)

2.3. Lubricant

flow rate

Equation (7) is used, in finite difference cant flow rate: f-l=

Ponhz3 6p(m-l)i=2

5

Hi3Ri SR _ Pi ,:-I

2.4. Frictional power The frictional power consumed f=

2npw2rz4 s

h,

0

or F=

’ R3 .I-

0

-

H

dR

form, to determine

’ R3 dR

H

as a result of pad rotation

the lubri-

(14)

is given by (15)

5

2.5, Smooth

surfaces

For smooth surfaces the bearing performance tained analytically as follows:

characteristics

are ob-

(16) Qs= fs =

1 + 0.5S(l

- Ro2)

(17)

ln(R,/R,)/Hi3 - In RI

npw2r24 2h,

l-RR,%

R14HI

Ro4

(18)

2.6. Computational details Computation was carried out on a 48K microcomputer with eight significant decimal digits. Test problems were used to evaluate the numerical method. In the test problems the performance of the smooth bearing (E = 0) is obtained using numerical solutions and compared with the values obtained analytically using eqns. (16) - (18). The calculations were made for certain bearing dimensions (R, = 0.05, RI = 0.3, HI = 5, S = 0 and 1) and for m = 20, 39, 77, 96, 191, 381 and 476. On the basis of the results of such numerical experiments and in order to minimize computation time, it was decided to use m = 191 in all calculations because acceptable accuracy is obtained in a reasonable time. The error in load was 1.7 X 10e4, in flow rate 4.2 X 10V3 and in frictional power 4.4 X 10P5. 3. Discussion of results

The effect of surface roughness E on the load-carrying capacity LR of a rough surface and the ratio LR/Ls of LR to the load-carrying capacity of a smooth surface for fl= 100, r0/r2 = 0.05, r1/r2 = 0.3, 0.5 and 0.7 and values of the speed parameter of 0,0.5 and 1 are given in Table 1 and Fig. 2. These results show that the bearing load L R decreases with increasing bearing roughness for bearings with a small recess radius ratio rl/r2. For higher values of rl/r2, the load-carrying capacity increases as the surface roughness increases. The effect of surface roughness e on LR is more significant at higher values of the speed parameter S. It is clear that the effect of surface roughness on bearing load is dependent on the recess radius ratio in a way similar to the effect of centrifugal inertia forces as indicated by Dowson [ 31. The effect of r,/r,, S and surface roughness E on the lubricant flow rate Qa is given in Table 2 and Fig. 3. The method of presentation is the same as that adopted in Table 1. The lubricant flow rate (in dimensionless form) increases as E increases up to 0.2 for bearings with a small recess radius ratio (rl/r2 < 0.3), especially at lower values of the speed parameter. For

6 TABLE

1

Bearing load-carrying

s

capacity

Values of LR and LRILs

E

rl/rz = 0.3

for the following

values of rl/r2

rl/r2 = 0.5

rI/r2 = 0.7

LR

LRILS

LR

LRILS

LRILS

0

0 0.1 0.2 0.3 0.4 0.5 0.6

0.37362 0.37439 0.37457 0.37422 0.37305 0.37045 0.36484

1.00016 1.00170 1.00219 1.00126 0.99612 0.99115 0.97616

0.52852 0.53327 0.53864 0.54430 0.54977 0.55430 0.55657

1.00023 1.00923 1.01939 1.03011 1.04045 1.04903 1.05333

0.68038 0.69172 0.70221 0.71153 0.71956 0.72533 0.72995

1.00035 1.01702 1.03244 1.04622 1.05795 1.06713 1.07324

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6

0.34226 0.34299 0.34319 0.34274 0.34127 0.33600 0.33099

1.00094 1.00297 1.00360 1.00229 0.99796 0.96843 0.96792

0.53600 0.54193 0.54663 0.55569 0.56251 0.56817 0.57100

1.00157 1.01265 1.02516 1.03836 1.05110 1.06167 1.06697

0.72642 0.74066 0.75335 0.76565 0.77571 0.76365 0.73690

1.00229 1.02194 1.04014 1.05642 1.07030 1.06126 1.03350

1

0 0.1 0.2 0.3 0.4 6.5 0.6

0.31074 0.31156 0.31181 0.31126 0.30949 0.30556 0.29713

1.00135 1.00450 1.00530 1.00354 0.99731 0.93515 0.95796

0.54349 0.55059 0.55362 0.56703 0.57526 0.58204 0.58543

1.00237 1.01597 1.03079 1.04641 1.06149 1.07400 1.03026

0.77245 0.78960 0.60549 0.81971 0.83185 0.34146 0.34734

1.00401 1.02630 1.04695 1.06544 1.06122 1.09370 1.10200

0 = 100, rO/rz = 0.05,

HI = 5.

jyk+ 01

0.2 0.3 ROUGHNESS

Fig. 2. Effect of surface o;---,s=o.5;---,S=l.O.

0.4 IE I

0.5

roughness

0.6

on bearing load-carrying

capacity

(0 = 100):

-,s=

7

TABLE2 flow rateQ

Lubricant

s

e

Values

of QR and QR/Q~

r-,/r2 =

0.3

s

for the following rjlrz =

values of rlirz r,lrz

0.5

= 0.7

QR

QRIQs

QR

QRIQs

QR

QRIQs

0

0.82389

0.2 0.3 0.4 0.5 0.6

0.85208 0.84906 0.81989 0.77273 0.71848 0.67105

1.00375 1.03809 1.03442 0.99888 0.94143 0.87533 0.81755

1.41015 1.36208 1.23924 1.05379 0.82458 0.57612 0.33739

1.00342 0.96921 0.88190 0.74984 0.58674 0.40995 0.24007

2.66983

0.1

2.25862 1.83345 1.41166 1.01300 0.65722 0.36290

1.00863 0.85328 0.69265 0.53330 0.38269 0.24829 0.13710

0.5 0 0.1 0.2 0.3 0.4 0.5 0.6

1.02523 1.06118 1.05812 1.02236 0.96413 0.89703 0.83846

0.99974 1.03479 1.03181 0.99694 0.94016 0.87473 0.81761

1.74721 1.68811 1.53549 1.30460 1.01900 0.70935 0.41191

0.99510 0.96144 0.87452 0.74302 0.58036 0.40400 0.23460

3.30642 2.78908 2.25418 1.72340 1.22144 0.77304 0.40147

0.99980 0.84336 0.68162 0.52112 0.36934 0.23375 0.12139

10

1.22657 1.27028 1.26718 1.22484 1.15553 1.07558 1.00588

0.99706 1.03258 1.03007 0.99565 0.93931 0.87432 0.81766

2.08427 2.01413 1.83174 1.55541 1.21342 0.84259 0.48643

0.98956 0.95625 0.86966 0.73847 0.57610 0.40004 0.23094

3.94301 3.31954 2.67490 2.03514 1.42987 0.88886 0.44004

0.99390 0.83675 0.67425 0.51299 0.36042 0.22405 0.11092

0

0.1 0.2 0.3 0.4 0.5 0.6 p = 100,r,/r2

=

0.05,Hl= 5.

4.0

FLOW

I 0.0

I

0.1

I

0.2

I

0.3 RMiCHNESS

0.k

0.5

t61

Fig.3. Effectof surface roughnes on lubricant flow rate@= 100):-,s=();--_--, s = 0.5;- * --,s= 1.0.

8

r,/rz > 0.3, the lubricant flow rate decreases as e increases. The effect of surface roughness is more pronounced at higher values of r,/rz, S and E. The effect of r,/rz and surface roughness on the frictional power is given in Table 3. The frictional power increases with decreasing r,/r, and increasing surface roughness E. These findings are clearly demonstrated in Fig. 4. TABLE Frictional

E

0 0.1 0.2 0.3 0.4 0.5 0.6

3 power Values -

of TR and T&Ts -r1fr2 = 0.3

for the following

values of r1/r2

r1/r2 = 0.5

rl/r2 = 0.7

TR

TdTs

TR

TRITs

TR

TR iTs

0.24841 0.25048 0.25523 0.26312 0.27497 0.29231 0.31808

1.00014 1 a00845 1.02760 1.05935 1.0706 1.17690 1.28062

0.23766 0.23955 0.24397 0.25135 0.26246 0.27875 0.30295

1.00070 1.00863 1.02727 1.05832 1.10511 1.17369 1.27561

0.20152 0.20314 0.20685 0.21299 0.22220 0.23567 0.25565

0.99773 1.00578 1.02415 1.05453 1.10013 1.16681 1.26576

p = 100, r,frz

= 0.05,

Fig. 4. Effect

of surface

H, = 5.

ROUGHNESS

(El

roughness

on frictional

power:

-

P= lOO;---,

,P=

200.

The effect of p or the number ~/2~ of roughness waves along the bearing surface on the bearing performance for r,/r, = 0.05, t-,/r2 = 0.5 and Hi = 5 is presented in Table 4 for E = 0.0 - 0.6. It is clear from the table that, as /3 increases, the roughness effects on the operating characteristics increase.

TABLE Rough

4 surface

bearing

performance

b

f

LR

LR1L.S

QR

QRIQs

TR

TR/Ts

100

0 0.1 0.2 0.3 0.4 0.5 0.6

0.53600 0.54193 0.54863 0.55569 0.56251 0.56817 0.57100

1.00157 1.01265 1.02516 1.03836 1.05110 1.06167 1.06697

1.74721 1.68811 1.53549 1.30460 1.01900 0.70935 0.41191

0.99510 0.96144 0.87452 0.74302 0.58036 0.40400 0.23460

0.23766 0.23955 0.24397 0.25135 0.26246 0.27875 0.30295

1.00070 1.00863 1.02727 1.05832 1.10511 1.17369 1.27561

200

0 0.1 0.2 0.3 0.4 0.5 0.6

0.53600 0.53862 0.54101 0.54243 0.54028 0.51895 0.69599

1.00157 1.00645 1.01093 1.01359 1.00956 0.96971 1.29883

1.74721 1.67530 1.51463 1.26937 0.94473 0.54775 0.45015

0.99510 0.95414 0.86264 0.72296 0.53806 0.31196 0.25764

0.23766 0.23891 0.24264 0.24926 0.25950 0.27476 0.20771

1.00070 1.00595 1.02168 1.04952 1.09264 1.15689 1.25351

4. Conclusions From the results obtained, the following conclusions can be drawn. (1) The roughness effects on bearing operating characteristics are more pronounced at lower values of the lubricant film thickness and large wavenumbers. (2) The surface roughness tends to decrease the load-carrying capacity of bearings with a recess radius ratio rl/rz < 0.3 and to increase the load for bearings with larger rl/rz. (3) Surface roughness reduces the lubricant flow rate and increases the frictional power.

References 1 2 3 4 5 6 7 8 9 10 11

A. Christ and M. Peron, Escherwyss News, 53 (1980) 40. F. A. Abdelhafez, Wear, 63 (1980) 71. D. Dowson, J. Basic Eng., 83 (1961) 227. M. K. Ghosh and B. C. Majumdar, J. Lubr. Technol., 104 (1982) 491. F. H. Rehsteiner and R. H. Cannon, J. Lubr. Technol., 94 (1972) 49. J. W. White,J. Lubr. Technol., 105 (1983) 131. J. W. White, J. Lubr. Technol., 102 (1980) 445. H. G. Elrod, J. Lubr. Technol., I01 (1979) 8. J. L. Teale and A. 0. Lebeck, J. Lubr. Technol., IO2 (1980) 360. K. Tdnder, ASLE Trans., 23 (1980) 326. M. A. El-Kadeem, E. A. Salem and H. R. El Sayed, Int. J. Mach. Tool Des. Res., (1972) 85.

12

10

Appendix f

F h H 1 L p P Q & r, 8,z R” S u, u P 6 E lu P w

A: Nomenclature frictional power fh2/2npdrz4, dimensionless frictional power film thickness h/h*, dimensionless film thickness bearing load-carrying capacity l/Ponrz2, dimensionless bearing loadcarrying capacity lubricant pressure p/P,, dimensionless pressure lubricant flow rate ~/~P~~h~3/6~}, dimensionless lubricant flow rate cylindrical coordinates roughness wavelength r/rZ, dimensionless radius 0.3p2r,/P,, speed factor velocity components in r and 0 directions dimensionless inverse of the sine wave roughness 2~r2/rw, length recess depth ratio of roughness amplitude to minimum bearing clearance lubricant viscosity lubricant density angular velocity of the rotor

Subscripts n nominal 0 inlet supply hole rough bearing surface R S smooth bearing surface recess radius 1 2 outer radius

wave-