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Surface finish effects on the performance of externally pressurized rectangular thrust bearings
Int. J. Math. Tool Des. Res.
Vol. 12, pp. 85-104.
Pergamon Press 1972 Printed in Great Britain
SURFACE FINISH EFFECTS ON THE PERFORMANCE OF EXTERNALLY PRESSURIZED R E C T A N G U L A R THRUST BEARINGS M. A. EL-KADEEM*, E. A. SALEMt and H. R. EL-SAYED~
(Received 22 October 1969) Abstraet--A beating surface may be considered to consist of one or more of the three known main features namely, mis-alignment, waviness and roughness. This study presents~theoretical and experimental investigations of the separate effects of both surface roughness and waviness, of the bearing pad surface, on the bearing performance. The bed surface is assumed to have irregularities of small magnitude to an extent that they can be neglected compared to the pad roughness.
The theoretical finditags are compared with those experimentally measured, and the results are presented in dimensionless groups. NOMENCLATURE
A E, Et
Projected ffrea of the bearing Bearing width Power consumption, total
F
~h Stiffness per unit supply pressure, - -
G
Shape factor
H
Stiffness per unit power, EdPs
B
PsA
3,/zll
L L o a d per unit area, pi A L L o a d per unit flow, - -
J
Qt ~-
L N P, P~, Ps
Q R
Ra Rmax a b d
l Total load carrying capacity Wave order Pressure, Inlet, Supply Flow rate Predominant m a x i m u m roughness height Centre line average height M a x i m u m peak-to-valley height o f surface irregularities Wave length Wave amplitude Capillary inside diameter
* Head of Production Engineering Department, Alexandria University, U.A.R. t Lecturer in Mechanical Engineering Department, Alexandria University, U.A.R. Assistant Lecturer in Production Engineering Department, Alexandria University, U.A.R. 85
86 h l I" m u, v, w x, y, z zl /~
M. A.
EL-KADF.EM, E.
A. SALEMand H. R. EL-SAYED
Film thickness Bearing length Capillary length Number of waves Linear velocity components Coordinates, rectangular External resistance Viscosity, absolute Stiffness
INTRODUCTION ON DEALINGwith the effects of the bearing surface finish on the performance of externally pressurized thrust bearings, little information exists. On the other hand, some attention is paid to the study of the effects of surface finish on the performance of hydrodynamic thrust bearings. Salama Ill presented a study of the effects of the bearing surface finish on the performance of circular thrust bearings. Fuller [2] explained the effects of surface finish on the performance of artificially roughened bearings. Boltz [3] reported on slide ways for a turret lathe bed which was constructed with a cross-directional scratch pattern. It is the purpose of this paper to discuss, theoretically and experimentally, the effects of the bearing surface finish on the performance characteristics of externally pressurized rectangular stationary thrust bearings, and to give reliable estimations of the magnitudes of these effects. E Q U A T I O N S G O V E R N I N G THE FLOW If the inertia and body forces of the incompressible lubricant are neglected, and steady state conditions prevail, Navier-Stoke's equations [4] reduce to d2u 1 dP dz 2 : /z dx"
(1)
E F F E C T OF PAD S U R F A C E R O U G H N E S S Figure l shows the bearing arrangement. The surface of the pad is assumed to have a regular triangular shape, of wave length a and amplitude b. It is also assumed that, there is a definite complete number of waves in the length (B2 - - B 1 ) . Hence, rr/=
B2 -- B1 a
where m is a positive integer. Only two equations are adopted to describe the film thickness in the bearing. The first one describes the film thickness under all the lines having positive slope in the first region (B1 ~< x ~ B2) and also those having negative slope in the second region ( - - B 1 ~ - x ~ - B2). The second equation describes the film thickness under the remaining lines.
Surface Finish Effectson the Performance of Externally Pressurized Rectangular Thrust Bearings
Ps,Qt,~Capillory ZI ~ Pi
87
restrictor
---V =!
I-I I"
_1
'T - - - - - .
'
I
f~Direction of flow
-t-.-
,--<. -(1-
! =-
"--1~-~
'
-[-"
"1
Supply
~
---4--
hole
_L._
FIG. 1. Since the bearing is stationary one region only of the bearing will be considered, namely, the region (B1 <~ x ~< Bz). Therefore, the film thickness under any line having positive slope is given by Fig. 1 hzl = h + 2b (x -- B1) -- 2 b ( N - - 1) a
(2)
and under any line having negative slope is given by Fig. 1 hzz = h + 2b {Ua -- (x -- B1)}.
(3)
a
Pressure distribution and total flow rate
Integrating equation (1) twice and substituting the following boundary conditions u=0
at z = 0
and
u=0
at z---=h
gives the velocity distribution u -- 2F
( Z 2 -- Zh)
and hence the total volume rate of flow is h
Qt = 2 f ul d Z 0
--
-
-
6p
1 lha dP _ _ . dx
(4)
88
M. A. EL-KADEEM, E. A. SALEM and H. R. EL-SAYED
The pressure distribution along the fluid film in the bearing, and at any point x is given by
/,;+c
f dx l P z _ -- 6tLQt
where hx is defined either by equation (2) or (3), and the constant of integration C can be determined from the b o u n d a r y conditions. (1) Pressure distribution under lines having positive slope at x = B~ at x = B1-~-
and
N = 1,
P = P~
and
N = 1,
P x l = Pa
a
2
therefore PxlN = Pc +
2lb
| (h + b) 2
h~
+
2b
+
.
(x--B1)
a
(2) Pressure distribution under lines having negative slope a at x = B a + 2 and N = 1,
-- 2 ( N - -
(5)
1)
Pxl=P3
therefore Px2N
= P i ~u
2Zb Qt (h + b) ~ . . . .
h2
-h+
2b {Na -- (x - - O 1 ) } a
2 •
(6)
Substituting the following conditions in equation (6) at x = B2
and
N =
B2 -- B1 a
,
Pz2Y = 0
therefore 2lbP1 Qt = 3treK1"
(7)
F r o m equations (5), (6) and (7), it follows that P x l N = Pc 1 + K1 { - ( h + b) 2 --
he
+ h+
1 {
2b z a (x -- B1) -- 2 b ( N - - 1)
2N
ex~u = e, 1 + K~ (-h~ b)2 h+
a
{Na-(~-B~)}
where
1
1
+ 2(Bz -- B1) -- a
2(B2
BD [1 _ a ...... Lh
2(Bz
]
1 ( h + b ) " J"
- - B1)
(8)
Surface Finish Effectson the Performance of Externally Pressurized Rectangular Thrust Bearings
89
The total load carrying capacity
Integrating the pressure of each line over the area of that line and by summation N=m
xlz
L = 2 P , B J + 21u~--=l f
xll
N=m
PxlN d x + 2l
~
N=I
x2~
f Px2N d x Xzt
where x l l = B1 + a ( N -- 1) X12
-=-
x~t = B1 q- a ( N -- ½) x2z = B1 -k Na
B1 + a(N -- ½)
therefore
(lo)
L =PI • AG
where G is a shape factor ~ ~-2 B I +
~ a-. N=I 2(B2 -- B1)~]
The stiffness
The bearing is compensated by a capillary restrictor, Fig. 1, whose resistance Zt is given by 128/~i zl -
~
[5].
The total volume rate of flow through the capillary is 1 (Ps - - P t )
Q : 21
[4].
(11)
Because of the continuity of the flow, equation (11) can be equated to equation (7), to give the pressure ratio as P~ _ 3/inK1 Ps 3t~aK1 q- 21bZl"
(12)
Since the stiffness of the bearing is the negative derivative of the load with respect to the film thickness, hence dL A-dh" From equations (10) and (12), the stiffness of the bearing is given by
/~ = 12/d,ps.4G(B2- gl)(h18
(h _]_1 b)S] [
2lbZx
]
[(3t~aK1 + 2blZ1)2]"
(13)
The power consumption
The power dissipated in the bearing Eb and the total power required to operate the bearing are given by 2lbp2i Eb = P~Qt -- 3tzaK1 Et = P, Qt -
21bP~Ps 3lzaKl"
(14)
(15)
90
M.A. EL-KADEEM,E. A. SALEMand H. R. ELoSAYED
The dimensionless groups
Four dimensionless groups, defined under NOMENCLATURE, were suggested to describe the bearing performance [6]. From equations (7), (10) and (15), the dimensionless groups of this bearing are (1) Load per unit area
Ufm{a
1 [B1 + I ~ B2
a2(2N -- I)t]
(16)
(2) Load per unit flow j - - 3aKaAG 2b
(17)
(3) Stiffness per unit supply pressure
[,
F1 ~ 12t~hG(B2 -- B1) ha
1 ] 2lbZ1 (h + b)3 (3t~aK1 + 2blZ1) 2"
(4) Stiffness per unit power
,]
241zB2G(B2 - B1) [ 1 H = (3tmK~ + 2blZ1)
[h3
(h + b) 3 "
08)
(19)
E F F E C T OF PAD SURFACE WAVINESS Figure 2 shows the bearing arrangement. The pad surface is assumed to have a wavy character. It is composed of a series of half sine waves of wave length a and amplitude b. It is also assumed that there is a definite complete number of waves of length (B2 -- B1), so that, m--
B2
-
-
B1
where m is a positive integer. Since the bearing is stationary, one region only of the bearing will be considered, namely, the region (B1 ~< x ~ B2). The film thickness in the bearing can be described as follows: (1) For parts of this region starting at Xl and ending at x2 the film thickness hx is given by 1)--B1}]
hxl=b[w+sinrr{x--a(N--a
where xl = Ba + a ( N - -
1)
x2 = B1 -b a(N -- ½) 14) ~
h --, b
(20)
Surface Finish Effectson the Performanceof Externally Pressurized Rectangular Thrust Bearings
Ps,Qt~.~/
91
Copilloryrestrictior"
~Y |
_r
-7 -l--
I I
-+--j
. D i r e c t i o n of flow
-4--
I
- - -
11
X
"~-i~Supply - hole
I
,
I_~ I. L-
~j
FIG. 2.
(2) For parts of this region starting at xa and ending at x4, the film thickness hx is given by
hx2 = b w + cos zr
(21)
a
where x3 :
Bl q- a ( N - - ½)
x4 :
Ba 4- aN.
Pressure distribution and total flow rate
The pressure distribution along the fluid film in the bearing and at any point x is given by Px -- -- 6tzQt ( d x z3 ÷ C l J hx
where hx is defined either by equation (20) or (21), and the constant of integration C can be evaluated from the boundary conditions. 1. Pressure distribution under parts starting at Xl and ending at xz
Ib 3
w ÷
sin rr
x
--
a(Na
1)
B1
92
M. A. EL-KAOEEM, A. E. SALEM a n d H. R. EL-SAYED
let,
let also, w + sin yl =
cos Y l ~
w2 -
1
(W - - s i n y )
(w e (w
1) 1/2 cos y - - sin y)
(w 2 -- 1)1/2 dya = (w ~ sin y) dy. Substituting in Pxl, therefore,
Pxl
-
-
-
-
61zQt lb 3 ~a . (w ~ _ 1 1)5/z f (w - sin y)2 dy.
Integrating and introducing the following b o u n d a r y conditions at x = B1
N = 1,
Px = Pc,
at x = Bl + a/2
N = 1,
P z = Pz
gives
exlN=P~+lb%r.(wZ~i)5/2Qt
w2+
N--sin-1
'Nl'si"'w2"lJ2il w
- ~'
+
2w
(2N
--
, --2 (N-l)
1) (w2 -
-- 1)1/2 w
--41{ 2 ( 2 N - l ) ( w 2 - 1 ) l / 2 w 2 - - s i n 2 ~ , } ] .
I
x--a(N--
let
cos
~,
} (22)
2. Pressure distribution under parts starting at xa and ending at x4
w + cos 7r
--
1)-a
BI+
Surface Finish Effects on the Performance of Externally Pressurized Rectangular Thrust Bearings
93
let also w2 -
1
w-q-COS y2-w--cos
4,
sin 4, = (w2 -- 1)1/2 sin 4, w - - c o s 4,
dy2 -- (w2 -- 1)1/2 de. (w -
cos 4,)
Substituting in .Px2 gives: Px2 -- --
6/zQt lb a
a " (w 2 - 1 1)5/2 f (w - cos 4,)2 d4,. ~r
Integrating and substituting with the following condition, at x = B1 + a/2
and
N = 1,
Px2 =
P3
therefore, 6/m
P:~2N ~ P* -4- ~ --
(N-
1
. ( w2 __ 1)5/2 Qt
1) sin -1 .(w2
[(
- - 1)1/2 w
w2 +
~){(
N sin -1
1)
4,}+2w{ (2N-1)
Zr -- ~ N (wz-1)l/2w
+ sin 4,}
41t22N 1,'w-m'l'2w sin2/]
(23)
Substituting the following boundary conditions in equation (23), at
x
=
B2,
N -- B2-
BI,,
Px2N :
0
a
hence lb3~r Qt : - Pi 6tza •
(w 2 --
1)5/2
~R1-
(24)
"
From equations (22), (23) and (24), it follows that: PxlN -= P,
l--R1
EI NE2 -- 2
--
-- y
+ 2 w { ( 2 N - - 1 ) E 4 -- cos y} --14 { 2 ( 2 N - - 1 ) E s -- sin 2y}]}} PXe~v : P~ 1
--
R1 EI N E 2 - Zr 2N--(N--1)E
(25)
a-4,
W 2w{(EN -- I)E4 + sin 4,} -- 1 (2(2N -- 1)E5 + sin 24,}] }}
(26)
94
M. A. EL-KADEEM, E. A. SALEMand H. R. EL-SAYED
where,
Ez
sin 1
1 W
Ea=sin
1( w 2 - 1)1/2 W
E4
(w2-
1)1/2 W
E5 : (w2 -- 1)1/2 W2
R1 --:- B2
a
B1
E1E2 --
2
E1 -- EIEa + 4 WE4 -- E5
¢ = sinl(W 2 - 1)l/2siny2. (w + cosyz)
~' = sin_ 1 1 q- w sin yl, w + sin yl The total load carrying capacity
The total load carrying capacity of the bearing can be expressed by the following equation
.... ( PzlN d x + 2l
L = 2PdB1 -~- 2l N=I
Px21v d x
X ll
(27)
X ~l
where P x l N and Px2N are defined by equation (25) and (26). However, Fig. 6 shows the pressure distribution in the bearing, which is independent of both a and b, and is the same as that of a plain pad bearing [6] which is a bearing having ideally flat surfaces. Therefore, the total load carrying capacity of this bearing is the same as that of a plain pad bearing [6] and may be expressed by (28)
L = PiAG
where, A = Projected area of the pad G=Shapefactor=2
'(
)
B1 1 +B2,.
The stiffness
Following the same procedure explained in considering roughness, the pressure ratio of the bearing is given by P~ _
6/zaR1
~Ps -- 6t~aR1 -- Z-llb%r(w 2 -- 1)~/2"
(29)
Surface Finish Effects on the Performance of Externally Pressurized Rectangular Thrust Bearings
95
From equations (28) and (29), the stiffness of the bearing is given by A:
6PsAGlzazllbZrr(w2--1)3/2[ [6liaR1zllba~(w 2 - - 1)5/2] 2
(w2 - -
dR1 1) dh
5RlW]
(30)
The power consumption
The power dissipated in the bearing Eb and the total power required to operate the bearing E t are Eb = e ~ Q t :
-- P~lbZzr(w~--
1)5/2
(31)
6/~R1 Et :
PsQt -
e~Pslb3~r(w2 -
1)5/2
(32)
61zaRx
The dimensionless groups
From equations (24), (28), (30) and (32), the dimensionless groups [6] of this bearing are 1. Load per unit area,
I----~1 ( 1 -q-~B1) .
(33)
2. Load per unit flow J=
--
6aAR1G
(34)
b3rr(w 2 - 1)5/2"
3. Stiffness per unit supply pressure F=
6hGt~azllb3~r(w 2 - -
[6vaR1 --
Zl*rb31(w 2 - -
1)3/2 [ dRldh 1)5/2]2 (w ~ -- 1)
5RlW] .
(35)
] 1) "
(36)
4. Stiffness per unit power - - 12B2Glza H - - 6p, a R l - - zllbZrr(w 2 - -
[dR1 1)5/2 [ dh
5Rlw b(w 2 --
EXPERIMENTAL VERIFICATIONS The presented experimental work was carried out to study the effect of bearing surface finish on bearing performance, and to verify the theoretical findings, sections (5) and (6). The experiments carried out were: 1. Measurement of the pressure distribution along the fluid film in the bearing. 2. Relation between the total volume rate of flow and the film thickness at constant inlet recess pressure. The bearing was mounted as shown in the test rig, Fig. 3, and the fluid and measuring systems are shown in Figs. 4 and 5.
Guide p l n ~ r ~ I
~
II
!
~DeoU
weights
I I/L°ad
application attachment
,It I,
/ Over
Jji ~-Drvin ig
f
shaft
°°°;r:V::'no~P°:~<,
bracket
|
i screws taps i
\~//~\"(//*~(//)x\~//X\~(/.,\.\y//'×\x(/ FIG. 3a. Guide pin
L
I Dead w e i g h t ' ~
~, N
i[
~',\~
/
.~
Load application attachment
Drawing nut
Annular contact bearing
t~
Driving shaft ~.~
\\ Shaft end
\ / /
i
~
Draw
bolt
Pad end taper
J-
T6 ", r / i / f / ] F t ~ . 3b.
in
SurfaceFinishEffectsonthePerformanceofExternallyPressurizedRectangularThrustBearings 97 /Bearing t Prate gaugePi Thermimeter
Gear pump Motor~, [ ~
o::oZ'e,_i l l
I
UU
Filter
~._.~ _-Regulating valve
filter
~'Oil tank FIG. 4.
Air nozzle Pressure Thermometer Capillary gaugePi Pressure ~ \ gouge Ps~ \ ~
t HMWI L NI Lq I11111", c
1'"Ii
egulati g volv
F"'erJ,.o.e,
\
1"-
II II
-"
":~"v':
,
Mercury manamete,
t
INI
.,~osp~e.,c pressure ,eve,
I~1
~
I Multi-tube mercury •
•
•
•
manometer Low pressure
FIG. 5.
I I
r'-'~ ~
r'-'~ r'--'n
Differential manometer High pressure
98
M. A. EL-KADEEM,E. A. SALEMand H. R. EL-SAYED
The following precautions were considered during the assembly of the test rig and the running of the tests: 1. The concentricity of pad and supply hole centres. 2. The perpendicularity between the pad face and the driving shaft axis. 3. The coincidence of pad and bed surfaces. 4. The axial application of load. 5. The easy dismounting of loads and pads. 6. Rigidity of mounting. The lubricant used was light oil "D.T.E. light", having the following viscosity temperature relation. Temperature (°F) Viscosity Redwood 1
100 120-145
140 60-70
200 35-45
212 33-42
The following measuring instruments were used in carrying out the experimental work; 1. Perth-O-Meter "Hommel Werke" and Roughness meter "Bruel and Kjaer" to measure the pad's surface finish. 2. Pneumatic comparator "Aeromess" contactless method to measure the film thickness. 3. Calibrated capillary tube and mercury in glass manometers to measure the lubricant flow rate. ~Ps ,Q t
!
~IZB, I~
i-
ZB2
<~Ps ,Q ,
,
'
-
~IZB,I-'-
i--zB2--i
I'0
/ -~--~ : 0 . 4 =0'5
\\\X
0.8
0.1
0-6
0.4
\ B2 =B2
0 . 2 - Pl =Pi o =0
I
b =b I 0
0.2
0.4
0.6 X BZ
FIG. 6.
0.8
bO
Surface Finish Effects on the Performance of Externally Pressurized Rectangular Thrust Bearings
99
4. O r d i n a r y thermometers to measure the lubricant temperature. 5. Mercury in glass m a n o m e t e r s to measure the pressure distribution along the fluid film in the bearing. The surface conditions of the bed was kept c o n s t a n t over all the tests, while those of the pad were changed by r e m a c h i n i n g its surface. Nine different m a c h i n i n g processes were chosen, those were. 1. Fine grinding. 2. Grinding. 3. Lapping. 4. Shaping along the recess length with a V-shaped single point tool (S.P.T.). 5. Shaping across the recess length with a V-shaped (S.P.T.). 6. Shaping along the recess length with a U-shaped (S.P.T.). 7. Shaping across the recess length with a U-shaped (S.P.T.). 8. T u r n i n g with a V-shaped (S.P.T.). 9. T u r n i n g with a U-shaped (S.P.T.). The conditions of the surface in each case are shown in table (1) a n d the test results in Tables 2 a n d 3, a n d Figs. 6-10. TABLE 1.
SURFACE CONDITIONS OF THE DIFFERENT PADS
A.
M A C H I N I N G PROCESSES
Pad dimensions (cm)
Code
,h
r
•
No.
Machining process
L
2B
R I4/
R
1 2 3 4 5 6 7 8 9
Find grinding Grinding Lapping V-Shaping along the recess V-Shaping across the recess U-Shaping along the recess U-Shaping across the recess V-Turning U-Turning
12 12 12 12 12 12 12 12 12
8 8 8 8 8 8 8 8 8
1 1 1 1 1 1 1 1 1
10 10 10 10 10 10 10 10 10
B . SURFACE CONDITIONS
Along the lay "micr"
Across the lay
Wave t"
Code No.
Rmax (tzm)
R
Ra
R
Ra
(/~m)
Rmax (t~m)
(/~m)
(~m)
2 4 12 20 28 18 21 35 30
2'81 7"62 16"90 28"10 29"50 25-40 29"60 49"40 42" 40
0"355 0"710 2-130 3"550 4"960 3"190 3'730 6'210 5" 320
2"5 6"0 14"0 62-0 46-0 38"0 27"0 34"0 42" 0
3"53 8"45 19'70 87-50 64-90 53'50 38'10 48'00 59' 50
•
(t~m)
Length (mm)
Amplitude (mm)
0-443 0"886 2"480 11"100 8-150 6-750 4-790 6-050 7"450
12'0 8"0 18-0 2'1 1'4 2'6 2'4 1-1 1" 1
0"0300 0"0020 0-0015 0-0130 0.0100 0'0150 0'0180 0'0200 0" 0230
a = a
Bj/B2
H = 50/.~
B 2 = 3 cm =0.3
LIB z =5
% m
o
I
0
4
8
12
16
20
24
28
b
FIG. 7. 0.4
I
4
I i
0.3 Plain
o.
\
\
c I
c B2=3cm 0.I -
0=o
~
Bl I B z = O , 3
k
L / B 2 =5
i
!
Z ~ = 4 0 kg s e c l c m 5
1
'
H =50/~
[
I I
0
4
8
12 b,
F]o. 8.
16 y.
20
24
28
Surface Finish Effects on the Performance of Externally Pressurized Rectangular Thrust Bearings TABLE 2. EFFECT O F
101
SURFACE FINISH DEGREE ON THE PRESSURE DISTRIBUTION IN PLAIN RECESSED BEARING A.
PAD DIMENSIONS AND MEASURING POINTS T
13
6
7
II
C
--- -
--'5-
2
I0
-
- - -
u cO
4
12 12 c m
h
B. PRESSUREDISTRIBUTIONP~
Measuring point 2 3 4 5 6 7 8 9 10 11 12 13
Machining process ^ 4 5 6
r I
2
3
0.670 0"655 0"660 0"920 0.208 0"322 0.322 0.875 0.460 0"525 0'230 0.236
0'675 0.675 0"675 0"965 0.160 0"296 0.296 0"970 0.490 0"530 0-265 0-250
0'655 0.680 0"685 0"980 0"203 0.335 0.355 0"980 0"570 0-555 0.235 0"350
TABLE 3.
0.675 0.645 0.660 0'865 0.275 0"375 0"469 0.890 0.540 0'425 0.260 0.297
0-630 0"675 0.655 0"950 0.235 0.340 0"340 0.950 0.568 0"591 0.284 0.284
0"630 0-640 0"660 0.855 0.189 0.311 0-311 0"845 0-545 0"485 0.471 0"239
7
8
9
0-639 0"630 0-608 0.845 0.163 0'289 0'374 0.865 0'570 0'570 0"245 0.237
0.653 0'610 0'630 0"820 0.196 0'319 0-312 0-798 0"479 0"472 0"254 0-254
0.610 0.615 0.620 0"875 0-132 0"387 0.270 0-895 0.585 0"510 0.286 0-396
EFFECTS OF SURFACE CONDITIONS ON FLOW RATE
(cma/sec) L = 12cm 2n= 8cm Surface conditions Rmax (t~m) Code No.
R~= lcm R z - - 10cm P, = 0" 6 kg/cm 2 Film thickness (t~m)
Pt = 0' 4 kg/cm 2 Film thickness (t~m)
Along
Across
Mean
50
I00
150
40
I00
150
2 4 12 20 28 18 21 35 30
2"5 6 14 62 46 38 27 34 42
2"25 5-00 13-00 41"00 37"00 28.00 24"00 34"50 36-00
1"000 1"050 0"900 2.600 1"725 1"100 1.300 1"850 1.700
2-475 2"625 2"325 5-300 4"200 2"652 2"830 3"750 3-750
3"975 4'225 3'725 7"950 6.500 4'250 4'400 5"750 5'825
0"725 0' 700 0'650 2-075 1"350 7"250 0"750 I'150 1"425
2'125 2- 200 2.000 4'700 3'750 2'155 2"250 2"750 3"375
3-550 3" 725 2-400 7"300 6"000 3"750 3"800 4"350 5'452
102
M. A. EL-KADEEM, E. A. SALEM and H. R. EL-SAYED 2'0
i
! I.,o',o
1'8
['6
1"4
% m
1.2
Z, o o. -2_
1.0--
i
O3 0"6
!
J
i
:
:
iii] i .... i\
0.4-
1
1
Bz=3cm
o =o
Bi/Bz=O'3
l/Bz=5
l
±=50 x l 0 -6 kg sec/cm 2 0,2
.
Z i = 4 0 kg sec/cm 5 H =5OFt
0
l
i
4
8
1
l 12
16
b,
20
\,
24
28
tz
Fie. 9.
TABLE 4.
S I G N I F I C A N C E TEST*
Measuring point
Mean value .~
Standard deviation S
Upper limit 2 + 3s
Lower limit ~ - 3s
No. of points outside the allowable limits
Results
2 3 4 5 6 7 8 9 10 11 12 13
0.6485 0'6471 0"7503 0-8861 0.1956 0"3304 0"3365 0"8964 0'5330 0'5181 0"2922 0'2936
0'0231 0'0261 0"0256 0-0590 0"0424 0'0331 0'0577 0.0602 0.0460 0'0517 0"0765 0-0600
0"6947 0"6993 0-7017 1.0041 0"2804 0.3966 0-4519 0'0168 0'6250 0"6215 0'4452 0.4136
0'6023 0"5949 0'5591 0'7681 0'1108 0"2642 0'2211 0'7760 0.4410 0-4147 0"1392 0"1736
0 0 0 0 0 0 1 0 0 0 1 0
Random Random Random Random Random Random Random Random Random Random Random Random
* O. 002 Significance level.
Surface Finish Effects on the Performance
of Externally Pressurized Rectangular Thrust Bearings
103
I IO
20 R mox .
30
40
P
FIG. 10.
DISCUSSION
The theoretically derived equations 8, 9, 25 and 26, which define the pressure distribution along the fluid film in the bearing in each of the considered cases, are shown in Fig. 6. The graph shows clearly that the pressure distribution is independent of the bearing surface finish, and is the same as that of a plain bearing working under the same conditions [6]. Table 2 shows the experimental pressure readings at the different measuring points for the considered nine processes. A significant test was made to check whether the variations of the pressure at the same measuring point were random or significant variations. The test results, Table 4, show that the variations of the pressure at the same point are random, and hence due to experimental error. The theoretical effects of the bearing surface finish on the load per unit flow, stiffness per unit supply pressure and stiffness per unit power are shown in Figs. 7-9. A comparison with those of a plain bearing working under the same cond.itions [6] is also included. The experimental measurements of flow rate are shown in Table (3) and Fig. 10. The figure shows that the lubricant flow rate increases as Rmax increases, which is a reasonable result, because the surface irregularities provide passages for the lubricant and Rm,, determines the maximum height of these passages.
104
M. A. EL-KADEEM,E. A. SALEMand H. R. EL-SAYED CONCLUSIONS
From the previous study the following conclusions can be drawn: I. Surface finish has no effect on the pressure distribution along the fluid film in plain stationary externally pressurized rectangular bearings. Consequently, surface finish has no effect on the load carrying capacity of such bearings. 2. The surface finish degree has a major increasing effect on the lubricant flow rate. 3. Surface finish reduces the load per unit flow and stiffness per unit power of the plain stationary externally pressurized bearings. 4. Surface finish improves the stiffness per unit supply pressure of the bearing to a certain value of surface irregularities height b, depending whether roughness or waviness is considered. As b is more increased, the stiffness is decreased.
REFERENCES [1] M. E. SALAMA,The effect of surface finish on the performance of parallel surface thrust bearings. Ph.D. Thesis, University of Manchester (1949). [2] D. D. FULLER,Theory and practice of lubrication for engineers. John Wiley, New York (1953). [3] R. W. BoL'rZ, Design considerations for manufacturing economy, Mech. Engng (December 1949). [4] O. PINKUSand B. STERNLICrrr,Theory of hydrodynamic lubrication. McGraw-Hill, New York (1961). [5] E. A. SALEM,Analysis and applications of externally pressurized bearings. Ph.D. Thesis, University of Manchester (1966). [6] M. A. EL KADEEM,E. A. SALEMand H. R. EL-SAYED,Effect of pad geometry and initial tilt on the performance of stationary externally, pressurized rectangular bearings. Bull. Faculty of Engineering, Alexandria University (1969).
Vol. 12, pp. 85-104.
Pergamon Press 1972 Printed in Great Britain
SURFACE FINISH EFFECTS ON THE PERFORMANCE OF EXTERNALLY PRESSURIZED R E C T A N G U L A R THRUST BEARINGS M. A. EL-KADEEM*, E. A. SALEMt and H. R. EL-SAYED~
(Received 22 October 1969) Abstraet--A beating surface may be considered to consist of one or more of the three known main features namely, mis-alignment, waviness and roughness. This study presents~theoretical and experimental investigations of the separate effects of both surface roughness and waviness, of the bearing pad surface, on the bearing performance. The bed surface is assumed to have irregularities of small magnitude to an extent that they can be neglected compared to the pad roughness.
The theoretical finditags are compared with those experimentally measured, and the results are presented in dimensionless groups. NOMENCLATURE
A E, Et
Projected ffrea of the bearing Bearing width Power consumption, total
F
~h Stiffness per unit supply pressure, - -
G
Shape factor
H
Stiffness per unit power, EdPs
B
PsA
3,/zll
L L o a d per unit area, pi A L L o a d per unit flow, - -
J
Qt ~-
L N P, P~, Ps
Q R
Ra Rmax a b d
l Total load carrying capacity Wave order Pressure, Inlet, Supply Flow rate Predominant m a x i m u m roughness height Centre line average height M a x i m u m peak-to-valley height o f surface irregularities Wave length Wave amplitude Capillary inside diameter
* Head of Production Engineering Department, Alexandria University, U.A.R. t Lecturer in Mechanical Engineering Department, Alexandria University, U.A.R. Assistant Lecturer in Production Engineering Department, Alexandria University, U.A.R. 85
86 h l I" m u, v, w x, y, z zl /~
M. A.
EL-KADF.EM, E.
A. SALEMand H. R. EL-SAYED
Film thickness Bearing length Capillary length Number of waves Linear velocity components Coordinates, rectangular External resistance Viscosity, absolute Stiffness
INTRODUCTION ON DEALINGwith the effects of the bearing surface finish on the performance of externally pressurized thrust bearings, little information exists. On the other hand, some attention is paid to the study of the effects of surface finish on the performance of hydrodynamic thrust bearings. Salama Ill presented a study of the effects of the bearing surface finish on the performance of circular thrust bearings. Fuller [2] explained the effects of surface finish on the performance of artificially roughened bearings. Boltz [3] reported on slide ways for a turret lathe bed which was constructed with a cross-directional scratch pattern. It is the purpose of this paper to discuss, theoretically and experimentally, the effects of the bearing surface finish on the performance characteristics of externally pressurized rectangular stationary thrust bearings, and to give reliable estimations of the magnitudes of these effects. E Q U A T I O N S G O V E R N I N G THE FLOW If the inertia and body forces of the incompressible lubricant are neglected, and steady state conditions prevail, Navier-Stoke's equations [4] reduce to d2u 1 dP dz 2 : /z dx"
(1)
E F F E C T OF PAD S U R F A C E R O U G H N E S S Figure l shows the bearing arrangement. The surface of the pad is assumed to have a regular triangular shape, of wave length a and amplitude b. It is also assumed that, there is a definite complete number of waves in the length (B2 - - B 1 ) . Hence, rr/=
B2 -- B1 a
where m is a positive integer. Only two equations are adopted to describe the film thickness in the bearing. The first one describes the film thickness under all the lines having positive slope in the first region (B1 ~< x ~ B2) and also those having negative slope in the second region ( - - B 1 ~ - x ~ - B2). The second equation describes the film thickness under the remaining lines.
Surface Finish Effectson the Performance of Externally Pressurized Rectangular Thrust Bearings
Ps,Qt,~Capillory ZI ~ Pi
87
restrictor
---V =!
I-I I"
_1
'T - - - - - .
'
I
f~Direction of flow
-t-.-
,--<. -(1-
! =-
"--1~-~
'
-[-"
"1
Supply
~
---4--
hole
_L._
FIG. 1. Since the bearing is stationary one region only of the bearing will be considered, namely, the region (B1 <~ x ~< Bz). Therefore, the film thickness under any line having positive slope is given by Fig. 1 hzl = h + 2b (x -- B1) -- 2 b ( N - - 1) a
(2)
and under any line having negative slope is given by Fig. 1 hzz = h + 2b {Ua -- (x -- B1)}.
(3)
a
Pressure distribution and total flow rate
Integrating equation (1) twice and substituting the following boundary conditions u=0
at z = 0
and
u=0
at z---=h
gives the velocity distribution u -- 2F
( Z 2 -- Zh)
and hence the total volume rate of flow is h
Qt = 2 f ul d Z 0
--
-
-
6p
1 lha dP _ _ . dx
(4)
88
M. A. EL-KADEEM, E. A. SALEM and H. R. EL-SAYED
The pressure distribution along the fluid film in the bearing, and at any point x is given by
/,;+c
f dx l P z _ -- 6tLQt
where hx is defined either by equation (2) or (3), and the constant of integration C can be determined from the b o u n d a r y conditions. (1) Pressure distribution under lines having positive slope at x = B~ at x = B1-~-
and
N = 1,
P = P~
and
N = 1,
P x l = Pa
a
2
therefore PxlN = Pc +
2lb
| (h + b) 2
h~
+
2b
+
.
(x--B1)
a
(2) Pressure distribution under lines having negative slope a at x = B a + 2 and N = 1,
-- 2 ( N - -
(5)
1)
Pxl=P3
therefore Px2N
= P i ~u
2Zb Qt (h + b) ~ . . . .
h2
-h+
2b {Na -- (x - - O 1 ) } a
2 •
(6)
Substituting the following conditions in equation (6) at x = B2
and
N =
B2 -- B1 a
,
Pz2Y = 0
therefore 2lbP1 Qt = 3treK1"
(7)
F r o m equations (5), (6) and (7), it follows that P x l N = Pc 1 + K1 { - ( h + b) 2 --
he
+ h+
1 {
2b z a (x -- B1) -- 2 b ( N - - 1)
2N
ex~u = e, 1 + K~ (-h~ b)2 h+
a
{Na-(~-B~)}
where
1
1
+ 2(Bz -- B1) -- a
2(B2
BD [1 _ a ...... Lh
2(Bz
]
1 ( h + b ) " J"
- - B1)
(8)
Surface Finish Effectson the Performance of Externally Pressurized Rectangular Thrust Bearings
89
The total load carrying capacity
Integrating the pressure of each line over the area of that line and by summation N=m
xlz
L = 2 P , B J + 21u~--=l f
xll
N=m
PxlN d x + 2l
~
N=I
x2~
f Px2N d x Xzt
where x l l = B1 + a ( N -- 1) X12
-=-
x~t = B1 q- a ( N -- ½) x2z = B1 -k Na
B1 + a(N -- ½)
therefore
(lo)
L =PI • AG
where G is a shape factor ~ ~-2 B I +
~ a-. N=I 2(B2 -- B1)~]
The stiffness
The bearing is compensated by a capillary restrictor, Fig. 1, whose resistance Zt is given by 128/~i zl -
~
[5].
The total volume rate of flow through the capillary is 1 (Ps - - P t )
Q : 21
[4].
(11)
Because of the continuity of the flow, equation (11) can be equated to equation (7), to give the pressure ratio as P~ _ 3/inK1 Ps 3t~aK1 q- 21bZl"
(12)
Since the stiffness of the bearing is the negative derivative of the load with respect to the film thickness, hence dL A-dh" From equations (10) and (12), the stiffness of the bearing is given by
/~ = 12/d,ps.4G(B2- gl)(h18
(h _]_1 b)S] [
2lbZx
]
[(3t~aK1 + 2blZ1)2]"
(13)
The power consumption
The power dissipated in the bearing Eb and the total power required to operate the bearing are given by 2lbp2i Eb = P~Qt -- 3tzaK1 Et = P, Qt -
21bP~Ps 3lzaKl"
(14)
(15)
90
M.A. EL-KADEEM,E. A. SALEMand H. R. ELoSAYED
The dimensionless groups
Four dimensionless groups, defined under NOMENCLATURE, were suggested to describe the bearing performance [6]. From equations (7), (10) and (15), the dimensionless groups of this bearing are (1) Load per unit area
Ufm{a
1 [B1 + I ~ B2
a2(2N -- I)t]
(16)
(2) Load per unit flow j - - 3aKaAG 2b
(17)
(3) Stiffness per unit supply pressure
[,
F1 ~ 12t~hG(B2 -- B1) ha
1 ] 2lbZ1 (h + b)3 (3t~aK1 + 2blZ1) 2"
(4) Stiffness per unit power
,]
241zB2G(B2 - B1) [ 1 H = (3tmK~ + 2blZ1)
[h3
(h + b) 3 "
08)
(19)
E F F E C T OF PAD SURFACE WAVINESS Figure 2 shows the bearing arrangement. The pad surface is assumed to have a wavy character. It is composed of a series of half sine waves of wave length a and amplitude b. It is also assumed that there is a definite complete number of waves of length (B2 -- B1), so that, m--
B2
-
-
B1
where m is a positive integer. Since the bearing is stationary, one region only of the bearing will be considered, namely, the region (B1 ~< x ~ B2). The film thickness in the bearing can be described as follows: (1) For parts of this region starting at Xl and ending at x2 the film thickness hx is given by 1)--B1}]
hxl=b[w+sinrr{x--a(N--a
where xl = Ba + a ( N - -
1)
x2 = B1 -b a(N -- ½) 14) ~
h --, b
(20)
Surface Finish Effectson the Performanceof Externally Pressurized Rectangular Thrust Bearings
Ps,Qt~.~/
91
Copilloryrestrictior"
~Y |
_r
-7 -l--
I I
-+--j
. D i r e c t i o n of flow
-4--
I
- - -
11
X
"~-i~Supply - hole
I
,
I_~ I. L-
~j
FIG. 2.
(2) For parts of this region starting at xa and ending at x4, the film thickness hx is given by
hx2 = b w + cos zr
(21)
a
where x3 :
Bl q- a ( N - - ½)
x4 :
Ba 4- aN.
Pressure distribution and total flow rate
The pressure distribution along the fluid film in the bearing and at any point x is given by Px -- -- 6tzQt ( d x z3 ÷ C l J hx
where hx is defined either by equation (20) or (21), and the constant of integration C can be evaluated from the boundary conditions. 1. Pressure distribution under parts starting at Xl and ending at xz
Ib 3
w ÷
sin rr
x
--
a(Na
1)
B1
92
M. A. EL-KAOEEM, A. E. SALEM a n d H. R. EL-SAYED
let,
let also, w + sin yl =
cos Y l ~
w2 -
1
(W - - s i n y )
(w e (w
1) 1/2 cos y - - sin y)
(w 2 -- 1)1/2 dya = (w ~ sin y) dy. Substituting in Pxl, therefore,
Pxl
-
-
-
-
61zQt lb 3 ~a . (w ~ _ 1 1)5/z f (w - sin y)2 dy.
Integrating and introducing the following b o u n d a r y conditions at x = B1
N = 1,
Px = Pc,
at x = Bl + a/2
N = 1,
P z = Pz
gives
exlN=P~+lb%r.(wZ~i)5/2Qt
w2+
N--sin-1
'Nl'si"'w2"lJ2il w
- ~'
+
2w
(2N
--
, --2 (N-l)
1) (w2 -
-- 1)1/2 w
--41{ 2 ( 2 N - l ) ( w 2 - 1 ) l / 2 w 2 - - s i n 2 ~ , } ] .
I
x--a(N--
let
cos
~,
} (22)
2. Pressure distribution under parts starting at xa and ending at x4
w + cos 7r
--
1)-a
BI+
Surface Finish Effects on the Performance of Externally Pressurized Rectangular Thrust Bearings
93
let also w2 -
1
w-q-COS y2-w--cos
4,
sin 4, = (w2 -- 1)1/2 sin 4, w - - c o s 4,
dy2 -- (w2 -- 1)1/2 de. (w -
cos 4,)
Substituting in .Px2 gives: Px2 -- --
6/zQt lb a
a " (w 2 - 1 1)5/2 f (w - cos 4,)2 d4,. ~r
Integrating and substituting with the following condition, at x = B1 + a/2
and
N = 1,
Px2 =
P3
therefore, 6/m
P:~2N ~ P* -4- ~ --
(N-
1
. ( w2 __ 1)5/2 Qt
1) sin -1 .(w2
[(
- - 1)1/2 w
w2 +
~){(
N sin -1
1)
4,}+2w{ (2N-1)
Zr -- ~ N (wz-1)l/2w
+ sin 4,}
41t22N 1,'w-m'l'2w sin2/]
(23)
Substituting the following boundary conditions in equation (23), at
x
=
B2,
N -- B2-
BI,,
Px2N :
0
a
hence lb3~r Qt : - Pi 6tza •
(w 2 --
1)5/2
~R1-
(24)
"
From equations (22), (23) and (24), it follows that: PxlN -= P,
l--R1
EI NE2 -- 2
--
-- y
+ 2 w { ( 2 N - - 1 ) E 4 -- cos y} --14 { 2 ( 2 N - - 1 ) E s -- sin 2y}]}} PXe~v : P~ 1
--
R1 EI N E 2 - Zr 2N--(N--1)E
(25)
a-4,
W 2w{(EN -- I)E4 + sin 4,} -- 1 (2(2N -- 1)E5 + sin 24,}] }}
(26)
94
M. A. EL-KADEEM, E. A. SALEMand H. R. EL-SAYED
where,
Ez
sin 1
1 W
Ea=sin
1( w 2 - 1)1/2 W
E4
(w2-
1)1/2 W
E5 : (w2 -- 1)1/2 W2
R1 --:- B2
a
B1
E1E2 --
2
E1 -- EIEa + 4 WE4 -- E5
¢ = sinl(W 2 - 1)l/2siny2. (w + cosyz)
~' = sin_ 1 1 q- w sin yl, w + sin yl The total load carrying capacity
The total load carrying capacity of the bearing can be expressed by the following equation
.... ( PzlN d x + 2l
L = 2PdB1 -~- 2l N=I
Px21v d x
X ll
(27)
X ~l
where P x l N and Px2N are defined by equation (25) and (26). However, Fig. 6 shows the pressure distribution in the bearing, which is independent of both a and b, and is the same as that of a plain pad bearing [6] which is a bearing having ideally flat surfaces. Therefore, the total load carrying capacity of this bearing is the same as that of a plain pad bearing [6] and may be expressed by (28)
L = PiAG
where, A = Projected area of the pad G=Shapefactor=2
'(
)
B1 1 +B2,.
The stiffness
Following the same procedure explained in considering roughness, the pressure ratio of the bearing is given by P~ _
6/zaR1
~Ps -- 6t~aR1 -- Z-llb%r(w 2 -- 1)~/2"
(29)
Surface Finish Effects on the Performance of Externally Pressurized Rectangular Thrust Bearings
95
From equations (28) and (29), the stiffness of the bearing is given by A:
6PsAGlzazllbZrr(w2--1)3/2[ [6liaR1zllba~(w 2 - - 1)5/2] 2
(w2 - -
dR1 1) dh
5RlW]
(30)
The power consumption
The power dissipated in the bearing Eb and the total power required to operate the bearing E t are Eb = e ~ Q t :
-- P~lbZzr(w~--
1)5/2
(31)
6/~R1 Et :
PsQt -
e~Pslb3~r(w2 -
1)5/2
(32)
61zaRx
The dimensionless groups
From equations (24), (28), (30) and (32), the dimensionless groups [6] of this bearing are 1. Load per unit area,
I----~1 ( 1 -q-~B1) .
(33)
2. Load per unit flow J=
--
6aAR1G
(34)
b3rr(w 2 - 1)5/2"
3. Stiffness per unit supply pressure F=
6hGt~azllb3~r(w 2 - -
[6vaR1 --
Zl*rb31(w 2 - -
1)3/2 [ dRldh 1)5/2]2 (w ~ -- 1)
5RlW] .
(35)
] 1) "
(36)
4. Stiffness per unit power - - 12B2Glza H - - 6p, a R l - - zllbZrr(w 2 - -
[dR1 1)5/2 [ dh
5Rlw b(w 2 --
EXPERIMENTAL VERIFICATIONS The presented experimental work was carried out to study the effect of bearing surface finish on bearing performance, and to verify the theoretical findings, sections (5) and (6). The experiments carried out were: 1. Measurement of the pressure distribution along the fluid film in the bearing. 2. Relation between the total volume rate of flow and the film thickness at constant inlet recess pressure. The bearing was mounted as shown in the test rig, Fig. 3, and the fluid and measuring systems are shown in Figs. 4 and 5.
Guide p l n ~ r ~ I
~
II
!
~DeoU
weights
I I/L°ad
application attachment
,It I,
/ Over
Jji ~-Drvin ig
f
shaft
°°°;r:V::'no~P°:~<,
bracket
|
i screws taps i
\~//~\"(//*~(//)x\~//X\~(/.,\.\y//'×\x(/ FIG. 3a. Guide pin
L
I Dead w e i g h t ' ~
~, N
i[
~',\~
/
.~
Load application attachment
Drawing nut
Annular contact bearing
t~
Driving shaft ~.~
\\ Shaft end
\ / /
i
~
Draw
bolt
Pad end taper
J-
T6 ", r / i / f / ] F t ~ . 3b.
in
SurfaceFinishEffectsonthePerformanceofExternallyPressurizedRectangularThrustBearings 97 /Bearing t Prate gaugePi Thermimeter
Gear pump Motor~, [ ~
o::oZ'e,_i l l
I
UU
Filter
~._.~ _-Regulating valve
filter
~'Oil tank FIG. 4.
Air nozzle Pressure Thermometer Capillary gaugePi Pressure ~ \ gouge Ps~ \ ~
t HMWI L NI Lq I11111", c
1'"Ii
egulati g volv
F"'erJ,.o.e,
\
1"-
II II
-"
":~"v':
,
Mercury manamete,
t
INI
.,~osp~e.,c pressure ,eve,
I~1
~
I Multi-tube mercury •
•
•
•
manometer Low pressure
FIG. 5.
I I
r'-'~ ~
r'-'~ r'--'n
Differential manometer High pressure
98
M. A. EL-KADEEM,E. A. SALEMand H. R. EL-SAYED
The following precautions were considered during the assembly of the test rig and the running of the tests: 1. The concentricity of pad and supply hole centres. 2. The perpendicularity between the pad face and the driving shaft axis. 3. The coincidence of pad and bed surfaces. 4. The axial application of load. 5. The easy dismounting of loads and pads. 6. Rigidity of mounting. The lubricant used was light oil "D.T.E. light", having the following viscosity temperature relation. Temperature (°F) Viscosity Redwood 1
100 120-145
140 60-70
200 35-45
212 33-42
The following measuring instruments were used in carrying out the experimental work; 1. Perth-O-Meter "Hommel Werke" and Roughness meter "Bruel and Kjaer" to measure the pad's surface finish. 2. Pneumatic comparator "Aeromess" contactless method to measure the film thickness. 3. Calibrated capillary tube and mercury in glass manometers to measure the lubricant flow rate. ~Ps ,Q t
!
~IZB, I~
i-
ZB2
<~Ps ,Q ,
,
'
-
~IZB,I-'-
i--zB2--i
I'0
/ -~--~ : 0 . 4 =0'5
\\\X
0.8
0.1
0-6
0.4
\ B2 =B2
0 . 2 - Pl =Pi o =0
I
b =b I 0
0.2
0.4
0.6 X BZ
FIG. 6.
0.8
bO
Surface Finish Effects on the Performance of Externally Pressurized Rectangular Thrust Bearings
99
4. O r d i n a r y thermometers to measure the lubricant temperature. 5. Mercury in glass m a n o m e t e r s to measure the pressure distribution along the fluid film in the bearing. The surface conditions of the bed was kept c o n s t a n t over all the tests, while those of the pad were changed by r e m a c h i n i n g its surface. Nine different m a c h i n i n g processes were chosen, those were. 1. Fine grinding. 2. Grinding. 3. Lapping. 4. Shaping along the recess length with a V-shaped single point tool (S.P.T.). 5. Shaping across the recess length with a V-shaped (S.P.T.). 6. Shaping along the recess length with a U-shaped (S.P.T.). 7. Shaping across the recess length with a U-shaped (S.P.T.). 8. T u r n i n g with a V-shaped (S.P.T.). 9. T u r n i n g with a U-shaped (S.P.T.). The conditions of the surface in each case are shown in table (1) a n d the test results in Tables 2 a n d 3, a n d Figs. 6-10. TABLE 1.
SURFACE CONDITIONS OF THE DIFFERENT PADS
A.
M A C H I N I N G PROCESSES
Pad dimensions (cm)
Code
,h
r
•
No.
Machining process
L
2B
R I4/
R
1 2 3 4 5 6 7 8 9
Find grinding Grinding Lapping V-Shaping along the recess V-Shaping across the recess U-Shaping along the recess U-Shaping across the recess V-Turning U-Turning
12 12 12 12 12 12 12 12 12
8 8 8 8 8 8 8 8 8
1 1 1 1 1 1 1 1 1
10 10 10 10 10 10 10 10 10
B . SURFACE CONDITIONS
Along the lay "micr"
Across the lay
Wave t"
Code No.
Rmax (tzm)
R
Ra
R
Ra
(/~m)
Rmax (t~m)
(/~m)
(~m)
2 4 12 20 28 18 21 35 30
2'81 7"62 16"90 28"10 29"50 25-40 29"60 49"40 42" 40
0"355 0"710 2-130 3"550 4"960 3"190 3'730 6'210 5" 320
2"5 6"0 14"0 62-0 46-0 38"0 27"0 34"0 42" 0
3"53 8"45 19'70 87-50 64-90 53'50 38'10 48'00 59' 50
•
(t~m)
Length (mm)
Amplitude (mm)
0-443 0"886 2"480 11"100 8-150 6-750 4-790 6-050 7"450
12'0 8"0 18-0 2'1 1'4 2'6 2'4 1-1 1" 1
0"0300 0"0020 0-0015 0-0130 0.0100 0'0150 0'0180 0'0200 0" 0230
a = a
Bj/B2
H = 50/.~
B 2 = 3 cm =0.3
LIB z =5
% m
o
I
0
4
8
12
16
20
24
28
b
FIG. 7. 0.4
I
4
I i
0.3 Plain
o.
\
\
c I
c B2=3cm 0.I -
0=o
~
Bl I B z = O , 3
k
L / B 2 =5
i
!
Z ~ = 4 0 kg s e c l c m 5
1
'
H =50/~
[
I I
0
4
8
12 b,
F]o. 8.
16 y.
20
24
28
Surface Finish Effects on the Performance of Externally Pressurized Rectangular Thrust Bearings TABLE 2. EFFECT O F
101
SURFACE FINISH DEGREE ON THE PRESSURE DISTRIBUTION IN PLAIN RECESSED BEARING A.
PAD DIMENSIONS AND MEASURING POINTS T
13
6
7
II
C
--- -
--'5-
2
I0
-
- - -
u cO
4
12 12 c m
h
B. PRESSUREDISTRIBUTIONP~
Measuring point 2 3 4 5 6 7 8 9 10 11 12 13
Machining process ^ 4 5 6
r I
2
3
0.670 0"655 0"660 0"920 0.208 0"322 0.322 0.875 0.460 0"525 0'230 0.236
0'675 0.675 0"675 0"965 0.160 0"296 0.296 0"970 0.490 0"530 0-265 0-250
0'655 0.680 0"685 0"980 0"203 0.335 0.355 0"980 0"570 0-555 0.235 0"350
TABLE 3.
0.675 0.645 0.660 0'865 0.275 0"375 0"469 0.890 0.540 0'425 0.260 0.297
0-630 0"675 0.655 0"950 0.235 0.340 0"340 0.950 0.568 0"591 0.284 0.284
0"630 0-640 0"660 0.855 0.189 0.311 0-311 0"845 0-545 0"485 0.471 0"239
7
8
9
0-639 0"630 0-608 0.845 0.163 0'289 0'374 0.865 0'570 0'570 0"245 0.237
0.653 0'610 0'630 0"820 0.196 0'319 0-312 0-798 0"479 0"472 0"254 0-254
0.610 0.615 0.620 0"875 0-132 0"387 0.270 0-895 0.585 0"510 0.286 0-396
EFFECTS OF SURFACE CONDITIONS ON FLOW RATE
(cma/sec) L = 12cm 2n= 8cm Surface conditions Rmax (t~m) Code No.
R~= lcm R z - - 10cm P, = 0" 6 kg/cm 2 Film thickness (t~m)
Pt = 0' 4 kg/cm 2 Film thickness (t~m)
Along
Across
Mean
50
I00
150
40
I00
150
2 4 12 20 28 18 21 35 30
2"5 6 14 62 46 38 27 34 42
2"25 5-00 13-00 41"00 37"00 28.00 24"00 34"50 36-00
1"000 1"050 0"900 2.600 1"725 1"100 1.300 1"850 1.700
2-475 2"625 2"325 5-300 4"200 2"652 2"830 3"750 3-750
3"975 4'225 3'725 7"950 6.500 4'250 4'400 5"750 5'825
0"725 0' 700 0'650 2-075 1"350 7"250 0"750 I'150 1"425
2'125 2- 200 2.000 4'700 3'750 2'155 2"250 2"750 3"375
3-550 3" 725 2-400 7"300 6"000 3"750 3"800 4"350 5'452
102
M. A. EL-KADEEM, E. A. SALEM and H. R. EL-SAYED 2'0
i
! I.,o',o
1'8
['6
1"4
% m
1.2
Z, o o. -2_
1.0--
i
O3 0"6
!
J
i
:
:
iii] i .... i\
0.4-
1
1
Bz=3cm
o =o
Bi/Bz=O'3
l/Bz=5
l
±=50 x l 0 -6 kg sec/cm 2 0,2
.
Z i = 4 0 kg sec/cm 5 H =5OFt
0
l
i
4
8
1
l 12
16
b,
20
\,
24
28
tz
Fie. 9.
TABLE 4.
S I G N I F I C A N C E TEST*
Measuring point
Mean value .~
Standard deviation S
Upper limit 2 + 3s
Lower limit ~ - 3s
No. of points outside the allowable limits
Results
2 3 4 5 6 7 8 9 10 11 12 13
0.6485 0'6471 0"7503 0-8861 0.1956 0"3304 0"3365 0"8964 0'5330 0'5181 0"2922 0'2936
0'0231 0'0261 0"0256 0-0590 0"0424 0'0331 0'0577 0.0602 0.0460 0'0517 0"0765 0-0600
0"6947 0"6993 0-7017 1.0041 0"2804 0.3966 0-4519 0'0168 0'6250 0"6215 0'4452 0.4136
0'6023 0"5949 0'5591 0'7681 0'1108 0"2642 0'2211 0'7760 0.4410 0-4147 0"1392 0"1736
0 0 0 0 0 0 1 0 0 0 1 0
Random Random Random Random Random Random Random Random Random Random Random Random
* O. 002 Significance level.
Surface Finish Effects on the Performance
of Externally Pressurized Rectangular Thrust Bearings
103
I IO
20 R mox .
30
40
P
FIG. 10.
DISCUSSION
The theoretically derived equations 8, 9, 25 and 26, which define the pressure distribution along the fluid film in the bearing in each of the considered cases, are shown in Fig. 6. The graph shows clearly that the pressure distribution is independent of the bearing surface finish, and is the same as that of a plain bearing working under the same conditions [6]. Table 2 shows the experimental pressure readings at the different measuring points for the considered nine processes. A significant test was made to check whether the variations of the pressure at the same measuring point were random or significant variations. The test results, Table 4, show that the variations of the pressure at the same point are random, and hence due to experimental error. The theoretical effects of the bearing surface finish on the load per unit flow, stiffness per unit supply pressure and stiffness per unit power are shown in Figs. 7-9. A comparison with those of a plain bearing working under the same cond.itions [6] is also included. The experimental measurements of flow rate are shown in Table (3) and Fig. 10. The figure shows that the lubricant flow rate increases as Rmax increases, which is a reasonable result, because the surface irregularities provide passages for the lubricant and Rm,, determines the maximum height of these passages.
104
M. A. EL-KADEEM,E. A. SALEMand H. R. EL-SAYED CONCLUSIONS
From the previous study the following conclusions can be drawn: I. Surface finish has no effect on the pressure distribution along the fluid film in plain stationary externally pressurized rectangular bearings. Consequently, surface finish has no effect on the load carrying capacity of such bearings. 2. The surface finish degree has a major increasing effect on the lubricant flow rate. 3. Surface finish reduces the load per unit flow and stiffness per unit power of the plain stationary externally pressurized bearings. 4. Surface finish improves the stiffness per unit supply pressure of the bearing to a certain value of surface irregularities height b, depending whether roughness or waviness is considered. As b is more increased, the stiffness is decreased.
REFERENCES [1] M. E. SALAMA,The effect of surface finish on the performance of parallel surface thrust bearings. Ph.D. Thesis, University of Manchester (1949). [2] D. D. FULLER,Theory and practice of lubrication for engineers. John Wiley, New York (1953). [3] R. W. BoL'rZ, Design considerations for manufacturing economy, Mech. Engng (December 1949). [4] O. PINKUSand B. STERNLICrrr,Theory of hydrodynamic lubrication. McGraw-Hill, New York (1961). [5] E. A. SALEM,Analysis and applications of externally pressurized bearings. Ph.D. Thesis, University of Manchester (1966). [6] M. A. EL KADEEM,E. A. SALEMand H. R. EL-SAYED,Effect of pad geometry and initial tilt on the performance of stationary externally, pressurized rectangular bearings. Bull. Faculty of Engineering, Alexandria University (1969).