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Starvation in Ball Bearing Lubricated by Oil and Air Lubrication System
Tribology for Energy Conservation / D. Dowson et al. (Editors) © 1998 Elsevier Science B.V. All rights reserved.
243
Starvation in Ball Bearing Lubricated by Oil and Air Lubrication System F. Itoigawa, T. Nakamura and T. Matsubara a aDepartment of Mechanical Engineering, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya 466 Japan An oil and air lubrication system for ball bearings supporting a high speed spindle maintains friction losses and temperature rises in low level comparing with other lubrication systems, e.g., oil jet or oil mist lubrication. In this study, rotating speeds of the ball in an angular contact ball bearing lubricated by the oil and air lubrication system are observed in various oil supply rates. In addition, quasi-static model analysis of the ball motion is carried out. Experiments indicate that the angular velocity of the ball varies with the oil supply rate even at the constant spindle speed. Furthermore, the model analysis suggests that the ball angular velocity is considerably concerned with an inlet film thickness. From the both results, a relationship between a starvation factor in the ball-race contacts and the oil supply rate is derived for the ball bearing under the oil and air lubrication.
1. I N T R O D U C T I O N In a last decade, many manufacturers of machine tools have employed an oil and air lubrication system for support bearings in a spindle. The oil and air lubrication is a method in which both a very small amount of oil, for example 10ram 3/min a bearing, and a large amount of compressed air are simultaneously supplied to the bearings through one nozzle for each bearing. Under the oil and ~ lubrication, it is known that an inlet to the conjunction between a ball and race is starved when the oil supply rate becomes smaller or the race rotational speed becomes higher. And a main cause of bearing temperature rise under the oil and air lubrication is shear heating of EHD films in the conjunctions owing to gyroscopic slip and spin slip, as drug loss and churning loss are negligibly small[l ]. So, when the bearing is running in the starved condition at low oil supply rate, a reduction of oil film thickness brings very low temperature rise. However, severe starvation induces metallic contacts or, in the worst case, bearing seizure. Therefore, an adequate oil supply rate must be selected in practical spindle bearings. Unfortunately, a variation of the degree of the starvation with the oil supply rate has not understood quantitatively under the oil and air lubrication. So, the oil supply rate has been selected empirically. There are many studies about the starvation condition in non-conformal contact. Wolveridge et al.[2]
and Archard and Baglin[3] semi-analytically presented an influence of the starvation on film thickness in line contact. Goksem and Hargreaves[4] introduced a starved film thickness including an effect of viscous shear heating. In point contact, a film thickness reduction factor caused by starvation was developed by Hamrock and Dowson[5]. These studies provided the film thickness in the starved condition as a function of an inlet meniscus length. More practical research on the starvation in point contact was developed by Chiu[6]. He predicted an effect of an oil replenishment of the rolling track. Furthermore, Olaru and Galitanu[7] and Pasdari and Gentle[8] investigated the starvation in the actual ball bearing. The former carried out the experiments under oil mist lubrication, the latter did under oil jet lubrication. An ultimate purpose of this study is to specify the optimum oil supply rate in the oil and air lubrication. In this paper, a relationship between the degree of the starvation and the oil supply rate is investigated by observing the ball motion in experiments and by quasistatic analyses of the ball motion under conditions in which the bearing is insufficiently lubricated.
2. OIL AND AIR LUBRICATION SYSTEM For the machine tools, the oil and air lubrication system has two advantages comparing with other lubrication systems, for example, an oil mist lubrica-
244
tion system, or an oil jet lubrication system. First, this lubrication system is superior to the other lubrication systems in size and cost because an oil cooler, oil recirculating devices and complicated piping are not necessary. Second, temperature rise caused by drug loss and churning loss is suppressed at relatively low level because of a small supplied oil amount. This is so valuable from a viewtx~int of precision preservation that many high speed and high precision machine tools have come to be equipped with the oil and air lubrication system. In addition, the oil and air lubrication is superior to grease lubrication with regard to maintainability, since new oil without deterioration by oxidizing is supplied continuously and stably. The oil and air lubrication system consists of a distributor and an oil pressure equipment as schematically shown in Fig. 1. The distributor discharges a very small amount of oil measmv.a by reciprocating motion of a constant-quantity piston into pressurized air. The reciprocating motion of the constantquantity piston is given by hydraulic pressure which is intermittently introduced into the piston by a solenoid valve. So, the oil supply rate is adjusted by changing an oil discharge interval. As a tube in which the oil and air flow is made of material which possesses low wettability against the oil such as PTFE, many small oil drops are formed in the tube. 3. TEST BEARING AND SPINDLE The test bearing is a high speed angular contact bali bearing(7010C) which has a bore diameter of 50ram and an outer race diameter of 80ram. 19 balls and a cage made of phenolic resin are assembled in the bearhag. Measured bearing geometries are listed in Table 1. Only one of the balls is magnetized in order to detect ball angular velocity by a search coil method proposed by Hirano[9] as shown in Fig. 2. A constant load is applied by a low stiffness coil spring in an axial direction only. Generally, the bearings are lightly pre-loaded in a practical high speed machine tools. So, in this study, the axial load is applied with a relatively light force of !62N. Under this load condition, mean contacting pressure is about 400MPa and a contact half width along a semi-major axis is about 0.4ram. The oil and air are injected into the inner race groove tlu'ough a nozzle mounted on the bearing
holder. The range of the oil supply rate is from 0.5mm3/min to 20mm3/min and an air flow rate is 20Nt/min regardless of the oil supply rate. A spindle can rotate at any speed up to 250rev/sec. In addition, outer race temperaaue is monitored by a thermocouple. Table l
Test bearing geometries
Ball Diameter Inner ring radius Outer ring radius Inner groove curvature radius Outer groove curvature radius
Air m~dm'l
i ~
"
! P:~..
8.7313 mm 28.115 mm 36.849 mm 4.4529 mm 4.5403 mm
Compre~e.d air Constant quantity
-.--' U ', piston
~ - i l
Fig. l
~ Solenoidvalve
Schematic illustration of oil and air lubrication system
Magnetizedball Spindle, " X
~ ttt Thermo-couple ~3-'~=j,./Bearing hol¢ter
~'i-: " ~ _ _ ~
Nozzle
Air
~ ~ '~ \ ' Thrustload ~.,--~'<~ i...-/'P--':.:Tj.--i ........... k \ adjustscrew
Fig. 2
Test al~aratus
4. VARIATION OF GYROSCOPIC SLIP WITH OIL SUPPLY RATE In the lightly loaded angular ball beating, a gyroscopic moment acting on the balls has a considerable effect on slip in the ball-race conjunction. In this case, ball motion substantially differs from that being predicted by a control theory[9][10]. Although the control theory expects no spinning motion on ether the
245 outer race or the inner race regardless of the rotational speed, the spinning motion, in fact, exists on the both races. In addition, spin velocities at the both contact points which generate large slip between the ball and races vary with the rotational speed. The slip at the contact point, what is called gyroscopic slip or gyroslip, can be easily confirmed by observing the ball angular velocity. Figures 3 and 4 show experimental results of the variations of angular velocity ratio of the ball to the inner race, wb/wi, caused by the gyroscopic slip with the inner ring speed. From these results, it is found that the velocity ratios, wb/wi, are far from the outer race control theory when the inner ring rotates at high speed. Furthermore, it is very noticeable that the magnitude ofthe velocity ratio, w~/wi obviously decreases and comes cross to the theory with decreasing in the oil supply rate. The magnitude of a separation from the curve of the outer race control theory is influenced by EHD traction in the ball-race conjunction. Therefore, the higher viscosity of the lubricant or the thinner the EHD film(the higher shear rate), the smaller this separation. As the experiments are carried out under same conditions except for the oil supply rate, it is considered that the reduction of the velocity ratio, wb/wi, with decreasing in the oil supply rate in Figs. 3, 4 shows the decrease in the EHD film thickness. Consequently, the oil supply rate dominates lack of the lubricant at the inlet, that is, the degree of the starvation, under the oil and air lubrication. Even if the oil is continuously supplied for a long time, that is, the total oil amount supplied to the bearing gradually increases, no change of the velocity ratio, wb/wi, appears. This means that the rate of the oil injection into bearing, Qin, is equal to that of the oil ejection from the bearing, Qout.
Q~,, Qo~,t =
(])
By the way, the ejection rate, Qout, is dominated by a formation rate of oil drops which part from side wall lubricant and size of the oil drop. Naturally, with increasing in the volume of the side wall, both the rate and the size also increase. In addition, a thickness of the side wall, Hb, influences the film thickness, ho, on the track after passage of the rolling ball as shown in Fig. 5. Chiu[6] derived this film thickness by taking flow-back from the side wall lubricant into consideration for ball and plane contact and Olaru and Gaff-
tanu[7] also did for the ball-race contact. They expressed a film thickness, h0, as the function of a time passed two successive balls in terms of the EHD film thickness, he, at a flat contact region,
Tt ho = he + Co 2rla
(2)
where T, ~ and a are the o i l - a i r interface surface tension, the viscosity of the oil and the half width along the semi-major axis of the contact ellipse, respectively. The coefficient Co, which was numerically obtained, varies with the side wall film thickness, Hb. From their calculating results, a variation of the coefficient, Co, with Hb is approximately linear, so next relationship is obtained. 3.80 °2.0 [114.0
_ ~
I n 10.0
vA
T20.0
Y
, oL t
A rn I
=
.'
. o
"
•
."
"°:'-"\
,
~
"
"
e
V~._
| 06.0
3.75I-"8.0
I
o 8
~ll~o
~
cont~oit.eo,y
3.65 ! ...........~ ~- ~U~l: ................................~................ 0 50 100 150 200 250 Inner ring rotational speed, rps Fig. 3 Gyroscopic slip variation with the inner ring rotational speed, oil viscosity = 0.027Pas; axial load = 162N. "
3.80
..... V :
"0.5
mm3/min
W ~
m|.o
'~ i
Q 3.0 a I0.0
~ 'v
~" 3.75 - v 14.0
"I
.......
II
o *
* .
Ii~
g
~
m
/
3.70
V
............................... ~ ' " l j ' ~
O
*
..
outer race
3 . 6 5 .......................................................................': 'con-tr°! t~orY~
0
F~g. 4
50 100 150 Inner ring rotational speed, rps Gyroscopic slip variation with the inner ring rotational speed, oil viscosity = 0.019Pas; axial load = 162N.
246
2r/a ( h o - h¢)-~7, o¢ Hb
(3)
Consequently, it is considered that not the total amount of oil supply but the oil supply rate controls the film thickness at the inlet oil layer. In other words, the oil supply rate can control the degree of the starvation under the oil and air lubrication. 5. ANALYSIS OF G Y R O S C O P I C SLIP MOTION IN STARVED EH1L CONDITION To estimate the reduction of the velocity ratio, wb/wi, caused by the starvation, quasi-static analysis of the ball motion is carried out. Before establishing the quasi-static model of the ball motion, a few assumptions are introduced as follows, I. A drug force and a churning force are ignored because of a very small oil amount. 2. There is no interactions between the ball and the cage, 3. Shear behavior of the lubricant in the conjunction is non-linear viscous because the product value of pressure-viscosity coefficient,c~ and average pressure, ~ is less than 13[ 11 ]. 4. All of the bearing element have same temperature and a uniform temperature distribution. From this assumption, effects of the thermal expansion are negligible. In addition, temperature rise at the contacting surface is ignored. 5. Compression heating and convection cooling in the EHD film are ignored. 6. An elastohydrodynamic pressure distribution assumes to be identical to a Hertzian pressure distribution.
Vb = Wb × rp
(4)
where Wb is a vector of the ball angular velocity and rp denotes a vector which locates any point in the contact ellipse relative to the ball center. On the other hands, a velocity vector of the race surface, v,., is, (wi - w e ) × (rm + rp + 8h,) (inner) - a ' e × (rrn + rp + ¢Iho) (outer)
"O r
(5)
where vector rrn locates the ball center relative to the race center and ~ih shows an incremental vector caused by the EHD film thickness. By making use of transformation matrix [T] from the azimuth frame to the local contact frame, a local slip velocity vector, Au, is given by Au
=
[T](v~ - v,)
(6)
ho=h0r+h0b ~ ,..~ . . . . . _ ~ ~ ~
.........................................
~
bal~
rate
t t
High oil supply rate
!
Fig. 5
Rebounding oil layer shape in lateral to the track. 2a
To describe the model, two coordinate frames are employed as shown in Fig. 6. One is an azimuth frame with x axis along the bearing axis and z axis pointing radially outward through the ball center. The other is a local contact frame Op-xpypzp with z axis normal to the contact surface and ~: and y axis in the tangential plane at the contact. 5.1. Force and Moment Acting on Ball A velocity vector of the ball surface at a point in the contact ellipse, vb, is,
Hertzian contact ellipse
Yr Y
xp Race ~ groove," /
Fig. 6
Coordinate frames to describe the ball motion,
247
Lubricant shear behavior is described by the Eyring Model[ 12][ 13], that is,
I Ave' Avu
OAvx=r,~=r__P.Osinh~ ) OAv';-
r---u= r--°-°sinh
~o)
(7)
where hc is EHD film thickness which is obtained by the formula proposed by Chittenden, et al.[14] to take the entrainment angle into consideration and 7"0, re and r1 denote the Eyring stress, the resultant shear stress(,v/r ~ + r~) and the viscosity, respectively. The viscosity of lubricant under high pressure is obtained by Roelands relationship[ 15]. From the assumption mentioned above, film temperature rise A0 at each point is calculated by N
l ~/T2 + TZ(hez - z2)dz AO= fo h~-~ere
(8)
where ke is the thermal conductivity of the lubricant. If the geometries, velocities and lubricant properties are given, the shear stresses, film thicknesses and film temperature rise can be obtained by solving Eqns. (4)--,(8) simultaneously. In calculation of the film thickness, mean temperature over a whole contact surface is employed. As the shear stresses being obtained, forces and moments locally acting the ball caused by the lubricant shear stresses are calculated by
dF= [T-t]
Pma~ ru il
_ ......... (~L) z
( bp)_.. 2
(9)
where m and d denote the mass of the ball and the moment of inertia, respectively, and Ai, Ao are the contact regions of both the inner and outer side, respectively. In addition, axial force balance of the inner race and geometric constraint are required to solve the ball motion.
Fa - / a d F . e,: = 0
(13)
where Fa is the axial load and ex denotes a unit vector along the axial direction.
rm,=Ri + ri - (7"/ - rb + ~e, -- he,)cos fli =Ro - ro + (to - rb + ~,o - hco)cos/30
(14)
where R, r, rb, 6e and/3 are a radius of the race, a radius of curvature of the race groove, a radius of ball, an elastic deflection and a contact angle, respectively. The radius of the inner race Ri varies with the inner race rotational velocity because of centrifugal expansion. If the inner ring is considered as the cylindrical element with an inner radius of rl and an outer radius of r2, the elastic expansion at the radius Ri is written as[16],
R~ = R~l~=0 + ~c pwi 2
x
(15)
/~ (3 + a ) ( l - ~)
8 1 +ff r12r22 rl + r22 + 1 - a R~ 2
E
l +or R2 ] 3+O" J
(16)
where p is the density of the inner race and E is the elastic modulus. The elastic deflection at the contact point is written as 2
d M = rp × d F
(10)
5.2. Equilibrium of Forces and Moments From the balance of the forces and moments acting on the ball, next equations are introduced.
[ o]
,
o
mrmw~
(11)
(12)
~e =
K~ sin/3
(17)
If the geometries of the contact bodies are known, Hertzian contact theory gives bearing contact stiffness, 1(~. Consequently, the ball motion can be determined by solving the equations described above simultaneously.
5.3. Starved Film Thickness As referring previous section, the film thickness, h0 at which the hydrodynamic pressure distribution commences decreases with decreasing in the oil supply rate under the oil and air lubrication. So, effects of
248
this film thickness, h0, on the ball motion should be revealed. The relationships between h0, a meniscus distance, Xi, and the central film thickness, hct were derived by the results of rigid, pseudo isoviscous theory for point contacts by Chiu[6]. Namely, next two equations were introduced. 5"5vr/c~/7"J1/2 12vrt°tXi - 1.0(18) hot3/2(3 + 2k) (3 + 2k)(hc ~+ Xi2/2t~) 2 2(2 + k) 2h0 = ~ h 0 3 + 2k
+
kvXi2 (3 + 2 k ) / ~
(19)
(
where b is contact half width along the semi-major axis. Starvation factor, ~b, is given by
( Xi/b - 1 ) °'29 X~/b- I
(21)
So, the film thickness, hst, under the starved condition can be written below
h,,t = r~he
10.0 ~E 5.0
where Ry is an equivalent radius of the contact along the rolling direction and k is the ratio of the equivalent radii of the contact. From these two equations, the meniscus distance, Xi, is obtained for various h0. The central film thickness, h~, obtained by above equations is not used in this analysis. The starved film thickness is obtained by the following method instead. Under fully flooded condition, the inlet meniscus distance is described by Hamrock and Dowson[5].
¢=
Log-linear plots of mean shear stress ~ under unidirectional uniform shear against the shear rate 5' provide the value of the Eyring stress by assurning an isothermal condition[ 18]. In Fig. 7, measured values of the Eyring stress are plotted.
(22)
The ball motion analysis under the starved condition can be carried out by making use of h~,t instead of he. 6. CALCULATION RESULTS AND DISCUSSION Calculations are performed in a velocity range !0rps-150rps about two kinds of oil(VG22, VG32). The value of Eyring stress which is required to estimate the shear stress distribution is measured by making use of a traction drive test apparatus[17].
o o
c
0 400
Fig. 7
600
80O
1000
Mean contact pressure, MPa Eyring stresses at several pressure conditions measured by simple traction drive
test. The shear forces and the moments are numerically integrated by S impson's rule. Equilibrium equations are solved by using Newton-Raphson method. Figure 8 shows the calculating results of Wb/wi, he, and AO at ambient temperature of 25°C and the axial force of 162N with the various values of the inlet film thickness, h0, where the values of h0 on the both races are assumed to be equal. Calculating results indicate that the reduction of velocity ratio, Wb/a;i, is significantly affected by the inlet film thickness, h0. When the inlet film thickness, h0, less than 0.6pro, the value of the inlet meniscus length, Xi/b, calculated by Eqns. (18) and (19) becomes less than unity with increasing in the rotational speed. In these cases, it is assumed that Xi = b. This seems to assume parched film lubrication proposed by Kingsbury[ 19]. From the calculating results of wb/a~i, it is found that when the thick inlet film exists, the reduction of the velocity ratio, a;b/o.;i does not occur, This is because that, at the thicker film more than 0.8pm, the gyroscopic slips have already occurred before beginning of starvation. The effects of the oil viscosity are shown in Fig. 9. It is clearly seen that the oil viscosity has a large effect on the gyroscopic slip. Starvation factors obtained under the same condition are also plotted in Fig. 9.
249
Increment of wb/wi as a function of the rotational speed in experimental results is steeper than that of these calculating results. In experiments, as there is remarkable ambient temperature rise when the gyroscopic slip occurs, the viscosity of the oil in the bearing seem to become lower in accordance with the temperature rise. The reduction of the viscosity makes increase in wb/wi as mentioned above. Then, at a few values of the ambient temperature, the velocity ratios are calculated as shown Fig. 10. Evidently, the velocity ratio increases with increasing in the ambient temperature rise. Accordingly, the gyroscopic slip can facilitate itself because of accompanying with considerable shear heat. From both the experimental results and the calculating results, a relationship between the oil supply rate and the starvation factor can be estimated. In Fig. 11, solid lines and dotted lines represent the velocity ratio as function of the starvation factor, which are analytically obtained. Also, the velocity ratio as function of the oil supply rate can be experimentally obtained at the same rotational speed. Therefore, the starvation factor with the oil supply rate can be obtained through the velocity ratio, although it is necessary to correct the former relationship with the ambient temperature rise. By means of this method, the starvation factor with the oil supply rate is estimated under the condition at which the rotational speed is 110rps as shown in Fig. 12. It is found that a strong correlation between the starvation factor and the oil supply rate. The relationship is approximately written as,
dp ,,~ (kQi)°"34"°"37
,... m . . . . . . . . . . . .
ho= 1.4 ~ ho= 1.2 p.m
m
J,J. . . . .
~
S
ho=o.s ~m-----..____~//// /
~0.4~m~// h~o.2~m'~//ZT//
3.7
.................
3.6
0
. .......-
.......
outer control theory
100 Inner ring rotational speed, rev/sec L
,
.............. 5 0
.
.=_......I
~
.__
150
(a) 1.2
r~
0.8 s
.~
o.4
50 10,0 Inner ring rotational speed, revlsec
150
(b)
....
r,.)
h : IAIxm n,~ t.2 ~m b'-
- Inner ring
Outerring
h0=!
~
.. "
4 / j4
2 ~
--"-
"
ho~0.6 ~
.
t_ "
,~0.2l.um ko=0.1 pm
[-, O
L
0
(24)
where k' is the coefficient which has dimension of reciprocal of volume. It is very interesting that an exponent of the fight hand of the eqn. (23) is similar value as that of the eqn. (24). This conclude that, under the oil and air lubrication, the oil amount in the ball bearing is almost proportional to the oil supply rate.
......
3.7
(23)
where k is the coefficient which has dimension of time and reciprocal of volume. Akagami and Aihara[20] carded out film thickness measurement in a ball bearing with minute amount of oil(0.0 l,,,20mm3). By estimating the relationship between the starvation factor and the oil amount V, not the supply rate, from their experimental results, the following is obtained. q~ ~, ( k / V ) 0"39
3.8
__
....~....:..._...I _.
.
50
.
L . .
.-
1O0 Inner ring rotational speed, rev/sec
150
(c) Fig. 8
Calculating results for viscosity of 0.027Pas. (a) angular velocity ratio ~b/~i, (b) film thickness developed on each race, (c) temperature rise at each race.
250
3.s. ................................. , ...... t
.
.
.
.
.
.
.
l .............. ,o.o
..
.
~%,.//?.?;~ /.,"-/i
3.75P
3"650 ....................5 ' 0
....
"
I~
"
Inner ring rotational speed, rev/sec (a)
................. '. . . .
lo 0
y , . "-.. ---._~
t6¢4
"
-i50
3"650
ho= /.0~1'[ ~o.~
100
50
150
Inner ring rotational speed, rev/se¢ (a)
lo 0
=o
",l:i
¢r$
r.,o
10-1 0
50 100 Inner ring rotational speed, rev/sec (b)
10-1
150
0
50 100 Inner ring rotational speed,roy/see (b) l~...
bo=.LO Im
100 \
¢.)
'.-..,
..,..,.
;
:..........~.
1o0
.-,.-..
X.
10-t
" - - . . . . . . . 0.018SPas
0
150
0
(0 Fig. 9
_
Viscosity effects on velocity ratio, ~ b / o ~ i , and starvation factor. (a) angular velocity ratio wb/o.,i, (b) starvation factor on inner race, (c) starvation factor on outer race.
Fig. 10
r
'q,
.-
.
.
= ......
.........
I
"~ll
. .
Illlll m
.../ho=
.
:....
t.olzm
i
""":~::':::2:2:::'::~.::;:
" " : . A j,: .....
-":L.~
10--I .............. ~ ° c , _-;. 350c
ho= 0 ! p m
50 100 Inner ring rotational speed, rev/sec
[
" ' ' ; L , , - ......
~~ r~
150
,
b0=0,~
h0=0.6lzm
.......
~.......
50 100 Inner ring rotational speed, rev/sec (c)
":"
150
Ambient temperature effects on velocity ratio, ~b/~i and starvation factor. (a) angular velocity ratio ¢Mb/Wi,(b) starvation factor on inner race, (c) starvation factor on outer race.
251
I 3 74~
4. Comparing the results of the model analysis with those of experiments, the variation of the starvation factor with the oil supply rate is estimated. In this experimental condition, the relationship between the starvation factor and the oil supply rate is represented as equation (23). These results suggest that the oil amount in the bearing is approximately proportional to the oil supply rate under oil and air lubrication.
............ Outer
L
0
Fig. I 1
.0
~
0.5
o
0.2
0.5
..... • •
0
C
• 0
0.05 0.2
0
ACKNOWLEDGEMENT The authors wish to acknowledge the financial support by Takahashi Foundation. The oil and air lubrication system is provided by Daido Metal Co.Ltd. This study is supported by a Grant-in-Aid for scientific Research(No. 0975166) from Japanese Ministry of Education.
0
REFERENCES
• Inner
m 0.1
Fig. 12
1
Starvation factor Relationship between velocity ratio and starvation factor with thermal correction for 110rps.
o Outer 0.5 1.0 2.0 5.0 10 Oil supply rate, mm 3/rain
[ [ 20
Relationship between velocity ratio and
starvation factor with thermal correction for 110rps. 7, CONCLUSIONS The conclusions drawn from this study are summarized as follows, 1. The gyroscopic slip at the ball-race contact in the angular contact ball bearing is strongly influenced by the oil supply rate under the oil and air lubrication. 2. The oil injection rate into the beating is equivalent to the oil ejection rate from the bearing. So, the total oil amount in the ball bearing has been kept constant. This oil amount is dependent on the oil supply rate. 3. The quasi-static model analysis of the ball motion in various values of the film thickness at inlet oil layer can predict the reduction of the velocity ratio of the ball to the inner race with oil supply rate.
1. Aramaki, H, Shoda, Y, Morishita, Y, and Sawamoto, T, "The Performance of Ball Bearings with Silicon Nitrite Ceramic Ball in High Speed Spindles for Machine Tools," ASME, J. Tribo., Vol. 110, 1988, pp.693-698. 2. Wolveridge, P. E., Baglin, K. P., and Archard, J. E, "The Starved Lubrication of Cylinders in Line Contact," Proc. Instn. Mech. Engrs., Vol. 185, 81/71, 1970, pp. 1159-1169. 3. Archard, J. E, and Baglin, K. P., "Nondimensional Presentation of Frictional Tractions in Elastohydrodynamic Lubrication- Part II: Starved Conditions," ASME J. Lub. Tech., Vol. 97, 1975, pp. 412-423. 4. Goksem, P. G., and Hargreaves, R. A., "The Effect of Viscous Shear heating on Both Film Thickness and Rolling Traction in an EHL Line Contact," ASME, J. Lub. Tech., Vol. 100, 1978, pp. 353-358 5. Hamrock, B. J., and Dowson, D., "Isothermal Elastohydrodynamic Lubrication of Point Contact." ASME, J. Lub. Tcch., Vol. 99, 1977, 15-23. 6. Chiu, Y. P., "An Analysis and Prediction of Lubricant Film Starvation in Roiling Contact System," ASLE Trans., Vol. 17, 1974, pp. 22-35. 7. Olaru, D. N., and Gafitanu, M. D., "Starvation in Ball Bearing," Wear, Vol. 170, 1993, pp.219-234. 8. Pasdari, M, and Gentle, C. R., "Effect of Lubricant Starvation on the Minimum Load Condition in a Thrust-Loaded Ball Bearing," ASLE Trans. Vol. 30, 1987, pp. 355-359. 9. Hirano, E, "Motion of a Ball in Angular-Contact Ball Bearing," ASLE Trans. Vol. 8, 1965, 425-434.
252
10. Kawamura, H., Touma, K., "Motion of Unbalanced Balls in High-Speed Angular Contact Ball Bearings," ASME, J. Trib., Vol. 112, 1990, pp. 105-110. 11. Ohno, N, Kuwano, N, and Hirano, E, "Diagrams for Estimation of the Solidified Film Thickness at High Pressure EHD Contacts," Proc. of the 20th Leeds-Lyon Symposium on Tribology, 1993, pp.507-518. 12. Johnson, K. L., Tevaarwerk, J. L., "Shear Behavior of Elastohydrodynamic Oil Films, " Proc. R. Sot., Vol. A356, 1977, pp. 215-236. 13. Houpert, L., "Fast Numerical Calculations of EHD Sfiding Traction Forces; Application to Rolling Bearings," ASME, J. Trib., Vol. 107, 1985, pp. 234-240. 14. Chittenden, R. L, Dowson, D., Dunn, J. F., and Taylor, C. M., "A Theoretical Analysis of Isothermal EHL Concentrated Contacts: Part II," Proc. R. Soc. Lond., Vol. A387, 1985, pp. 271-294. 15. Houpert, L., "New Results of Traction Force Calcularions in Elastohydrodynamic Contacts," ASME, J. Tribo., Vol. 107, pp.241-248. 16. Gupta, P. K., "Advanced Dynamics of Rolling Elements," Springer-Verlag, 1984, pp. 212 17. Nakamura, T., Itoigawa, F., and Matsubara, T., "Estimarion of Pressure Viscosity Coefficient by High Speed and High Pressure Traction Drive Test," Proc. of the 23rd Leeds-Lyon Symposium on Tribology, 1996, pp. 253-259. 18. Evans, C. R., Johnson, K. L., "Regimes of Traction in Elastohydrodynamic Lubrication," Proc. instn. Mech. Engrs., Vo!. 200, C5, 1986, pp.313-324. !9. Kingsbury, E., "Parched Elastohydrodynamic Lubrication, "ASME, J. Trib., Vol. 107, 1985, 229-233. 20. Akagami, K, and Aihara, S., "Experimental Study on Starved EHL in a Small Ball Bearing Lubricated with Minute Amount of Oil," Proc. Int. Trib. Conf. in Yokohama, 1995, pp. 1023-1028.
243
Starvation in Ball Bearing Lubricated by Oil and Air Lubrication System F. Itoigawa, T. Nakamura and T. Matsubara a aDepartment of Mechanical Engineering, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya 466 Japan An oil and air lubrication system for ball bearings supporting a high speed spindle maintains friction losses and temperature rises in low level comparing with other lubrication systems, e.g., oil jet or oil mist lubrication. In this study, rotating speeds of the ball in an angular contact ball bearing lubricated by the oil and air lubrication system are observed in various oil supply rates. In addition, quasi-static model analysis of the ball motion is carried out. Experiments indicate that the angular velocity of the ball varies with the oil supply rate even at the constant spindle speed. Furthermore, the model analysis suggests that the ball angular velocity is considerably concerned with an inlet film thickness. From the both results, a relationship between a starvation factor in the ball-race contacts and the oil supply rate is derived for the ball bearing under the oil and air lubrication.
1. I N T R O D U C T I O N In a last decade, many manufacturers of machine tools have employed an oil and air lubrication system for support bearings in a spindle. The oil and air lubrication is a method in which both a very small amount of oil, for example 10ram 3/min a bearing, and a large amount of compressed air are simultaneously supplied to the bearings through one nozzle for each bearing. Under the oil and ~ lubrication, it is known that an inlet to the conjunction between a ball and race is starved when the oil supply rate becomes smaller or the race rotational speed becomes higher. And a main cause of bearing temperature rise under the oil and air lubrication is shear heating of EHD films in the conjunctions owing to gyroscopic slip and spin slip, as drug loss and churning loss are negligibly small[l ]. So, when the bearing is running in the starved condition at low oil supply rate, a reduction of oil film thickness brings very low temperature rise. However, severe starvation induces metallic contacts or, in the worst case, bearing seizure. Therefore, an adequate oil supply rate must be selected in practical spindle bearings. Unfortunately, a variation of the degree of the starvation with the oil supply rate has not understood quantitatively under the oil and air lubrication. So, the oil supply rate has been selected empirically. There are many studies about the starvation condition in non-conformal contact. Wolveridge et al.[2]
and Archard and Baglin[3] semi-analytically presented an influence of the starvation on film thickness in line contact. Goksem and Hargreaves[4] introduced a starved film thickness including an effect of viscous shear heating. In point contact, a film thickness reduction factor caused by starvation was developed by Hamrock and Dowson[5]. These studies provided the film thickness in the starved condition as a function of an inlet meniscus length. More practical research on the starvation in point contact was developed by Chiu[6]. He predicted an effect of an oil replenishment of the rolling track. Furthermore, Olaru and Galitanu[7] and Pasdari and Gentle[8] investigated the starvation in the actual ball bearing. The former carried out the experiments under oil mist lubrication, the latter did under oil jet lubrication. An ultimate purpose of this study is to specify the optimum oil supply rate in the oil and air lubrication. In this paper, a relationship between the degree of the starvation and the oil supply rate is investigated by observing the ball motion in experiments and by quasistatic analyses of the ball motion under conditions in which the bearing is insufficiently lubricated.
2. OIL AND AIR LUBRICATION SYSTEM For the machine tools, the oil and air lubrication system has two advantages comparing with other lubrication systems, for example, an oil mist lubrica-
244
tion system, or an oil jet lubrication system. First, this lubrication system is superior to the other lubrication systems in size and cost because an oil cooler, oil recirculating devices and complicated piping are not necessary. Second, temperature rise caused by drug loss and churning loss is suppressed at relatively low level because of a small supplied oil amount. This is so valuable from a viewtx~int of precision preservation that many high speed and high precision machine tools have come to be equipped with the oil and air lubrication system. In addition, the oil and air lubrication is superior to grease lubrication with regard to maintainability, since new oil without deterioration by oxidizing is supplied continuously and stably. The oil and air lubrication system consists of a distributor and an oil pressure equipment as schematically shown in Fig. 1. The distributor discharges a very small amount of oil measmv.a by reciprocating motion of a constant-quantity piston into pressurized air. The reciprocating motion of the constantquantity piston is given by hydraulic pressure which is intermittently introduced into the piston by a solenoid valve. So, the oil supply rate is adjusted by changing an oil discharge interval. As a tube in which the oil and air flow is made of material which possesses low wettability against the oil such as PTFE, many small oil drops are formed in the tube. 3. TEST BEARING AND SPINDLE The test bearing is a high speed angular contact bali bearing(7010C) which has a bore diameter of 50ram and an outer race diameter of 80ram. 19 balls and a cage made of phenolic resin are assembled in the bearhag. Measured bearing geometries are listed in Table 1. Only one of the balls is magnetized in order to detect ball angular velocity by a search coil method proposed by Hirano[9] as shown in Fig. 2. A constant load is applied by a low stiffness coil spring in an axial direction only. Generally, the bearings are lightly pre-loaded in a practical high speed machine tools. So, in this study, the axial load is applied with a relatively light force of !62N. Under this load condition, mean contacting pressure is about 400MPa and a contact half width along a semi-major axis is about 0.4ram. The oil and air are injected into the inner race groove tlu'ough a nozzle mounted on the bearing
holder. The range of the oil supply rate is from 0.5mm3/min to 20mm3/min and an air flow rate is 20Nt/min regardless of the oil supply rate. A spindle can rotate at any speed up to 250rev/sec. In addition, outer race temperaaue is monitored by a thermocouple. Table l
Test bearing geometries
Ball Diameter Inner ring radius Outer ring radius Inner groove curvature radius Outer groove curvature radius
Air m~dm'l
i ~
"
! P:~..
8.7313 mm 28.115 mm 36.849 mm 4.4529 mm 4.5403 mm
Compre~e.d air Constant quantity
-.--' U ', piston
~ - i l
Fig. l
~ Solenoidvalve
Schematic illustration of oil and air lubrication system
Magnetizedball Spindle, " X
~ ttt Thermo-couple ~3-'~=j,./Bearing hol¢ter
~'i-: " ~ _ _ ~
Nozzle
Air
~ ~ '~ \ ' Thrustload ~.,--~'<~ i...-/'P--':.:Tj.--i ........... k \ adjustscrew
Fig. 2
Test al~aratus
4. VARIATION OF GYROSCOPIC SLIP WITH OIL SUPPLY RATE In the lightly loaded angular ball beating, a gyroscopic moment acting on the balls has a considerable effect on slip in the ball-race conjunction. In this case, ball motion substantially differs from that being predicted by a control theory[9][10]. Although the control theory expects no spinning motion on ether the
245 outer race or the inner race regardless of the rotational speed, the spinning motion, in fact, exists on the both races. In addition, spin velocities at the both contact points which generate large slip between the ball and races vary with the rotational speed. The slip at the contact point, what is called gyroscopic slip or gyroslip, can be easily confirmed by observing the ball angular velocity. Figures 3 and 4 show experimental results of the variations of angular velocity ratio of the ball to the inner race, wb/wi, caused by the gyroscopic slip with the inner ring speed. From these results, it is found that the velocity ratios, wb/wi, are far from the outer race control theory when the inner ring rotates at high speed. Furthermore, it is very noticeable that the magnitude ofthe velocity ratio, w~/wi obviously decreases and comes cross to the theory with decreasing in the oil supply rate. The magnitude of a separation from the curve of the outer race control theory is influenced by EHD traction in the ball-race conjunction. Therefore, the higher viscosity of the lubricant or the thinner the EHD film(the higher shear rate), the smaller this separation. As the experiments are carried out under same conditions except for the oil supply rate, it is considered that the reduction of the velocity ratio, wb/wi, with decreasing in the oil supply rate in Figs. 3, 4 shows the decrease in the EHD film thickness. Consequently, the oil supply rate dominates lack of the lubricant at the inlet, that is, the degree of the starvation, under the oil and air lubrication. Even if the oil is continuously supplied for a long time, that is, the total oil amount supplied to the bearing gradually increases, no change of the velocity ratio, wb/wi, appears. This means that the rate of the oil injection into bearing, Qin, is equal to that of the oil ejection from the bearing, Qout.
Q~,, Qo~,t =
(])
By the way, the ejection rate, Qout, is dominated by a formation rate of oil drops which part from side wall lubricant and size of the oil drop. Naturally, with increasing in the volume of the side wall, both the rate and the size also increase. In addition, a thickness of the side wall, Hb, influences the film thickness, ho, on the track after passage of the rolling ball as shown in Fig. 5. Chiu[6] derived this film thickness by taking flow-back from the side wall lubricant into consideration for ball and plane contact and Olaru and Gaff-
tanu[7] also did for the ball-race contact. They expressed a film thickness, h0, as the function of a time passed two successive balls in terms of the EHD film thickness, he, at a flat contact region,
Tt ho = he + Co 2rla
(2)
where T, ~ and a are the o i l - a i r interface surface tension, the viscosity of the oil and the half width along the semi-major axis of the contact ellipse, respectively. The coefficient Co, which was numerically obtained, varies with the side wall film thickness, Hb. From their calculating results, a variation of the coefficient, Co, with Hb is approximately linear, so next relationship is obtained. 3.80 °2.0 [114.0
_ ~
I n 10.0
vA
T20.0
Y
, oL t
A rn I
=
.'
. o
"
•
."
"°:'-"\
,
~
"
"
e
V~._
| 06.0
3.75I-"8.0
I
o 8
~ll~o
~
cont~oit.eo,y
3.65 ! ...........~ ~- ~U~l: ................................~................ 0 50 100 150 200 250 Inner ring rotational speed, rps Fig. 3 Gyroscopic slip variation with the inner ring rotational speed, oil viscosity = 0.027Pas; axial load = 162N. "
3.80
..... V :
"0.5
mm3/min
W ~
m|.o
'~ i
Q 3.0 a I0.0
~ 'v
~" 3.75 - v 14.0
"I
.......
II
o *
* .
Ii~
g
~
m
/
3.70
V
............................... ~ ' " l j ' ~
O
*
..
outer race
3 . 6 5 .......................................................................': 'con-tr°! t~orY~
0
F~g. 4
50 100 150 Inner ring rotational speed, rps Gyroscopic slip variation with the inner ring rotational speed, oil viscosity = 0.019Pas; axial load = 162N.
246
2r/a ( h o - h¢)-~7, o¢ Hb
(3)
Consequently, it is considered that not the total amount of oil supply but the oil supply rate controls the film thickness at the inlet oil layer. In other words, the oil supply rate can control the degree of the starvation under the oil and air lubrication. 5. ANALYSIS OF G Y R O S C O P I C SLIP MOTION IN STARVED EH1L CONDITION To estimate the reduction of the velocity ratio, wb/wi, caused by the starvation, quasi-static analysis of the ball motion is carried out. Before establishing the quasi-static model of the ball motion, a few assumptions are introduced as follows, I. A drug force and a churning force are ignored because of a very small oil amount. 2. There is no interactions between the ball and the cage, 3. Shear behavior of the lubricant in the conjunction is non-linear viscous because the product value of pressure-viscosity coefficient,c~ and average pressure, ~ is less than 13[ 11 ]. 4. All of the bearing element have same temperature and a uniform temperature distribution. From this assumption, effects of the thermal expansion are negligible. In addition, temperature rise at the contacting surface is ignored. 5. Compression heating and convection cooling in the EHD film are ignored. 6. An elastohydrodynamic pressure distribution assumes to be identical to a Hertzian pressure distribution.
Vb = Wb × rp
(4)
where Wb is a vector of the ball angular velocity and rp denotes a vector which locates any point in the contact ellipse relative to the ball center. On the other hands, a velocity vector of the race surface, v,., is, (wi - w e ) × (rm + rp + 8h,) (inner) - a ' e × (rrn + rp + ¢Iho) (outer)
"O r
(5)
where vector rrn locates the ball center relative to the race center and ~ih shows an incremental vector caused by the EHD film thickness. By making use of transformation matrix [T] from the azimuth frame to the local contact frame, a local slip velocity vector, Au, is given by Au
=
[T](v~ - v,)
(6)
ho=h0r+h0b ~ ,..~ . . . . . _ ~ ~ ~
.........................................
~
bal~
rate
t t
High oil supply rate
!
Fig. 5
Rebounding oil layer shape in lateral to the track. 2a
To describe the model, two coordinate frames are employed as shown in Fig. 6. One is an azimuth frame with x axis along the bearing axis and z axis pointing radially outward through the ball center. The other is a local contact frame Op-xpypzp with z axis normal to the contact surface and ~: and y axis in the tangential plane at the contact. 5.1. Force and Moment Acting on Ball A velocity vector of the ball surface at a point in the contact ellipse, vb, is,
Hertzian contact ellipse
Yr Y
xp Race ~ groove," /
Fig. 6
Coordinate frames to describe the ball motion,
247
Lubricant shear behavior is described by the Eyring Model[ 12][ 13], that is,
I Ave' Avu
OAvx=r,~=r__P.Osinh~ ) OAv';-
r---u= r--°-°sinh
~o)
(7)
where hc is EHD film thickness which is obtained by the formula proposed by Chittenden, et al.[14] to take the entrainment angle into consideration and 7"0, re and r1 denote the Eyring stress, the resultant shear stress(,v/r ~ + r~) and the viscosity, respectively. The viscosity of lubricant under high pressure is obtained by Roelands relationship[ 15]. From the assumption mentioned above, film temperature rise A0 at each point is calculated by N
l ~/T2 + TZ(hez - z2)dz AO= fo h~-~ere
(8)
where ke is the thermal conductivity of the lubricant. If the geometries, velocities and lubricant properties are given, the shear stresses, film thicknesses and film temperature rise can be obtained by solving Eqns. (4)--,(8) simultaneously. In calculation of the film thickness, mean temperature over a whole contact surface is employed. As the shear stresses being obtained, forces and moments locally acting the ball caused by the lubricant shear stresses are calculated by
dF= [T-t]
Pma~ ru il
_ ......... (~L) z
( bp)_.. 2
(9)
where m and d denote the mass of the ball and the moment of inertia, respectively, and Ai, Ao are the contact regions of both the inner and outer side, respectively. In addition, axial force balance of the inner race and geometric constraint are required to solve the ball motion.
Fa - / a d F . e,: = 0
(13)
where Fa is the axial load and ex denotes a unit vector along the axial direction.
rm,=Ri + ri - (7"/ - rb + ~e, -- he,)cos fli =Ro - ro + (to - rb + ~,o - hco)cos/30
(14)
where R, r, rb, 6e and/3 are a radius of the race, a radius of curvature of the race groove, a radius of ball, an elastic deflection and a contact angle, respectively. The radius of the inner race Ri varies with the inner race rotational velocity because of centrifugal expansion. If the inner ring is considered as the cylindrical element with an inner radius of rl and an outer radius of r2, the elastic expansion at the radius Ri is written as[16],
R~ = R~l~=0 + ~c pwi 2
x
(15)
/~ (3 + a ) ( l - ~)
8 1 +ff r12r22 rl + r22 + 1 - a R~ 2
E
l +or R2 ] 3+O" J
(16)
where p is the density of the inner race and E is the elastic modulus. The elastic deflection at the contact point is written as 2
d M = rp × d F
(10)
5.2. Equilibrium of Forces and Moments From the balance of the forces and moments acting on the ball, next equations are introduced.
[ o]
,
o
mrmw~
(11)
(12)
~e =
K~ sin/3
(17)
If the geometries of the contact bodies are known, Hertzian contact theory gives bearing contact stiffness, 1(~. Consequently, the ball motion can be determined by solving the equations described above simultaneously.
5.3. Starved Film Thickness As referring previous section, the film thickness, h0 at which the hydrodynamic pressure distribution commences decreases with decreasing in the oil supply rate under the oil and air lubrication. So, effects of
248
this film thickness, h0, on the ball motion should be revealed. The relationships between h0, a meniscus distance, Xi, and the central film thickness, hct were derived by the results of rigid, pseudo isoviscous theory for point contacts by Chiu[6]. Namely, next two equations were introduced. 5"5vr/c~/7"J1/2 12vrt°tXi - 1.0(18) hot3/2(3 + 2k) (3 + 2k)(hc ~+ Xi2/2t~) 2 2(2 + k) 2h0 = ~ h 0 3 + 2k
+
kvXi2 (3 + 2 k ) / ~
(19)
(
where b is contact half width along the semi-major axis. Starvation factor, ~b, is given by
( Xi/b - 1 ) °'29 X~/b- I
(21)
So, the film thickness, hst, under the starved condition can be written below
h,,t = r~he
10.0 ~E 5.0
where Ry is an equivalent radius of the contact along the rolling direction and k is the ratio of the equivalent radii of the contact. From these two equations, the meniscus distance, Xi, is obtained for various h0. The central film thickness, h~, obtained by above equations is not used in this analysis. The starved film thickness is obtained by the following method instead. Under fully flooded condition, the inlet meniscus distance is described by Hamrock and Dowson[5].
¢=
Log-linear plots of mean shear stress ~ under unidirectional uniform shear against the shear rate 5' provide the value of the Eyring stress by assurning an isothermal condition[ 18]. In Fig. 7, measured values of the Eyring stress are plotted.
(22)
The ball motion analysis under the starved condition can be carried out by making use of h~,t instead of he. 6. CALCULATION RESULTS AND DISCUSSION Calculations are performed in a velocity range !0rps-150rps about two kinds of oil(VG22, VG32). The value of Eyring stress which is required to estimate the shear stress distribution is measured by making use of a traction drive test apparatus[17].
o o
c
0 400
Fig. 7
600
80O
1000
Mean contact pressure, MPa Eyring stresses at several pressure conditions measured by simple traction drive
test. The shear forces and the moments are numerically integrated by S impson's rule. Equilibrium equations are solved by using Newton-Raphson method. Figure 8 shows the calculating results of Wb/wi, he, and AO at ambient temperature of 25°C and the axial force of 162N with the various values of the inlet film thickness, h0, where the values of h0 on the both races are assumed to be equal. Calculating results indicate that the reduction of velocity ratio, Wb/a;i, is significantly affected by the inlet film thickness, h0. When the inlet film thickness, h0, less than 0.6pro, the value of the inlet meniscus length, Xi/b, calculated by Eqns. (18) and (19) becomes less than unity with increasing in the rotational speed. In these cases, it is assumed that Xi = b. This seems to assume parched film lubrication proposed by Kingsbury[ 19]. From the calculating results of wb/a~i, it is found that when the thick inlet film exists, the reduction of the velocity ratio, a;b/o.;i does not occur, This is because that, at the thicker film more than 0.8pm, the gyroscopic slips have already occurred before beginning of starvation. The effects of the oil viscosity are shown in Fig. 9. It is clearly seen that the oil viscosity has a large effect on the gyroscopic slip. Starvation factors obtained under the same condition are also plotted in Fig. 9.
249
Increment of wb/wi as a function of the rotational speed in experimental results is steeper than that of these calculating results. In experiments, as there is remarkable ambient temperature rise when the gyroscopic slip occurs, the viscosity of the oil in the bearing seem to become lower in accordance with the temperature rise. The reduction of the viscosity makes increase in wb/wi as mentioned above. Then, at a few values of the ambient temperature, the velocity ratios are calculated as shown Fig. 10. Evidently, the velocity ratio increases with increasing in the ambient temperature rise. Accordingly, the gyroscopic slip can facilitate itself because of accompanying with considerable shear heat. From both the experimental results and the calculating results, a relationship between the oil supply rate and the starvation factor can be estimated. In Fig. 11, solid lines and dotted lines represent the velocity ratio as function of the starvation factor, which are analytically obtained. Also, the velocity ratio as function of the oil supply rate can be experimentally obtained at the same rotational speed. Therefore, the starvation factor with the oil supply rate can be obtained through the velocity ratio, although it is necessary to correct the former relationship with the ambient temperature rise. By means of this method, the starvation factor with the oil supply rate is estimated under the condition at which the rotational speed is 110rps as shown in Fig. 12. It is found that a strong correlation between the starvation factor and the oil supply rate. The relationship is approximately written as,
dp ,,~ (kQi)°"34"°"37
,... m . . . . . . . . . . . .
ho= 1.4 ~ ho= 1.2 p.m
m
J,J. . . . .
~
S
ho=o.s ~m-----..____~//// /
~0.4~m~// h~o.2~m'~//ZT//
3.7
.................
3.6
0
. .......-
.......
outer control theory
100 Inner ring rotational speed, rev/sec L
,
.............. 5 0
.
.=_......I
~
.__
150
(a) 1.2
r~
0.8 s
.~
o.4
50 10,0 Inner ring rotational speed, revlsec
150
(b)
....
r,.)
h : IAIxm n,~ t.2 ~m b'-
- Inner ring
Outerring
h0=!
~
.. "
4 / j4
2 ~
--"-
"
ho~0.6 ~
.
t_ "
,~0.2l.um ko=0.1 pm
[-, O
L
0
(24)
where k' is the coefficient which has dimension of reciprocal of volume. It is very interesting that an exponent of the fight hand of the eqn. (23) is similar value as that of the eqn. (24). This conclude that, under the oil and air lubrication, the oil amount in the ball bearing is almost proportional to the oil supply rate.
......
3.7
(23)
where k is the coefficient which has dimension of time and reciprocal of volume. Akagami and Aihara[20] carded out film thickness measurement in a ball bearing with minute amount of oil(0.0 l,,,20mm3). By estimating the relationship between the starvation factor and the oil amount V, not the supply rate, from their experimental results, the following is obtained. q~ ~, ( k / V ) 0"39
3.8
__
....~....:..._...I _.
.
50
.
L . .
.-
1O0 Inner ring rotational speed, rev/sec
150
(c) Fig. 8
Calculating results for viscosity of 0.027Pas. (a) angular velocity ratio ~b/~i, (b) film thickness developed on each race, (c) temperature rise at each race.
250
3.s. ................................. , ...... t
.
.
.
.
.
.
.
l .............. ,o.o
..
.
~%,.//?.?;~ /.,"-/i
3.75P
3"650 ....................5 ' 0
....
"
I~
"
Inner ring rotational speed, rev/sec (a)
................. '. . . .
lo 0
y , . "-.. ---._~
t6¢4
"
-i50
3"650
ho= /.0~1'[ ~o.~
100
50
150
Inner ring rotational speed, rev/se¢ (a)
lo 0
=o
",l:i
¢r$
r.,o
10-1 0
50 100 Inner ring rotational speed, rev/sec (b)
10-1
150
0
50 100 Inner ring rotational speed,roy/see (b) l~...
bo=.LO Im
100 \
¢.)
'.-..,
..,..,.
;
:..........~.
1o0
.-,.-..
X.
10-t
" - - . . . . . . . 0.018SPas
0
150
0
(0 Fig. 9
_
Viscosity effects on velocity ratio, ~ b / o ~ i , and starvation factor. (a) angular velocity ratio wb/o.,i, (b) starvation factor on inner race, (c) starvation factor on outer race.
Fig. 10
r
'q,
.-
.
.
= ......
.........
I
"~ll
. .
Illlll m
.../ho=
.
:....
t.olzm
i
""":~::':::2:2:::'::~.::;:
" " : . A j,: .....
-":L.~
10--I .............. ~ ° c , _-;. 350c
ho= 0 ! p m
50 100 Inner ring rotational speed, rev/sec
[
" ' ' ; L , , - ......
~~ r~
150
,
b0=0,~
h0=0.6lzm
.......
~.......
50 100 Inner ring rotational speed, rev/sec (c)
":"
150
Ambient temperature effects on velocity ratio, ~b/~i and starvation factor. (a) angular velocity ratio ¢Mb/Wi,(b) starvation factor on inner race, (c) starvation factor on outer race.
251
I 3 74~
4. Comparing the results of the model analysis with those of experiments, the variation of the starvation factor with the oil supply rate is estimated. In this experimental condition, the relationship between the starvation factor and the oil supply rate is represented as equation (23). These results suggest that the oil amount in the bearing is approximately proportional to the oil supply rate under oil and air lubrication.
............ Outer
L
0
Fig. I 1
.0
~
0.5
o
0.2
0.5
..... • •
0
C
• 0
0.05 0.2
0
ACKNOWLEDGEMENT The authors wish to acknowledge the financial support by Takahashi Foundation. The oil and air lubrication system is provided by Daido Metal Co.Ltd. This study is supported by a Grant-in-Aid for scientific Research(No. 0975166) from Japanese Ministry of Education.
0
REFERENCES
• Inner
m 0.1
Fig. 12
1
Starvation factor Relationship between velocity ratio and starvation factor with thermal correction for 110rps.
o Outer 0.5 1.0 2.0 5.0 10 Oil supply rate, mm 3/rain
[ [ 20
Relationship between velocity ratio and
starvation factor with thermal correction for 110rps. 7, CONCLUSIONS The conclusions drawn from this study are summarized as follows, 1. The gyroscopic slip at the ball-race contact in the angular contact ball bearing is strongly influenced by the oil supply rate under the oil and air lubrication. 2. The oil injection rate into the beating is equivalent to the oil ejection rate from the bearing. So, the total oil amount in the ball bearing has been kept constant. This oil amount is dependent on the oil supply rate. 3. The quasi-static model analysis of the ball motion in various values of the film thickness at inlet oil layer can predict the reduction of the velocity ratio of the ball to the inner race with oil supply rate.
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