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Motion of a ball in a ball bearing
WEAR
MOTION
OF A BALL F. HIRANO K_@shii
IN A BALL
BEARING*
AND H. TANOUE
University,
(Received
I77
Fukuoka
October
IO,
(Japan) 1960)
SUMMARY A magnetized ball rolling in a ball bearing induces an alternating current in a coil wound on the outer ring. The period of the current represents one revolution of the ball. Deviation from ideal motion of the ball occurs as slip and spin. The former is shown by the prolongation of the period. The latter causes fluctuation of the rolling axis relative to the magnetic axis of the ball. It can be easily estimated by measuring the variation of the amplitude of the current. Under radial load a considerable slip occurs at the unloaded side. It is found that the slip decreases exponentially with increase in speed. The magnitude and the direction of load, the lubicating condition and the radial clearance also affect the slip. In the special case where the ratio of the number of revolutions of the ball to that of the cage becomes nearly equal to an integral value, the deviation of the rolling axis shows a sharp increase, which often exceeds 70”. This deviation is due to the spinning motion of the ball during slipping. ZUSAMMESFASSUNG Eine in einem Kugellager rotierende magnetisierte Kugel induziert einen Wechselstrom in einer urn den Aussenring gewickelten Spule. Die Periode des Stroms entspricht einer Rotation der Kugel. Abweichung von der idealen Bewegung der Kugel entsteht in zwei Formen: Gleiten und Kreiselbewegung. Gleiten wird sichtbar an einer Verlangerung der Periode. Die Kreiselbewegung verursacht eine relative Schwankung der Drehachse zur magnetischen Achse der Kugel und ist durch Messung der Veranderung der Amplitude des Stromes leicht zu finden. Bei Radialbelastung tritt an der unbelasteten Seite ein betrachtliches Gleiten ein, das mit zunehmender Rotationsgeschwindigkeit rasch abnimmt. Ferner wird Gleiten durch GrSsse und Richtung der Belastung, durch den Schmierzustand und die Radialspielung des Lagers beeinflusst. Wenn das Verhlltniss der Drehzahl der Kugel zu der des Kafigs annahrend gleich einer ganze Zahl ist, nimmt die Richtungslnderung der Drehachse erheblich iu, manchmali mehr als 70”. Ursache ist das Auftreten einer Kreiselbewegung der Kugel beim Gleiten. INTRODUCTION
The motion motion.
of rolling
Depending
elements
on operating
in a rolling conditions,
deviation
should have an important
elements.
Therefore,
basic information bearing,
because
deviates
influence upon the friction,
an investigation
on rolling bearings. observation
bearing
somewhat
wear and life of rolling
of the motion of rolling elements The situation
of the motion
from ideal
slip and spinning motion take place. This is required
for
is simple in the case of a roller
is easy. HAMPP~
reported
the results
of
* Published in Japanese in J. Japan. Sm. Lubrication Engrs., 5 (1960) 175 and Tvans. Japan. Sot. Mech. Engrs., 27 (1961) (in the press). The last section of the paper has not been published previously. IFear, 4 (1961) r77-197
17s
1:. t111<.\\0,
measurement capacitance.
of the motion of rollers in ;I connecting
tracing
motion
on the other
and observation
previously
the motion
and special
rod bearing 1,~. mr~ans of ~~ai-iabl~~
and analyst$d the slip of the roller.
In the cast’ of ball bearings, dimensional
l.\SOl’l~.
SasitKrz recorded the locus of a point on the edge of 3 roller I)\. using 311
optical method,
the authors
I-1.
reported
of the motion
that a magnetized
of the balls.
attention
hand, balls have the freedom
Recently
becomes
more difficult.
ball could successfully
the experiments
has been given to the spinning
of t lm.t?-
hail
motion
One ot
I9c used
for
been rtbsumecl ant1
of balls.
The present
paper deals mainly with the results under radial load. The case of thrust ant1 cmnbinml load is now under investigation
and will be reported .S(~MET<‘I . A
=
am~iit~de
,>
on
later.
YI-I’RE
of oscillo~~m
(mm)
=
residual magnetism
1
shaft speed or speed of inner ring (revjminj
of ball {C)
=
load (kg)
=- radius of inner raceway
(mm)
.= radius of outer raceway
(mm)
:
mean slip r’,;)
--
number of revolutions
=
number
of ball relative to cage (rev/mm)
of revolutions
of cage or number of revolutions
of ball
ccntre (rev/mm) :
frequency
=
radius of ball (mm)
of fluctuation
of amplitude
(c/minj
= angle between rolling axis and magnetic
axis of ball (deg.)
--_ change oflw (deg.) =
phase shift of a definite point ou the ball for each revolution
of
the cage {radians) =
angular displacemel~t of spin (radians or deg.)
=
theoretical
angular
velocity
of the ball relative
to
the
cage
to thr
cage
(radian/set) --= actual
mean angular
velocity
of the ball relative
(radian/set) =
theorc?tical
=
actual mean angular velocity
angular
velocity
=
circular frequency
the motion
of cage or ball centrc
of fluctuation
EXPERIMENTAL
In order to observe
of cage or halt centrc
(ra~ia~~~s~~~) (radianjsec)
of (?c(radianjsec)
METHOD
of a ball in a ball bearing
without
disturbance,
change in flux due to the rolling motion
of the magnetized
The bearing steel is sufficiently
for this purpose. When the magnetized
inserted
magnetic
into a pocket hole of the cage is rolling between
ball has been observed.
the raceways
ball
the magnetic
MOTION
poles also revolve. current
Consequently,
the change in magnetic
I79
flux induces an alternating
in the circuit of the coil wound on the outer ring, as shown in Fig. I. The in-
duced current current
OF A BALL IN A BALL BEARING
is recorded on an oscillograph.
corresponds
Comparing
to the period
the frequency
As will be shown later, the period of the
of the rotation
of the ball relative
to the cage.
of the actual change in flux with the ideal one, the rate of
Ip
Fig. 1. Principle of mcasurcmcnt.
Fig. 2. Bearing testing machine. B, Bearing; C, Bakelite cover; D, Drop-feed oiling tube; I;, Frame of loading apparatus; LI, Loading lever; TA, Lever for measuring friction; M, Magnet
slip is easily estimated. maximum vanishes,
Moreover,
when the magnetic however,
can be estimated
the amplitude
of the current
axis is perpendicular
when both axes coincide.
from the fluctuations
is found to be at the
to the rolling axis. The current
Thus, the angle oc between
the two axes
in amplitude. Wear,
L+
(1961) I,~-I~T
IS0
I;. HIH,\;VO,
H. 'T:\KOl'E
E‘xperimental conditions The bearing
tester
with the parallellogram
lever system
shown in Fig. z was usc~l
for radial loads below 300 kg and for speeds below 5,000 revjmin. were carried below
out mainly
goo rev/min
bearings
kg. For thrust
system
at the rate of
of lubricant
were also used. The bearing loading apparatus,
IOO
the loading
were lubricated
find the influence
under
IO
of the four-ball
with the bakelite I
tester
was available.
Thc~
cc/min, mainly with spindle oil. In order to
upon the motion
as shown in Figs.
The experiments
loads below 92 kg and for speeds
of the ball, cylinder
oil and greascl
bobbin was fitted into the frame of the
and 2. The outer ring was fixed with the bakclitc>
cover plates.
HEARINGSUSE” IN THE EXPERIMEh'TS
so.
6307
3.5 x 80 x 21
No.
7307
3,j X
X0
X
21
solid brass pressed steel
17132 * s
solid brass
17/32 x 1 I x ‘3
I?/32
cageless
s!,,. G30i;
30 x 72 x rg
pressed steel
3’/04
No.
40 x 80 x
pressed steel
I5/32 x 9
6208
18
x 8
* Theoretical values (~b*/m* calculated by using Rz/r. The bearings experiments
used in the present research
are listed in Table I. The main part of the
was carried out on the deep-grooved
brass solid cage. Its radial clearance
bearing
No. 6307 furnished
with a
was zz ,u.
Magfletization of ball In order to obtain a cha!ge
in flux strong enough to be recorded on the oscillogram,
it is necessary
to give the ball a residual magnetism
coil consisted
of 460 turns of copper wire,
diameter,
34 mm inner diameter,
1.8
of about 200 G. The magnetizing
mm diam.
and is 95 mm in length).
(the coil has 76 mm outer The ball to be magnetized
was placed at the centre of the coil and then the magnetizing
0
Fig. 3.
20
current
60 I (n) magnetism of 17/32 in. ball plotted against magnetizing
was applied for
40
current (I A =
ho
Oc)
MOTION OF A BALL IN A BALL BEARING z seconds. Fig. 3 shows the residual magnetism against
the magnetizing
current.
applying a current exceeding
181
B, of the ball (17/32 in. diam.) plotted
The necessary
value B, = 200 G was obtained
by
50 A. Balls of various other sizes were also magnetized
to
the same value. Scatter
of this residual magnetism
residual magnetism of the
bearing.
running
5
was reduced to a sufficiently
was found to decrease
Calibrating
experiments
104 revolutions
narrow range. The
from its initial value during the operation ceased
after
of the inner ring and that the final value became
showed
that
the
decrease
B, =
140 G, i.e. 7oyb of the initial value under radial load above IOO kg. Therefore,
in order
to obtain
at least
reliable results,
the experiments
must be carried out after running
5 . 10~ revolutions. Recording of rotation of the ball In order to facilitate
the mounting
and dismounting
of the magnetized
pieces of the cage were fixed with tap bolts of small diameter. consisted
of 200 turns of enamelled
as shown in Fig. I. The bobbin
ball, both
The measuring
wire, 0.3 mm diam., wound on a bakelite
was fitted into the space between
coil
bobbin,
the outer ring and
the frame of the loading apparatus. The coil was led to the vibrator current
had a constant
The number of revolutions steel piece attached
of the electromagnetic
sensitivity
to the cage with the non-magnetic
poles of the magnet
The induced
range o-200
of the cage was recorded using a fixed magnet
and 2. The change of flux induced at the moment revolutions
oscillograph.
of 30 mm/mA in the frequency
served
to record
bridge,
Hz.
and a small
as shown in Figs.
I
when the steel piece passed by the
the revolutions
of the cage. The number
of
of the inner ring or shaft was also measured by the same method.
Prelim.inary experiment Owing to the freedom rolling axis relative fore to determine
of three-dimensional
to the magnetic
the angle a between
purpose the following preliminary Balls
of the bearing
go’ were prepared. and inserted
No. 6307
with
Fig. 4 shows examples (a) gives the maximum
however, amplitude function
definite
angles a = o’, 22.5’,
and for this 45”, 67.5” and
in the preliminary
flux change corresponding with that of the rotation
the flux change is negligibly A is proportional
1(b).
Thus,
like a roller in a roller bearing. experiment.
Oscillogram
to 01 = 90~. The slight fluctuation
is caused by the slip of the ball at the unloaded coincides
to the ball
the cage, as shown in Fig.
and it behaved
of the oscillograms
of its there-
was carried out.
The rolling axis was formed by short wires soldered
the motion of the ball was constrained
the fluctuation
of the ball, the position
the axes from the oscillogram,
experiment
into small holes drilled through
of the amplitude
motion
axis can change with time. It is necessary
side. The period of
of the cage. In the case of (x: = o’,
small, as shown in (c). It was found that the
to sin ix at constant
speed.
of the shaft speed N. Hence, for other bearings
Fig. 5 shows A /sin cy as a the measurements
were car-
ried out only for o( = 90~. Wenr, 4
(1961)
177-197
Denoting
the angular
of the measuring
\~elocity of the ball relati\,r to the cagt’, tire numh~r oi tinII>
coil, its resistance
the half amplitude
and inductance
1~~.PIN, ic, I\’ and I,, rrspectivel\-
of the induced current can be written as Ii,
iiWi,(lJl,’, II) I~il.C~hi2
11
50
ii20 A : $10 8 6 5
where 00 is the amplitude proportional
of the effective
to sin 0~). These results
residual magnetism
could be detected.
into 10 mA using the sensitivity
flux change
were obtained
In Fig. 5 the amplitude
unloaded side, owing to slip, the amplitude (I). On
to the coil (@CIis decrease
in the
4 mm is also converted
30 mm/mA of the vibrator.
Since the actual value of ~1)~shows an appreciable ing to eqn.
interlinked
after no further
decrease under radial load at the
A or I0 decreases
correspondingly
accord
the other hand, there is no slip of the ball at the loaded side, so that
the value of wo coincides
with the ideal one, giving maximum
within the period of the cage. Dividing
the measured
value of the amplitude
amplitude
by =I /sin x found in
Fig. 5 at the known speed S, the \ralue of sin x or 2 can be easily obtained.
MOTION
OF A BALL IN A BALL BEARING
183
RESULTS
Interpretation
o/ records
The oscillograms
shown in Fig. 6 are examples
The short period
T,,represents
not an absolute
motion
one revolution
of the ball but
of records taken under radial load.
of the ball relative
the relative
motion
to the cage. It is
to the cage that
the
Fig. 6. Explanation of oscillogram. (a) No. 6307, P = IOO kg, N = 1290 rev/min. (b) No. 7307 P : IOO kg, .V = 7jo rev/min. T: Period of revolution of inner ring or shaft; Tb: Period of rolling ball, Tbmax: Maximum of Tb; Te: Period of revolution of cage or ball centre; T,: Period of fluctuation of amplitude.
oscillogram
indicates,
because the motion of the ball fixed to the cage does not induce
any change in flux.
T, shows the period of a revolution of the cage. The ball centre travels around the inner ring at a period T,.As mentioned above, the fluctuation of the amplitude with the period T, is caused by the slip of the ball at the unloaded side of the bearing. An extremely unloaded period
The oscillogram
(b) is a record of the angular contact
large slip represented
by the prolonged
side owing to the absence
able. The amplitude angle LSbetween mentary
value.
(a) the fluctuation A,,
with the long period
and Amin correspond
the magnetic
axis of rolling is calculated
7-10
are the oscillograms
by using the calibration
in speed the slip decreases.
&lax
-
of the bearings
No. 6208. It is noted that considerable
and minimum
of the
curves in Fig. 5 :
(2)
ami,
No. 6307,
No. 7307,
No. 6306
and
slip occurs at low speed and that with increase
The regular fluctuation
several records. The condition
T,like a beat is remark-
to the maximum
and the rolling axes of the ball. The change of the mo-
Aoc = Figs.
bearing No. 7307.
Tbmaxis produced at the
of a thrust load. At the loaded side, however, the
Tb coincides to the theoretical
In the oscillogram
period
for its occurrence
of the angle OLis shown clearly in
will be discussed later.
184
I.. HllC\SO.
H. -t’.\5rOl‘H
MOTION
Fig.
8. Oscillograms
Fig.
9. Oscillograms
OF A BALL
IN A BALL
BEARING
185
of No. 7307 with solid cage, P = 100 kg, spindle oil IO cc/min. (a) 305, (b) 500. (c) 1000, (d) 4000, (e) 1600 (P = 0).
of cageless
bearing No. 7307. P = IOO kg, spindle (a) 500, (b) 1320, (c) 3000.
oil IO cc/mix
WU’,
4 (1961)
Rev/min:
Rev/min:
177-197
IS0
lb.
Using the symbols relation is obtained
Hlli.\hO,Il.
as shown in Fig. II and under EOMENCLATUKE,the following
for the ideal motion of the rolling elements
The values of cc)b*/wC*are listed in Table measured
on the oscillogram
mb fluctuates
1.\>(111~~
deviate
I. The actual angular
from the theoretical
with the period re. The angular
oC* in the present experiments. The mean value of (Ob/(,_)c =
velocity
velocities
OJ~and W-
values. As shown in Fig. 6. we, however,
coincides
with
In other words, there is no slip of the cage. cob/we
*
is easily obtained
waves included in a definite number of revolutions
by counting
the number
of
of the cage. Then the mean slip is
giLTen by the definition s :
(CW,* Wb)lW,*
, (wb*,lc!!lc*)((l/b:(‘J~)‘/(~~Ob*/f’)i*/
(4)
ir’?ctr, , (1001) L;/ 1’1;
RIOTIoN OF A BALL IN A BALL BEARING l‘he maximum
slip corresponding
is impossible, considered Fig.
however,
without
to
Wb
2n/Tb max is also significant.
=
an appreciable
error.
Hence
An estimate
only the mean slip is
in the following discussion.
12 shows the slip in the bearing
speed. At 5,000 increase
187
rev/min the motion
in radial load. The effect
small. However,
becomes
nearly
of viscosity
normal.
of the lubricant
with increase S decreases
in shaft also with
is found to be rather
the higher the oil-feed rate, the lower the slip (see Fig. 12(a)). In the
case of grease lubrication
0
No. 6307. It decreases
0
much lower slip is observed.
3000
1000
0
3000 Nkev/min)
ICUO
N (rev/mid
N(revlmin) b
a
C
Fig.
IL. Mean slip of No. 6307. (a) Effect of lubrication. P = IOO kg; spindle oil l , IO cc!min cc/min; cylinder oil A 9 cc/min, A 0.8 cc/min; x grease I cc. (b) Effect of load. Spindle oil IO cc/min. o, P = o kg; 0, IO kg; ., IOO kg: 0, 300 kg. (c) Effect of reduction of ball size,. Amount of reduction = 20 p. 0, P = IOO kg, normal size; 0, o kg, 4, 100 kg, all reduced six; A., o kg, A, IOOkg, ox reduced size. The bearing used in the experiment (h) was diffxezt from that in (a). 0,
I
Expressing
the position of the ball centre by the angle measured
of the load applied in the direction 90’,
gradually
develops
abruptly
disappears
raceways
increases
Consequently,
of the rotation
at the unloaded
at 320-330a4. at the unloaded
the contact
a maximum
clearance
Thus the radial clearance
side owing to the application
of radial
load.
loose. This causes
load the ball always keeps contact
is considered
the clearance,
slightly
balls of No. 6307. As a result, clearance
to have a marked smaller
increased
influence
with the
higher speed in the case of no load. Substituting magnetized
ball, between
neighbouring
the slip increases the magnetized
balls were of normal
its circumference
further.
to 60~ compared
imperfect
with the value of except
at
smaller ball only for the
It is reasonable
size. An experiment
for ordinary
considerably
the slightly
ball and the raceway
showed that
upon slip. In
balls were substituted
22 ,u for the ordinary balls. Fig. IZ (c) shows that slip increases
around
and
the ball and the
Here slip does not occur even at low speed (see Fig. 7 (j)).
order to change
clearance
at 29W300”
between
of the ball with the inner race becomes
slip of the ball. But under pure thrust raceways.
of the shaft, slip begins at nearly
side, attains
The actual
from the position
to consider
was increased
that
the
because
the
in which a ball was ground
sphericity
resulted
in a somewhat
ISS irregular
H.
b’. HlRANO.
rolling
motion.
There
l.\SOl!li
was no significant
difference
betne~n
the result5
using the solid cage and the pressed cage. The data Though
of the angular
the bearing
contact
bearing
of this type should
No. 7307
be properly
are summarized operated
in Fig.
under the action
1.3. of
N (rev/min)
Fig. 13, Mean slip of No. 7307. Spindle oil IO cc/min. 0, P L o kg, l , 100kg, solid cage ; A, o kg, A , 1oo kg, cag~lc~s.
thrust load, the experiment contact
between
deep-grooved
Fig. 14. Oscillograms of netized balls, P = LOOkg, (a) L magnetized balls in rcv/min. (b) S magnetized
No. 1,307 with magspindle oil, I cc/min. opposite holes, 3010 balls, 3050 rcv!mil1.
was carried out under pure radial load. Consequently
the balls and the raceways
became
type. This leads to the remarkable
much looser compared
increase
13 balls for IO in the ordinary
Fig. 13. The extraordinarily by adjacent
the cage and
type No. 7307. The result is also plotted in
high value of the slip is attributed
to the braking
action
balls.
The experimental
points of the bearing No. 6306 fall near the upper curve in Fig. IZ
(c). The slip in the bearing
bearings
with thr
of the slip even at higher
speed as shown in Fig. 8. The cageless bearing was prepared by removing substituting
the
were provided
after repeated
mounting
No. 6208 was at most 3qi0, except at very low speed. These
with the pressed steel cages. Since deformation and dismounting
data was inferior compared
was inevitable,
of the cages
the reproducibility
of the
with the case of the solid cage.
Change ilz rolling axis o/ ball The fluctuation maintain
of the amplitude
with the period ‘r, as shown in Figs. C-IO tends to
its wave form for a long time or after repeated
shaft speed. Further,
increase
it has been verified that the behaviour
same except for its phase. The resultant regular form shown in the oscillograms
and decrease of the
of each ball is exactly
current induced by the magnetized in Fig. 14.
the
balls has a
MOTION
Denoting
the frequency
189
OF A BALL IN A BALL BEARING
of the fluctuation
ratio CO~/CO~ is equal to the difference
by ns c/min or ws rad/sec = 24T,,
the
of wb/wCfrom the nearest integer m. The relation
can be written W&C = I(oe/wC) - ml or n&c = I(?+,) where %b =
2nw~/60
and vt, = 27~~()~/60.Because
6307 as in Table I, the value of m is 5 with a few exceptions In the case of No. 7307, however, 2,3
and 4 frequently
ml
(5)
where considerable
corresponding
slip is observed,
to m = 4.
several values
appear beside m = 5. This means that the period T, is determined
by the number of revolutions the ball becomes
-
e&*/e&* = 5.30 in the case of No.
until the accumulation
of the phase lag of the motion of
2~.
0 Fig.
1000
2000
3000 4000 N(rev/min)
15. Change of rolling axis plotted
5000
against
shaft speed.
10
2
3
4b
5, b/nTj
5
Fig. 16. Change of rolling axis plotted against nO/n,. (a) No. 6307. P = IOO kg. ., spindle oil IO cc/min; 0, cylinder oil 9 cc/min; o cylinder oil 0.8 cc/min; A spindle oil + o. I oh quartz powder; x grease I cc. (b) No. 7307. 0 0, P = IOO kg; o, o kg. (c) No. 6306, P = IOO kg. (d) NO. 6208, ., P = Iookg; o, okg.
As explained
above, the fluctuation
of the amplitude
angle between the rolling axis and the magnetic fizsis that of the fluctuation
of the instantaneous
represents
axis. Accordingly,
the change of the the frequency
rolling axis relative Wear,
os or
to the ball. For 4 (1961)
177-797
the sake of brwity Is, calculated
it will bc drsi~inated as /ls in the following description.
by means of the calibration
cur\cs show resonance
curve in Fig. 5, against
as in Fig. 15. The dependence
relation becomes different
of ,l x on speed is not unicl~i~~.‘I‘h(a
if the slip varies owing to the change of load orvtlicr
It is readily seen that thcx peaks of the curves in Fig. 15 correspond values of the ratio ,nb/wzc= w,;~oc. Hence, plotting 70’- mar the point MI :=
I?( = 2, 3. 4 and 5 are cltarfy pi-oportioli~tlit~
i.i’., .h . Sill interpreted
observed
j and
witi1
to the integral
in the cast of the So.
4. Four peaks corresponding
to
in the cast of So. 7307. 7’1~~cmves show ~10s~~
to icoscc ~~~,)~,~(I~~) :- cosec (TW~,~/~C)~). l’hc
(7f(tJ8jlrJc~]iimiw?s
f;~c‘tors
,/~ccagainst the ratio, the esperimen-
tal points fall on a unique cur\:e as shown in Fig. 16. Especially 0307 the change esceeds
l’lottinr:
the shaft speed, 111,.
dip
.i‘. In
the
pi-oportio~lalit~
constant,
following wction this I-t’I&$On \ViIlbi’
tlworttticall~~. I)ISCIISSIt)N
(‘mtsr
o,iclznngt~in
rolling uxis o,i hall
Since at the beginning to be constant
of the slip the position
under given operating
conditions,
surface of the ball, e.g. the displacement
the ball centre, bearing,
may be treated
the point CC) makes
revolution
of the magnetic
as follows.
If the ball behaves
CIJ~~/W C revolutions
about
of a point Co on the
pole after each revolution
of
as a roller in a roller
the x-axis
at the end of one
of the ball centre and comes to the point Co’. The shift of the phase angle is
then expressed
by p -- X((W,/W,.)
After successive revolutions /_C00x
d of the ball in Fig. 17 is considered the displacement
= J__CO’OX =:
MZ)~ or #q = znw,/w?
((j)
of the ball centre the point travels to Co”, CO”‘, . . Because
. =z 2 in this case, the fluctuation
of the amplitude
does not
occur. In order to explain
the fluctuation
from the path CO‘CO“CO”’
of the amplitude,
a deviation
of the point Cct
. should be assumed. Let the position of Cn after the deGa-
tion beC1’. It may be reasonable
to assume that the deviation
is caused by the angular
displacement E about the diameter AB (y-axis) passing the contact point A. At the unloaded side tht ball is set fret! from the contact with the inner race, though the contact with the outer race is still maintained here owing to the centrifugal fowl acting on the ball. Thus the effect of the cage! on the motion of the ball becomes signiIicant at the unloaded side. Actually, there is always more or ltzss dv\riation from tlw ideal contact condition. Especially, the contact point of the ball with tht: pock(~t holt~ of the cage or with the groove of thv outer race may have a slight offset from the plant. of syrnrnvtr!. of the groove. The offset cauws a spinning moment about then asis AI3. ‘I‘hii; suggests the origin of thv displacemrnt c. i. c.. t IH spin.
Fig. IS illustrate5 tile motion of the point C‘” as a result of the succcssiv~ displawment ,4 and F. Actnall!., the ball rolls with the thcorctical angular velocity c~J,,* at the loaded side and then slips at the unloaded siclc. Howlver, it is assumed for brc\. ity that the rolling velocity is c~lual to the constant myan angular \-ctlocit!, CII~nntl that the displacement F takes plaw instantaneously at the point A. Starting from the point CO, ij-displacement transfers it to C‘“‘, \vhich comvs to (‘I’ after F-displacement. \\‘hen the displacements arc repeated the point travels to (‘I’, (‘2”. (‘3”‘. , C‘,,(n).If the point Co is the magnetic pole, the angle /_,xOC,(n) is equal to the angle z for each revolution of the cage. The amplitude of the oscillogram is reprcscnt4 bv sin 1._AX‘,(“). The difference bctwccn ,zmax and Ymin shown in Fig-. IS is th(s c.hangc~ of the rolling axis rclati\~c to the co-ordinatc~ s\,stcrn fiwd to thv ball.
192
Theoreticul
camideration
analysis the position of the point on the batl is cxprt%sed b\- tlttz
In the following
unit vectors
cl”’ : Yn
,(n)
(x,,.
Gn) and C.,J., :
1’1,
, 1’,,-I
i.T,,..I.
,1’,,
I,
u^,,-1)
:7
where fl = ~,a, _ _ . The displacements /Iand R are represented by the transformations of the vectors from ~~-1to ~~~‘-1 and from ~~‘-1to r,, respectively. Those tr~~ilsformations are also written as the matricrs x =
I ‘: p .-; 0 COh ] 0 sin fl
;]
and Y :- [_$::
1
T]
(S)
- (YXP ]Z]
(9)
The relation between the vectors is then expressed 1~~ = Y*[z;]
Now, it is known that the resultant transformation T = YX is equivalent angular displacement y about the unit vector ~(a, p, Y)“. T is written as COSy + 12(r-ctts yrl T =
vsin
-V sin y + .Ajt(r-cos
y + puii(~~--cmT/I)
i -p sin y + vA( I
ws 7p)
7p)
to the
Q sin y $ Av(r--- cos y)
cos y + y"(r---cosy') ----a sin ?/,+ /LV(I cos y)
CI,S 1,’ +
7. sin y --C vh( I- cos y)
iJOI
L’( I-- (‘OS y1).1
where A = ~0s (42) v
-
-sin
sin (~/z)~s~n (ytk),
(e/2) sin (~~z)/sin (y/r)
cm (y/2)
I__ cm (F/L) cm (P/L)
.- sin (F/Z) cm
~~~2~~~~1~
(y$z)
md
(I rl
or sin (y/z)
--
1:1--4zos’ {p/2) cm2 (/3/L)
Since the repeated transformation (YX)n = Tnrepresents the angular displacement ny about the e-vector, it is derived from (IO) 1~~;substituting 1ny1for y: WV)
(12!
T(V)
Hence from (g), (IO) and (12)
= il (,4x0 + j4yf, + ~70) -t cos mp[xo --- A (Ax0 + jhuy,,T w-a) + sin q
(pa
vy0)
('3)
Denoting by LeOx and i_eOro the angle between r-vector and x-axis and that bctween e and ro-vector, respectively, eqn. (13) is written as x,, = ~0s (~~ec>x) ~0s (LPOQ) -I- sin (jot)
sin
(jr(h)
~0s 14
(14)
where CDS(LPor) 7 3.. cos (/ r0vo) = 3,s”+ /&Jo-‘- YZi.8 anit 21-= 8211’ --- sin -1 (.LG~ 1 ly~~)/sin (/ .20x) sin (/_eOro) are substituted.
(15)
MOTION OF A BALL IN A BALL BEARING
Since
LxOCn(n)
After rewriting
=
LxOr,
=
that
u takes discrete
u behaves
permitted,
LeOx) cos2 (u/z) + cos(LeOr0
values with increasing
as a continuous
variable.
xn is equal
to cos 0~.
and minimum
(k = integer).
+ LeOx) sin2 (U/Z)
12. However,
This assumption
except in the case where y/n is expressed
the maximum tively
mentioned,
(14))
cos OL= cos (LeOro -
The variable
CL, as previously
193
of LXare attained
(16)
it is assumed
of simplification
here
may be
by a ratio of simple integers. Then
at u = (zk + I) 7c and u = okn, respec-
Thus mrnax= LeOro + LeOx and ~(,,,i,, = 1LeOro -
LeOxl
(17)
Hence -‘ahin)/ LeOx = (L%lIax
or LeOx
The expressions
AB, where A and B are the projections
axis. The angular co-ordinate
(16). The ordinate
graphically
LeOx)
(18)
LeOx)
(18)’
in Fig. 19. Take two
max and amin on the unit circle. Then describe to OL
Fig. 19. Diagrammatic
of the circle AB is the variable
representation
of fluctuation
the
of A’ and B’ on the u in (14) or
of amplitude.
of the point on the circle A ‘B’ is found to be sin L-X from (14) and (17).
Since the amplitude
of the oscillogram
sented by the figure obtained Special
(for LeOvo <
(16), (17) and (18) are represented
circle with the diameter
abscissa
(for //eOro >
da/a
= (amax + bin)/
points A’ and B’ corresponding horizontal
=
is proportional
by plotting
to sin LX,the wave form is repre-
the ordinate
of the circle A’B’ against
the
u. attention
must be paid to the expression
amin may be taken in the negative
(IS’), which shows that the angle
sense, as shown in Fig. 19 by B”. Even if the change Wear, 4 (1961) 177-197
L
in is the same
fluctuation
as in the normal
of the amplitude
Fig. 19(b), the amplitude is an example treated
HIH.XISO,H.
I’.
19-i
difference
the circle .I “H”
correspondin@y.
observed
(II)
smaller.
the period1‘,. The oscillogram
record?;.
These
‘11~.
CJO“as in l:ig. ~(ki
two cases must be tlistincti\~c~l~~ however, is rather rare and tllta
case is readily apparent.
cos(~.&)x)
= ?., eqn. (18) can be written sin (LIr+./2)--
Substituting
becomt5
When *xrnnxescerds
of the data. Their occurrence,
from the ordinary
Now, considering
case (IS),
decreases
has two maximawithin
of actually
in the analysis
T.\XOl‘K
as
1 I---L”
for ii, we obtain sin (.iln/r)
= sin (r/l)/sin
(y/zj
(IQ1
The period of the change of a is represented
by ny =
according to eqn. (13). On the
other hand, the period T, in Fig. 6 contains
n revolutions
2x
of the cage, i.e., T, = ~27;.
Hence ?/I~7rn/lz Then the relation
rrzTe/Ta
-=
mw&u<
(19) becomes sin (.&z/2) -7 sin (c/r)/sin (nw,/wp) = sin (r/r)/fsin (nwb/toc); The expression
where (5) or (6) is substituted.
shown in Fig. 16. A more detailed discussion
(20)
verifies
the experimental
(20) results
of the quantitati~~e relation between the
spin E and the slip S will be developed in the next section. Relation
between sli$ artd chaagt: i?z rolling CL&
As mentioned assuming
above, the change in the rolling axis of the ball is clearly explained
the existence
of spin associated
2Or
20
(~2)
15 ~
E
.
5
s r‘
0.
0
10 slip
25
1 (b)
E
l*
10
by
with slip. In order to find the factors affect-
P,ZO
0
10
20
30
s
k)
40 50 60 Slip S PM
.
20 E
.
15
10 :
5
5
l*
l
LL 0
10
20
30 40 SP?%)
c (d)
l *
IO
l2kffKc
0
1
2
3
SW*)
Fig. 20. Relation between slip and spin. (a) So. 6307. P - IOO kg. (b) No. 7307. 0, P == ukg, 0, IOO kg, solid cage; A, o kg, A, IOOkg, cageless. (c) No. 11306. P = ran kg, (d) So. 6~08. 0. 3’~~ o kg; 0. TOOkg.
MOTION
ing spin,
it must
OF A BALL
be calculated
(or ws/oe) and AX into relation S it is finally confirmed The proportionality and on the operating metric contact
first
constant,
however,
conditions.
Since
hole in the direction the assembled
is specified
to S, as in Fig.
0.15
cage into the bearing
originates
20.
in the asym-
of the bearing.
this effect.
One of
The amount
of the
mm in the moderate
The position
case. The results
of the hole in which the magnet-
by the angle measured
of the revolution
of WC/CU~
the mean slip
hole of the cage or with the groove of the
was carried out, exaggerating
21.
data
against
is found to depend on the type of bearing
case and
in the former case are shown in Fig.
the observed
the spin probably
holes was offset in the axial direction
offset was 0.3 mm in the extreme
195
the results
that the spin E is proportional
of the ball with the pocket
ized ball was inserted
BEARIKG
by substituting
(20). On plotting
outer ring, the following experiment the pocket
IN A BALL
from the centre of the offset
of the cage. The axial deviation was shown also in Fig.
21.
of the holes in
The holes o” (offset
N (rev/mid
Fig.
21. Slip Of balls in cage with offs.3 hole. P =
hole), 45” and 180’ occupied 270’ the internal
position.
the external
position
The cage was supported
x/min. IOO kg, spindle oil 10 G
and the holes 9o”, 135”,225’ and by the balls in three external
The four internal
balls were supposed
This is confirmed
by the fact that the slip occurs at the internal
Fig.
21,
and that at the external
to be free from contact
holes negative
holes.
with the inner race. holes, as shown in
values are obtained.
The negative
slip
Wear, 4 (1961) 177-197
1:. HIRANO,
196 represents
the overrun
H. TANOUE
of tfre ball owing to the offset of the ball centre.
In the inter-
mediate case of the hole 315”, the ball shows small positive slip at lower speed. Further,
it was found that the spin E calculated
holes 90” and
225’
one may conclude
22
the greater is the spin of the bail. The result of the smaller offset,
mm, also supports
looser except
this conclusion.
at the offset hole. Negative
it is shown that an appreciable
Fig.
spin associated operation negative rings,
of the bearing,
Here,
slip is considered
unfavourable
effect
under thrust
and combined
slip. The
hole.
to have no harmful
contact
influence
upon the
side. On the contrary,
the
of the ball with the grooves of the
of the spin becomes
load were disregarded.
effect of slip and spin, however,
out under thrust
with the negative
because it occurs at the unloaded the harmful
of the balls was much
only at the offset hole. In Fig.
spin is also associated
slip is caused by the asymmetric the experiments
the contact
slip occurred
~2. Spin of balls in cage with offset
with positive
and consequently
research
Consequentl!..
22.
that the larger the offset of the ball centre from the plane of sym-
metry of the bearing, 0.15
by using eqn. (20) is larger at the,
than at the holes 135” and 270°, as shown in Fig.
further
inevitable.
In this
In order to find the
investigation
should be carried
loads. CONCLUSIONS
By using a magnetized ball bearing
ball it was possible to follow the behaviour
under radial load. The motion
sional and associated
of the ball in a
of the ball was found to be three-dimen-
with slip and spin. The conclusions
are as follows:
I. Slip of the ball occurs at the unloaded side, where its contact
with the raceways is
looser. Slip decreases with increase in speed and load. It increases with radial clearance. 2.
The instantaneous
rolling axis relative
to the ball changes regularly.
Its change
is closely related to the slip. 3. The cause of the change placement
in the rolling axis is spin during slip. The angular
of the spin is proportional
4. The spin increases
dis-
to the mean slip.
with the asymmetry
of the contact
of the ball with the cage
or with the grooves of the races. 5. Further
increase in asymmetry
slip is observed;
causes negative
in this case the contact
is considered
slip. Under pure thrust to be still maintained.
load no
197
MOTIONOFABALLINABALLBEARING ACKNOWLEDGEMENTS
The authors wish to thank the members of Research Laboratory, Nippon Seiko Co., Fujisawa, Japan, especially Dr. T. HATTORI and Mr. S. YAMAMOTO for their kind support.
FEFERENCES 1 W. HAMPP, Igr.-Arch., 12 (1941) 6. 2 T. SASAKI et al., Trans. Japan. Sot. Mech. Engrs., 8 (1942) 1-64 (in Japanese). 3 F. HIRANO AND H. YOSHINARI, Machine Science, 2 (1950) 31 (in Japanese). 4 K. IIDA AND T. IGARASHI, J. Japan. Sot. Mech. Engrs., 62 (1959) 369 (in Japanese). 5 E. T. WHITTAKER, A Treatise on the Analytical Dynamics, 4th ed., Cambridge University 1937.
Press,
Wea+‘, 4 (1961) 177-197
MOTION
OF A BALL F. HIRANO K_@shii
IN A BALL
BEARING*
AND H. TANOUE
University,
(Received
I77
Fukuoka
October
IO,
(Japan) 1960)
SUMMARY A magnetized ball rolling in a ball bearing induces an alternating current in a coil wound on the outer ring. The period of the current represents one revolution of the ball. Deviation from ideal motion of the ball occurs as slip and spin. The former is shown by the prolongation of the period. The latter causes fluctuation of the rolling axis relative to the magnetic axis of the ball. It can be easily estimated by measuring the variation of the amplitude of the current. Under radial load a considerable slip occurs at the unloaded side. It is found that the slip decreases exponentially with increase in speed. The magnitude and the direction of load, the lubicating condition and the radial clearance also affect the slip. In the special case where the ratio of the number of revolutions of the ball to that of the cage becomes nearly equal to an integral value, the deviation of the rolling axis shows a sharp increase, which often exceeds 70”. This deviation is due to the spinning motion of the ball during slipping. ZUSAMMESFASSUNG Eine in einem Kugellager rotierende magnetisierte Kugel induziert einen Wechselstrom in einer urn den Aussenring gewickelten Spule. Die Periode des Stroms entspricht einer Rotation der Kugel. Abweichung von der idealen Bewegung der Kugel entsteht in zwei Formen: Gleiten und Kreiselbewegung. Gleiten wird sichtbar an einer Verlangerung der Periode. Die Kreiselbewegung verursacht eine relative Schwankung der Drehachse zur magnetischen Achse der Kugel und ist durch Messung der Veranderung der Amplitude des Stromes leicht zu finden. Bei Radialbelastung tritt an der unbelasteten Seite ein betrachtliches Gleiten ein, das mit zunehmender Rotationsgeschwindigkeit rasch abnimmt. Ferner wird Gleiten durch GrSsse und Richtung der Belastung, durch den Schmierzustand und die Radialspielung des Lagers beeinflusst. Wenn das Verhlltniss der Drehzahl der Kugel zu der des Kafigs annahrend gleich einer ganze Zahl ist, nimmt die Richtungslnderung der Drehachse erheblich iu, manchmali mehr als 70”. Ursache ist das Auftreten einer Kreiselbewegung der Kugel beim Gleiten. INTRODUCTION
The motion motion.
of rolling
Depending
elements
on operating
in a rolling conditions,
deviation
should have an important
elements.
Therefore,
basic information bearing,
because
deviates
influence upon the friction,
an investigation
on rolling bearings. observation
bearing
somewhat
wear and life of rolling
of the motion of rolling elements The situation
of the motion
from ideal
slip and spinning motion take place. This is required
for
is simple in the case of a roller
is easy. HAMPP~
reported
the results
of
* Published in Japanese in J. Japan. Sm. Lubrication Engrs., 5 (1960) 175 and Tvans. Japan. Sot. Mech. Engrs., 27 (1961) (in the press). The last section of the paper has not been published previously. IFear, 4 (1961) r77-197
17s
1:. t111<.\\0,
measurement capacitance.
of the motion of rollers in ;I connecting
tracing
motion
on the other
and observation
previously
the motion
and special
rod bearing 1,~. mr~ans of ~~ai-iabl~~
and analyst$d the slip of the roller.
In the cast’ of ball bearings, dimensional
l.\SOl’l~.
SasitKrz recorded the locus of a point on the edge of 3 roller I)\. using 311
optical method,
the authors
I-1.
reported
of the motion
that a magnetized
of the balls.
attention
hand, balls have the freedom
Recently
becomes
more difficult.
ball could successfully
the experiments
has been given to the spinning
of t lm.t?-
hail
motion
One ot
I9c used
for
been rtbsumecl ant1
of balls.
The present
paper deals mainly with the results under radial load. The case of thrust ant1 cmnbinml load is now under investigation
and will be reported .S(~MET<‘I . A
=
am~iit~de
,>
on
later.
YI-I’RE
of oscillo~~m
(mm)
=
residual magnetism
1
shaft speed or speed of inner ring (revjminj
of ball {C)
=
load (kg)
=- radius of inner raceway
(mm)
.= radius of outer raceway
(mm)
:
mean slip r’,;)
--
number of revolutions
=
number
of ball relative to cage (rev/mm)
of revolutions
of cage or number of revolutions
of ball
ccntre (rev/mm) :
frequency
=
radius of ball (mm)
of fluctuation
of amplitude
(c/minj
= angle between rolling axis and magnetic
axis of ball (deg.)
--_ change oflw (deg.) =
phase shift of a definite point ou the ball for each revolution
of
the cage {radians) =
angular displacemel~t of spin (radians or deg.)
=
theoretical
angular
velocity
of the ball relative
to
the
cage
to thr
cage
(radian/set) --= actual
mean angular
velocity
of the ball relative
(radian/set) =
theorc?tical
=
actual mean angular velocity
angular
velocity
=
circular frequency
the motion
of cage or ball centrc
of fluctuation
EXPERIMENTAL
In order to observe
of cage or halt centrc
(ra~ia~~~s~~~) (radianjsec)
of (?c(radianjsec)
METHOD
of a ball in a ball bearing
without
disturbance,
change in flux due to the rolling motion
of the magnetized
The bearing steel is sufficiently
for this purpose. When the magnetized
inserted
magnetic
into a pocket hole of the cage is rolling between
ball has been observed.
the raceways
ball
the magnetic
MOTION
poles also revolve. current
Consequently,
the change in magnetic
I79
flux induces an alternating
in the circuit of the coil wound on the outer ring, as shown in Fig. I. The in-
duced current current
OF A BALL IN A BALL BEARING
is recorded on an oscillograph.
corresponds
Comparing
to the period
the frequency
As will be shown later, the period of the
of the rotation
of the ball relative
to the cage.
of the actual change in flux with the ideal one, the rate of
Ip
Fig. 1. Principle of mcasurcmcnt.
Fig. 2. Bearing testing machine. B, Bearing; C, Bakelite cover; D, Drop-feed oiling tube; I;, Frame of loading apparatus; LI, Loading lever; TA, Lever for measuring friction; M, Magnet
slip is easily estimated. maximum vanishes,
Moreover,
when the magnetic however,
can be estimated
the amplitude
of the current
axis is perpendicular
when both axes coincide.
from the fluctuations
is found to be at the
to the rolling axis. The current
Thus, the angle oc between
the two axes
in amplitude. Wear,
L+
(1961) I,~-I~T
IS0
I;. HIH,\;VO,
H. 'T:\KOl'E
E‘xperimental conditions The bearing
tester
with the parallellogram
lever system
shown in Fig. z was usc~l
for radial loads below 300 kg and for speeds below 5,000 revjmin. were carried below
out mainly
goo rev/min
bearings
kg. For thrust
system
at the rate of
of lubricant
were also used. The bearing loading apparatus,
IOO
the loading
were lubricated
find the influence
under
IO
of the four-ball
with the bakelite I
tester
was available.
Thc~
cc/min, mainly with spindle oil. In order to
upon the motion
as shown in Figs.
The experiments
loads below 92 kg and for speeds
of the ball, cylinder
oil and greascl
bobbin was fitted into the frame of the
and 2. The outer ring was fixed with the bakclitc>
cover plates.
HEARINGSUSE” IN THE EXPERIMEh'TS
so.
6307
3.5 x 80 x 21
No.
7307
3,j X
X0
X
21
solid brass pressed steel
17132 * s
solid brass
17/32 x 1 I x ‘3
I?/32
cageless
s!,,. G30i;
30 x 72 x rg
pressed steel
3’/04
No.
40 x 80 x
pressed steel
I5/32 x 9
6208
18
x 8
* Theoretical values (~b*/m* calculated by using Rz/r. The bearings experiments
used in the present research
are listed in Table I. The main part of the
was carried out on the deep-grooved
brass solid cage. Its radial clearance
bearing
No. 6307 furnished
with a
was zz ,u.
Magfletization of ball In order to obtain a cha!ge
in flux strong enough to be recorded on the oscillogram,
it is necessary
to give the ball a residual magnetism
coil consisted
of 460 turns of copper wire,
diameter,
34 mm inner diameter,
1.8
of about 200 G. The magnetizing
mm diam.
and is 95 mm in length).
(the coil has 76 mm outer The ball to be magnetized
was placed at the centre of the coil and then the magnetizing
0
Fig. 3.
20
current
60 I (n) magnetism of 17/32 in. ball plotted against magnetizing
was applied for
40
current (I A =
ho
Oc)
MOTION OF A BALL IN A BALL BEARING z seconds. Fig. 3 shows the residual magnetism against
the magnetizing
current.
applying a current exceeding
181
B, of the ball (17/32 in. diam.) plotted
The necessary
value B, = 200 G was obtained
by
50 A. Balls of various other sizes were also magnetized
to
the same value. Scatter
of this residual magnetism
residual magnetism of the
bearing.
running
5
was reduced to a sufficiently
was found to decrease
Calibrating
experiments
104 revolutions
narrow range. The
from its initial value during the operation ceased
after
of the inner ring and that the final value became
showed
that
the
decrease
B, =
140 G, i.e. 7oyb of the initial value under radial load above IOO kg. Therefore,
in order
to obtain
at least
reliable results,
the experiments
must be carried out after running
5 . 10~ revolutions. Recording of rotation of the ball In order to facilitate
the mounting
and dismounting
of the magnetized
pieces of the cage were fixed with tap bolts of small diameter. consisted
of 200 turns of enamelled
as shown in Fig. I. The bobbin
ball, both
The measuring
wire, 0.3 mm diam., wound on a bakelite
was fitted into the space between
coil
bobbin,
the outer ring and
the frame of the loading apparatus. The coil was led to the vibrator current
had a constant
The number of revolutions steel piece attached
of the electromagnetic
sensitivity
to the cage with the non-magnetic
poles of the magnet
The induced
range o-200
of the cage was recorded using a fixed magnet
and 2. The change of flux induced at the moment revolutions
oscillograph.
of 30 mm/mA in the frequency
served
to record
bridge,
Hz.
and a small
as shown in Figs.
I
when the steel piece passed by the
the revolutions
of the cage. The number
of
of the inner ring or shaft was also measured by the same method.
Prelim.inary experiment Owing to the freedom rolling axis relative fore to determine
of three-dimensional
to the magnetic
the angle a between
purpose the following preliminary Balls
of the bearing
go’ were prepared. and inserted
No. 6307
with
Fig. 4 shows examples (a) gives the maximum
however, amplitude function
definite
angles a = o’, 22.5’,
and for this 45”, 67.5” and
in the preliminary
flux change corresponding with that of the rotation
the flux change is negligibly A is proportional
1(b).
Thus,
like a roller in a roller bearing. experiment.
Oscillogram
to 01 = 90~. The slight fluctuation
is caused by the slip of the ball at the unloaded coincides
to the ball
the cage, as shown in Fig.
and it behaved
of the oscillograms
of its there-
was carried out.
The rolling axis was formed by short wires soldered
the motion of the ball was constrained
the fluctuation
of the ball, the position
the axes from the oscillogram,
experiment
into small holes drilled through
of the amplitude
motion
axis can change with time. It is necessary
side. The period of
of the cage. In the case of (x: = o’,
small, as shown in (c). It was found that the
to sin ix at constant
speed.
of the shaft speed N. Hence, for other bearings
Fig. 5 shows A /sin cy as a the measurements
were car-
ried out only for o( = 90~. Wenr, 4
(1961)
177-197
Denoting
the angular
of the measuring
\~elocity of the ball relati\,r to the cagt’, tire numh~r oi tinII>
coil, its resistance
the half amplitude
and inductance
1~~.PIN, ic, I\’ and I,, rrspectivel\-
of the induced current can be written as Ii,
iiWi,(lJl,’, II) I~il.C~hi2
11
50
ii20 A : $10 8 6 5
where 00 is the amplitude proportional
of the effective
to sin 0~). These results
residual magnetism
could be detected.
into 10 mA using the sensitivity
flux change
were obtained
In Fig. 5 the amplitude
unloaded side, owing to slip, the amplitude (I). On
to the coil (@CIis decrease
in the
4 mm is also converted
30 mm/mA of the vibrator.
Since the actual value of ~1)~shows an appreciable ing to eqn.
interlinked
after no further
decrease under radial load at the
A or I0 decreases
correspondingly
accord
the other hand, there is no slip of the ball at the loaded side, so that
the value of wo coincides
with the ideal one, giving maximum
within the period of the cage. Dividing
the measured
value of the amplitude
amplitude
by =I /sin x found in
Fig. 5 at the known speed S, the \ralue of sin x or 2 can be easily obtained.
MOTION
OF A BALL IN A BALL BEARING
183
RESULTS
Interpretation
o/ records
The oscillograms
shown in Fig. 6 are examples
The short period
T,,represents
not an absolute
motion
one revolution
of the ball but
of records taken under radial load.
of the ball relative
the relative
motion
to the cage. It is
to the cage that
the
Fig. 6. Explanation of oscillogram. (a) No. 6307, P = IOO kg, N = 1290 rev/min. (b) No. 7307 P : IOO kg, .V = 7jo rev/min. T: Period of revolution of inner ring or shaft; Tb: Period of rolling ball, Tbmax: Maximum of Tb; Te: Period of revolution of cage or ball centre; T,: Period of fluctuation of amplitude.
oscillogram
indicates,
because the motion of the ball fixed to the cage does not induce
any change in flux.
T, shows the period of a revolution of the cage. The ball centre travels around the inner ring at a period T,.As mentioned above, the fluctuation of the amplitude with the period T, is caused by the slip of the ball at the unloaded side of the bearing. An extremely unloaded period
The oscillogram
(b) is a record of the angular contact
large slip represented
by the prolonged
side owing to the absence
able. The amplitude angle LSbetween mentary
value.
(a) the fluctuation A,,
with the long period
and Amin correspond
the magnetic
axis of rolling is calculated
7-10
are the oscillograms
by using the calibration
in speed the slip decreases.
&lax
-
of the bearings
No. 6208. It is noted that considerable
and minimum
of the
curves in Fig. 5 :
(2)
ami,
No. 6307,
No. 7307,
No. 6306
and
slip occurs at low speed and that with increase
The regular fluctuation
several records. The condition
T,like a beat is remark-
to the maximum
and the rolling axes of the ball. The change of the mo-
Aoc = Figs.
bearing No. 7307.
Tbmaxis produced at the
of a thrust load. At the loaded side, however, the
Tb coincides to the theoretical
In the oscillogram
period
for its occurrence
of the angle OLis shown clearly in
will be discussed later.
184
I.. HllC\SO.
H. -t’.\5rOl‘H
MOTION
Fig.
8. Oscillograms
Fig.
9. Oscillograms
OF A BALL
IN A BALL
BEARING
185
of No. 7307 with solid cage, P = 100 kg, spindle oil IO cc/min. (a) 305, (b) 500. (c) 1000, (d) 4000, (e) 1600 (P = 0).
of cageless
bearing No. 7307. P = IOO kg, spindle (a) 500, (b) 1320, (c) 3000.
oil IO cc/mix
WU’,
4 (1961)
Rev/min:
Rev/min:
177-197
IS0
lb.
Using the symbols relation is obtained
Hlli.\hO,Il.
as shown in Fig. II and under EOMENCLATUKE,the following
for the ideal motion of the rolling elements
The values of cc)b*/wC*are listed in Table measured
on the oscillogram
mb fluctuates
1.\>(111~~
deviate
I. The actual angular
from the theoretical
with the period re. The angular
oC* in the present experiments. The mean value of (Ob/(,_)c =
velocity
velocities
OJ~and W-
values. As shown in Fig. 6. we, however,
coincides
with
In other words, there is no slip of the cage. cob/we
*
is easily obtained
waves included in a definite number of revolutions
by counting
the number
of
of the cage. Then the mean slip is
giLTen by the definition s :
(CW,* Wb)lW,*
, (wb*,lc!!lc*)((l/b:(‘J~)‘/(~~Ob*/f’)i*/
(4)
ir’?ctr, , (1001) L;/ 1’1;
RIOTIoN OF A BALL IN A BALL BEARING l‘he maximum
slip corresponding
is impossible, considered Fig.
however,
without
to
Wb
2n/Tb max is also significant.
=
an appreciable
error.
Hence
An estimate
only the mean slip is
in the following discussion.
12 shows the slip in the bearing
speed. At 5,000 increase
187
rev/min the motion
in radial load. The effect
small. However,
becomes
nearly
of viscosity
normal.
of the lubricant
with increase S decreases
in shaft also with
is found to be rather
the higher the oil-feed rate, the lower the slip (see Fig. 12(a)). In the
case of grease lubrication
0
No. 6307. It decreases
0
much lower slip is observed.
3000
1000
0
3000 Nkev/min)
ICUO
N (rev/mid
N(revlmin) b
a
C
Fig.
IL. Mean slip of No. 6307. (a) Effect of lubrication. P = IOO kg; spindle oil l , IO cc!min cc/min; cylinder oil A 9 cc/min, A 0.8 cc/min; x grease I cc. (b) Effect of load. Spindle oil IO cc/min. o, P = o kg; 0, IO kg; ., IOO kg: 0, 300 kg. (c) Effect of reduction of ball size,. Amount of reduction = 20 p. 0, P = IOO kg, normal size; 0, o kg, 4, 100 kg, all reduced six; A., o kg, A, IOOkg, ox reduced size. The bearing used in the experiment (h) was diffxezt from that in (a). 0,
I
Expressing
the position of the ball centre by the angle measured
of the load applied in the direction 90’,
gradually
develops
abruptly
disappears
raceways
increases
Consequently,
of the rotation
at the unloaded
at 320-330a4. at the unloaded
the contact
a maximum
clearance
Thus the radial clearance
side owing to the application
of radial
load.
loose. This causes
load the ball always keeps contact
is considered
the clearance,
slightly
balls of No. 6307. As a result, clearance
to have a marked smaller
increased
influence
with the
higher speed in the case of no load. Substituting magnetized
ball, between
neighbouring
the slip increases the magnetized
balls were of normal
its circumference
further.
to 60~ compared
imperfect
with the value of except
at
smaller ball only for the
It is reasonable
size. An experiment
for ordinary
considerably
the slightly
ball and the raceway
showed that
upon slip. In
balls were substituted
22 ,u for the ordinary balls. Fig. IZ (c) shows that slip increases
around
and
the ball and the
Here slip does not occur even at low speed (see Fig. 7 (j)).
order to change
clearance
at 29W300”
between
of the ball with the inner race becomes
slip of the ball. But under pure thrust raceways.
of the shaft, slip begins at nearly
side, attains
The actual
from the position
to consider
was increased
that
the
because
the
in which a ball was ground
sphericity
resulted
in a somewhat
ISS irregular
H.
b’. HlRANO.
rolling
motion.
There
l.\SOl!li
was no significant
difference
betne~n
the result5
using the solid cage and the pressed cage. The data Though
of the angular
the bearing
contact
bearing
of this type should
No. 7307
be properly
are summarized operated
in Fig.
under the action
1.3. of
N (rev/min)
Fig. 13, Mean slip of No. 7307. Spindle oil IO cc/min. 0, P L o kg, l , 100kg, solid cage ; A, o kg, A , 1oo kg, cag~lc~s.
thrust load, the experiment contact
between
deep-grooved
Fig. 14. Oscillograms of netized balls, P = LOOkg, (a) L magnetized balls in rcv/min. (b) S magnetized
No. 1,307 with magspindle oil, I cc/min. opposite holes, 3010 balls, 3050 rcv!mil1.
was carried out under pure radial load. Consequently
the balls and the raceways
became
type. This leads to the remarkable
much looser compared
increase
13 balls for IO in the ordinary
Fig. 13. The extraordinarily by adjacent
the cage and
type No. 7307. The result is also plotted in
high value of the slip is attributed
to the braking
action
balls.
The experimental
points of the bearing No. 6306 fall near the upper curve in Fig. IZ
(c). The slip in the bearing
bearings
with thr
of the slip even at higher
speed as shown in Fig. 8. The cageless bearing was prepared by removing substituting
the
were provided
after repeated
mounting
No. 6208 was at most 3qi0, except at very low speed. These
with the pressed steel cages. Since deformation and dismounting
data was inferior compared
was inevitable,
of the cages
the reproducibility
of the
with the case of the solid cage.
Change ilz rolling axis o/ ball The fluctuation maintain
of the amplitude
with the period ‘r, as shown in Figs. C-IO tends to
its wave form for a long time or after repeated
shaft speed. Further,
increase
it has been verified that the behaviour
same except for its phase. The resultant regular form shown in the oscillograms
and decrease of the
of each ball is exactly
current induced by the magnetized in Fig. 14.
the
balls has a
MOTION
Denoting
the frequency
189
OF A BALL IN A BALL BEARING
of the fluctuation
ratio CO~/CO~ is equal to the difference
by ns c/min or ws rad/sec = 24T,,
the
of wb/wCfrom the nearest integer m. The relation
can be written W&C = I(oe/wC) - ml or n&c = I(?+,) where %b =
2nw~/60
and vt, = 27~~()~/60.Because
6307 as in Table I, the value of m is 5 with a few exceptions In the case of No. 7307, however, 2,3
and 4 frequently
ml
(5)
where considerable
corresponding
slip is observed,
to m = 4.
several values
appear beside m = 5. This means that the period T, is determined
by the number of revolutions the ball becomes
-
e&*/e&* = 5.30 in the case of No.
until the accumulation
of the phase lag of the motion of
2~.
0 Fig.
1000
2000
3000 4000 N(rev/min)
15. Change of rolling axis plotted
5000
against
shaft speed.
10
2
3
4b
5, b/nTj
5
Fig. 16. Change of rolling axis plotted against nO/n,. (a) No. 6307. P = IOO kg. ., spindle oil IO cc/min; 0, cylinder oil 9 cc/min; o cylinder oil 0.8 cc/min; A spindle oil + o. I oh quartz powder; x grease I cc. (b) No. 7307. 0 0, P = IOO kg; o, o kg. (c) No. 6306, P = IOO kg. (d) NO. 6208, ., P = Iookg; o, okg.
As explained
above, the fluctuation
of the amplitude
angle between the rolling axis and the magnetic fizsis that of the fluctuation
of the instantaneous
represents
axis. Accordingly,
the change of the the frequency
rolling axis relative Wear,
os or
to the ball. For 4 (1961)
177-797
the sake of brwity Is, calculated
it will bc drsi~inated as /ls in the following description.
by means of the calibration
cur\cs show resonance
curve in Fig. 5, against
as in Fig. 15. The dependence
relation becomes different
of ,l x on speed is not unicl~i~~.‘I‘h(a
if the slip varies owing to the change of load orvtlicr
It is readily seen that thcx peaks of the curves in Fig. 15 correspond values of the ratio ,nb/wzc= w,;~oc. Hence, plotting 70’- mar the point MI :=
I?( = 2, 3. 4 and 5 are cltarfy pi-oportioli~tlit~
i.i’., .h . Sill interpreted
observed
j and
witi1
to the integral
in the cast of the So.
4. Four peaks corresponding
to
in the cast of So. 7307. 7’1~~cmves show ~10s~~
to icoscc ~~~,)~,~(I~~) :- cosec (TW~,~/~C)~). l’hc
(7f(tJ8jlrJc~]iimiw?s
f;~c‘tors
,/~ccagainst the ratio, the esperimen-
tal points fall on a unique cur\:e as shown in Fig. 16. Especially 0307 the change esceeds
l’lottinr:
the shaft speed, 111,.
dip
.i‘. In
the
pi-oportio~lalit~
constant,
following wction this I-t’I&$On \ViIlbi’
tlworttticall~~. I)ISCIISSIt)N
(‘mtsr
o,iclznngt~in
rolling uxis o,i hall
Since at the beginning to be constant
of the slip the position
under given operating
conditions,
surface of the ball, e.g. the displacement
the ball centre, bearing,
may be treated
the point CC) makes
revolution
of the magnetic
as follows.
If the ball behaves
CIJ~~/W C revolutions
about
of a point Co on the
pole after each revolution
of
as a roller in a roller
the x-axis
at the end of one
of the ball centre and comes to the point Co’. The shift of the phase angle is
then expressed
by p -- X((W,/W,.)
After successive revolutions /_C00x
d of the ball in Fig. 17 is considered the displacement
= J__CO’OX =:
MZ)~ or #q = znw,/w?
((j)
of the ball centre the point travels to Co”, CO”‘, . . Because
. =z 2 in this case, the fluctuation
of the amplitude
does not
occur. In order to explain
the fluctuation
from the path CO‘CO“CO”’
of the amplitude,
a deviation
of the point Cct
. should be assumed. Let the position of Cn after the deGa-
tion beC1’. It may be reasonable
to assume that the deviation
is caused by the angular
displacement E about the diameter AB (y-axis) passing the contact point A. At the unloaded side tht ball is set fret! from the contact with the inner race, though the contact with the outer race is still maintained here owing to the centrifugal fowl acting on the ball. Thus the effect of the cage! on the motion of the ball becomes signiIicant at the unloaded side. Actually, there is always more or ltzss dv\riation from tlw ideal contact condition. Especially, the contact point of the ball with tht: pock(~t holt~ of the cage or with the groove of thv outer race may have a slight offset from the plant. of syrnrnvtr!. of the groove. The offset cauws a spinning moment about then asis AI3. ‘I‘hii; suggests the origin of thv displacemrnt c. i. c.. t IH spin.
Fig. IS illustrate5 tile motion of the point C‘” as a result of the succcssiv~ displawment ,4 and F. Actnall!., the ball rolls with the thcorctical angular velocity c~J,,* at the loaded side and then slips at the unloaded siclc. Howlver, it is assumed for brc\. ity that the rolling velocity is c~lual to the constant myan angular \-ctlocit!, CII~nntl that the displacement F takes plaw instantaneously at the point A. Starting from the point CO, ij-displacement transfers it to C‘“‘, \vhich comvs to (‘I’ after F-displacement. \\‘hen the displacements arc repeated the point travels to (‘I’, (‘2”. (‘3”‘. , C‘,,(n).If the point Co is the magnetic pole, the angle /_,xOC,(n) is equal to the angle z for each revolution of the cage. The amplitude of the oscillogram is reprcscnt4 bv sin 1._AX‘,(“). The difference bctwccn ,zmax and Ymin shown in Fig-. IS is th(s c.hangc~ of the rolling axis rclati\~c to the co-ordinatc~ s\,stcrn fiwd to thv ball.
192
Theoreticul
camideration
analysis the position of the point on the batl is cxprt%sed b\- tlttz
In the following
unit vectors
cl”’ : Yn
,(n)
(x,,.
Gn) and C.,J., :
1’1,
, 1’,,-I
i.T,,..I.
,1’,,
I,
u^,,-1)
:7
where fl = ~,a, _ _ . The displacements /Iand R are represented by the transformations of the vectors from ~~-1to ~~~‘-1 and from ~~‘-1to r,, respectively. Those tr~~ilsformations are also written as the matricrs x =
I ‘: p .-; 0 COh ] 0 sin fl
;]
and Y :- [_$::
1
T]
(S)
- (YXP ]Z]
(9)
The relation between the vectors is then expressed 1~~ = Y*[z;]
Now, it is known that the resultant transformation T = YX is equivalent angular displacement y about the unit vector ~(a, p, Y)“. T is written as COSy + 12(r-ctts yrl T =
vsin
-V sin y + .Ajt(r-cos
y + puii(~~--cmT/I)
i -p sin y + vA( I
ws 7p)
7p)
to the
Q sin y $ Av(r--- cos y)
cos y + y"(r---cosy') ----a sin ?/,+ /LV(I cos y)
CI,S 1,’ +
7. sin y --C vh( I- cos y)
iJOI
L’( I-- (‘OS y1).1
where A = ~0s (42) v
-
-sin
sin (~/z)~s~n (ytk),
(e/2) sin (~~z)/sin (y/r)
cm (y/2)
I__ cm (F/L) cm (P/L)
.- sin (F/Z) cm
~~~2~~~~1~
(y$z)
md
(I rl
or sin (y/z)
--
1:1--4zos’ {p/2) cm2 (/3/L)
Since the repeated transformation (YX)n = Tnrepresents the angular displacement ny about the e-vector, it is derived from (IO) 1~~;substituting 1ny1for y: WV)
(12!
T(V)
Hence from (g), (IO) and (12)
= il (,4x0 + j4yf, + ~70) -t cos mp[xo --- A (Ax0 + jhuy,,T w-a) + sin q
(pa
vy0)
('3)
Denoting by LeOx and i_eOro the angle between r-vector and x-axis and that bctween e and ro-vector, respectively, eqn. (13) is written as x,, = ~0s (~~ec>x) ~0s (LPOQ) -I- sin (jot)
sin
(jr(h)
~0s 14
(14)
where CDS(LPor) 7 3.. cos (/ r0vo) = 3,s”+ /&Jo-‘- YZi.8 anit 21-= 8211’ --- sin -1 (.LG~ 1 ly~~)/sin (/ .20x) sin (/_eOro) are substituted.
(15)
MOTION OF A BALL IN A BALL BEARING
Since
LxOCn(n)
After rewriting
=
LxOr,
=
that
u takes discrete
u behaves
permitted,
LeOx) cos2 (u/z) + cos(LeOr0
values with increasing
as a continuous
variable.
xn is equal
to cos 0~.
and minimum
(k = integer).
+ LeOx) sin2 (U/Z)
12. However,
This assumption
except in the case where y/n is expressed
the maximum tively
mentioned,
(14))
cos OL= cos (LeOro -
The variable
CL, as previously
193
of LXare attained
(16)
it is assumed
of simplification
here
may be
by a ratio of simple integers. Then
at u = (zk + I) 7c and u = okn, respec-
Thus mrnax= LeOro + LeOx and ~(,,,i,, = 1LeOro -
LeOxl
(17)
Hence -‘ahin)/ LeOx = (L%lIax
or LeOx
The expressions
AB, where A and B are the projections
axis. The angular co-ordinate
(16). The ordinate
graphically
LeOx)
(18)
LeOx)
(18)’
in Fig. 19. Take two
max and amin on the unit circle. Then describe to OL
Fig. 19. Diagrammatic
of the circle AB is the variable
representation
of fluctuation
the
of A’ and B’ on the u in (14) or
of amplitude.
of the point on the circle A ‘B’ is found to be sin L-X from (14) and (17).
Since the amplitude
of the oscillogram
sented by the figure obtained Special
(for LeOvo <
(16), (17) and (18) are represented
circle with the diameter
abscissa
(for //eOro >
da/a
= (amax + bin)/
points A’ and B’ corresponding horizontal
=
is proportional
by plotting
to sin LX,the wave form is repre-
the ordinate
of the circle A’B’ against
the
u. attention
must be paid to the expression
amin may be taken in the negative
(IS’), which shows that the angle
sense, as shown in Fig. 19 by B”. Even if the change Wear, 4 (1961) 177-197
L
in is the same
fluctuation
as in the normal
of the amplitude
Fig. 19(b), the amplitude is an example treated
HIH.XISO,H.
I’.
19-i
difference
the circle .I “H”
correspondin@y.
observed
(II)
smaller.
the period1‘,. The oscillogram
record?;.
These
‘11~.
CJO“as in l:ig. ~(ki
two cases must be tlistincti\~c~l~~ however, is rather rare and tllta
case is readily apparent.
cos(~.&)x)
= ?., eqn. (18) can be written sin (LIr+./2)--
Substituting
becomt5
When *xrnnxescerds
of the data. Their occurrence,
from the ordinary
Now, considering
case (IS),
decreases
has two maximawithin
of actually
in the analysis
T.\XOl‘K
as
1 I---L”
for ii, we obtain sin (.iln/r)
= sin (r/l)/sin
(y/zj
(IQ1
The period of the change of a is represented
by ny =
according to eqn. (13). On the
other hand, the period T, in Fig. 6 contains
n revolutions
2x
of the cage, i.e., T, = ~27;.
Hence ?/I~7rn/lz Then the relation
rrzTe/Ta
-=
mw&u<
(19) becomes sin (.&z/2) -7 sin (c/r)/sin (nw,/wp) = sin (r/r)/fsin (nwb/toc); The expression
where (5) or (6) is substituted.
shown in Fig. 16. A more detailed discussion
(20)
verifies
the experimental
(20) results
of the quantitati~~e relation between the
spin E and the slip S will be developed in the next section. Relation
between sli$ artd chaagt: i?z rolling CL&
As mentioned assuming
above, the change in the rolling axis of the ball is clearly explained
the existence
of spin associated
2Or
20
(~2)
15 ~
E
.
5
s r‘
0.
0
10 slip
25
1 (b)
E
l*
10
by
with slip. In order to find the factors affect-
P,ZO
0
10
20
30
s
k)
40 50 60 Slip S PM
.
20 E
.
15
10 :
5
5
l*
l
LL 0
10
20
30 40 SP?%)
c (d)
l *
IO
l2kffKc
0
1
2
3
SW*)
Fig. 20. Relation between slip and spin. (a) So. 6307. P - IOO kg. (b) No. 7307. 0, P == ukg, 0, IOO kg, solid cage; A, o kg, A, IOOkg, cageless. (c) No. 11306. P = ran kg, (d) So. 6~08. 0. 3’~~ o kg; 0. TOOkg.
MOTION
ing spin,
it must
OF A BALL
be calculated
(or ws/oe) and AX into relation S it is finally confirmed The proportionality and on the operating metric contact
first
constant,
however,
conditions.
Since
hole in the direction the assembled
is specified
to S, as in Fig.
0.15
cage into the bearing
originates
20.
in the asym-
of the bearing.
this effect.
One of
The amount
of the
mm in the moderate
The position
case. The results
of the hole in which the magnet-
by the angle measured
of the revolution
of WC/CU~
the mean slip
hole of the cage or with the groove of the
was carried out, exaggerating
21.
data
against
is found to depend on the type of bearing
case and
in the former case are shown in Fig.
the observed
the spin probably
holes was offset in the axial direction
offset was 0.3 mm in the extreme
195
the results
that the spin E is proportional
of the ball with the pocket
ized ball was inserted
BEARIKG
by substituting
(20). On plotting
outer ring, the following experiment the pocket
IN A BALL
from the centre of the offset
of the cage. The axial deviation was shown also in Fig.
21.
of the holes in
The holes o” (offset
N (rev/mid
Fig.
21. Slip Of balls in cage with offs.3 hole. P =
hole), 45” and 180’ occupied 270’ the internal
position.
the external
position
The cage was supported
x/min. IOO kg, spindle oil 10 G
and the holes 9o”, 135”,225’ and by the balls in three external
The four internal
balls were supposed
This is confirmed
by the fact that the slip occurs at the internal
Fig.
21,
and that at the external
to be free from contact
holes negative
holes.
with the inner race. holes, as shown in
values are obtained.
The negative
slip
Wear, 4 (1961) 177-197
1:. HIRANO,
196 represents
the overrun
H. TANOUE
of tfre ball owing to the offset of the ball centre.
In the inter-
mediate case of the hole 315”, the ball shows small positive slip at lower speed. Further,
it was found that the spin E calculated
holes 90” and
225’
one may conclude
22
the greater is the spin of the bail. The result of the smaller offset,
mm, also supports
looser except
this conclusion.
at the offset hole. Negative
it is shown that an appreciable
Fig.
spin associated operation negative rings,
of the bearing,
Here,
slip is considered
unfavourable
effect
under thrust
and combined
slip. The
hole.
to have no harmful
contact
influence
upon the
side. On the contrary,
the
of the ball with the grooves of the
of the spin becomes
load were disregarded.
effect of slip and spin, however,
out under thrust
with the negative
because it occurs at the unloaded the harmful
of the balls was much
only at the offset hole. In Fig.
spin is also associated
slip is caused by the asymmetric the experiments
the contact
slip occurred
~2. Spin of balls in cage with offset
with positive
and consequently
research
Consequentl!..
22.
that the larger the offset of the ball centre from the plane of sym-
metry of the bearing, 0.15
by using eqn. (20) is larger at the,
than at the holes 135” and 270°, as shown in Fig.
further
inevitable.
In this
In order to find the
investigation
should be carried
loads. CONCLUSIONS
By using a magnetized ball bearing
ball it was possible to follow the behaviour
under radial load. The motion
sional and associated
of the ball in a
of the ball was found to be three-dimen-
with slip and spin. The conclusions
are as follows:
I. Slip of the ball occurs at the unloaded side, where its contact
with the raceways is
looser. Slip decreases with increase in speed and load. It increases with radial clearance. 2.
The instantaneous
rolling axis relative
to the ball changes regularly.
Its change
is closely related to the slip. 3. The cause of the change placement
in the rolling axis is spin during slip. The angular
of the spin is proportional
4. The spin increases
dis-
to the mean slip.
with the asymmetry
of the contact
of the ball with the cage
or with the grooves of the races. 5. Further
increase in asymmetry
slip is observed;
causes negative
in this case the contact
is considered
slip. Under pure thrust to be still maintained.
load no
197
MOTIONOFABALLINABALLBEARING ACKNOWLEDGEMENTS
The authors wish to thank the members of Research Laboratory, Nippon Seiko Co., Fujisawa, Japan, especially Dr. T. HATTORI and Mr. S. YAMAMOTO for their kind support.
FEFERENCES 1 W. HAMPP, Igr.-Arch., 12 (1941) 6. 2 T. SASAKI et al., Trans. Japan. Sot. Mech. Engrs., 8 (1942) 1-64 (in Japanese). 3 F. HIRANO AND H. YOSHINARI, Machine Science, 2 (1950) 31 (in Japanese). 4 K. IIDA AND T. IGARASHI, J. Japan. Sot. Mech. Engrs., 62 (1959) 369 (in Japanese). 5 E. T. WHITTAKER, A Treatise on the Analytical Dynamics, 4th ed., Cambridge University 1937.
Press,
Wea+‘, 4 (1961) 177-197