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Rehfeldt, H. F.
The coefficient of friction
of ball bearings and horse
Digitized by the Internet Archive
in 2009 with funding from
CARL!: Consortium of Academic and Research Libraries in Illinois
http://www.archive.org/details/coefficientoffriOOrehf
The Coefficient of Friction of Ball
Bearings and Horse Power to Drive
A thesis
PRESENTED BY
H. F. REHFELDT
TO THE
PRESIDENT AND FACULTY
OF
ARMOUR INSTITUTE OF TECHNOLOGY
FOR THE DEGREE OF
BACHELOR OF SCIENCE
IN
MECHANICAL ENGINEERING
MAY 29, 1919
APPROVED
ILLINOIS INSTITUTE OF TECHNOLOGV
PAUL V, GALVIN LIBRARY
35 WEST 33RD STREET
CHICAGO, iL mm
Professor of Mechanical Engineering
Dean of EuEfineering Studies
Dean of Cultural Studies
lEDEX
Apparatus, Description of • 2 - 5
Calibration of Machine 5 - 12
Coefficient of Friction, Elements Involved., IE - 14
Conclusion 20 - 21
Design of Bearing Surfaces • 17 - 18
Method of Procedure 15 - 16
ITeirr Desi gn of Housing 18
OlDOPGt 11
APPEl^IX
Axial Load Curves \. 27 - 29
Axial Load Data 35 - 38
Radial Load Curves 24 - 26
Radial Load Data 30 - 35
Sample Calculations 22
28126
DETSRinNATTOlI OF 'THE COEFI'ICIENT OF ERICTIOK
OF BAIL BEARIiTGS AND THE HORSE POWER REQUIRED TO DRIVE
The olDJect of this investigation is the deter-
mination of the coefficient of friction of "ball bear-
ings and the horse power required to drive them, mider
various loads and speeds.
The machine for making this investigation was
designed "by Prof. G. E. Gebhardt of the Armour In-
stitute of Technology and huilt in the shops of the
school. It consists essentially of four hall
bearings and a shaft, mounted as sha7n in figure 1.
The ti7o outer bearings support the shaft, v;hile the
two inner ones are merely hung on it at equal dis-
tances from the outer bearings, ihe machine is so
made as to nermit the placing of axial as well as
radial loads on all four bearings, either independ-
ently or in any combination of the two.
The details of the machine will not be taken
up. It consists of a base A, figure 1, on which are
3.
moimted two Hess-3right #6309 iDall "bearizigs, B
and C. The two central "bearings, also Hess-Briglit
#6309, D and E, are mounted on the slfaft, hut not
on the hase; the housing heing kept from rotating hy
the casting P, shovrn more clearly in figure 2.
The shaft is connected hy means of a flange coupl-
ing to a sensitive electric cradle dynamometer.
The lock rings G-G keep the "bearings in a fised
position relative to the shaft.
The method of applying the radial load can "be
seen in figure 1. The pin H is screwed into the
"base, and is the fulcrum for the lever J, The ratio;
of the lever arms is 20 to 1, therefore any load
applied on the end of the lever will induce a load
20 times this on the hearing. The method of
fastening the rod K into the housing can he seen in
figure 2. The rod L passes through thehDusing of
the hearing and also through the short arm of the
i.
lever M, teing held ' in place "by two conical
headed nuts. Thus a load applied at the end of
the lever M v;ill "be transmitted to the housing of
the "bearing, whibh in turn presses against the
outer race of the hall hearing, the inner race
being pressed against the collar on the shaft.
This latter arratig eiiie nt can he f ollov'ed out quite
easily in figure 3. The ratio of the arms on lever
M is 10 to 1, thus any load placed on the end of
the lever will transmit a load ten times as great
to the hearing.
In conducting this investigation, the machine
had to he calibrated before any tests could be made,
which consisted of finding the dead v/eights of the
levers (this includes the weight tods on the ends
of the levers.) To accomplish this a bell crank
was made, as shcr^n in figure 4. This was placed on
the end of the base with the aid of two small
brackets, as shown in figiare 5, so that the hole B
would line up v;ith tlie bolt C. A v;ire was then
4>.
8.
f
passed through the hole B, and fastened to C,
Another v/ire was passed throng the hole E, so
that the weights could "be placed on the end of it.
Y/eights were now placed on the end of the ann D,
until a halance accurred.
Knowing the weight of the aim D, and the load
on it, a moment equation can "be set up as follows:
5 W = 5w + £ Jf w
W a w + .588 w fl)
Since w is so small compared with the other
weights, the term •588w may he omitted leaving Y/ ~ w.
In following out the ahove procedure it was
found that the levers with 12 inch weight rods gave
an 11 pound pull at the "bearing, while the levers
with 24 inch weight rods produced a 15 pound pull
at the hearing. Enough weights were added to ifchS
weight rods to produce a 100 pound axial load on the
bearings.
10.
The radial load dead weights were foimd Tdj
weighing the radial load levers and weight rods, and
hy finding the center of gravity of the levers.
Knov;ing these values, the value of P, f jgujre 6, can
TDe calculated from the following equation:
P X 1 = R X T; + 10 s w (2)
In whicQi -
P - puU on the hearing
R - distance of pivot to center of gravity
Vif - weight of lever
w - weight of weight rod.
For one lever the follov/ing values were ob-
tained, R - 8 5/8", W r 2.6#, w s 1#
Suhstituting these values in equation (2) we
get P = 42. 4#.
For the other lever the following values were
obtained R » 8 l/4", W = 2.3^f, w = 1#, P = 39#.
Enough weights were added to the weight rods
to produce a 50 pound radial load on each of the
hearings.
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The dynamoneter ^vas imcoupled from the macaine,
and its zero load determined, vrhich was made 0.5
pounds. The length of the dynamometer arm was fonnd
to be 21^-", The dynamometer was recoupled to the
machine, the apparatus noxr heing ready for the first
run.
Bgfore entering into the actual testing let us
see what detemines the coefficient of friction of
a ball hearing. First, we must have a definition
of the coefficient of friction, which is the ratio
_of the force required to slide one surface over
another, to the total load on the surfaces. In a
"ball hearing, or any hearing for that matter, it
is a torque which tends to do the sliding. The
torque required to drive a hall hearing is con-
stand for a fised load and speed, hut the size
of the force producing this torque varies according
to the distance from the shaft. Since the co-
/:?.
efficient of friction depends directly on the
force, and only indirectly on the torque, some
means should he made for determining this force
viiich vrould he universal.
It can easily he seen that if the torque arm
say, is measured to the surface of the outer race,
the force, and therefore the coefficient of friction
•would he smaller than if measured say to the surface
of the innter race. This question of measuring the
torque arm offers a suhject of much discussion
and confusion amoiig hall hearing manufacturers, and
also hetween the manufacturers and huyers.
let us assume that all manufactuers hased
their coefficients of friction on the distance from
the center of the sliaft to the surface of the inner
race. This still would not solve the prohlem,
hecause some manufacturers may malae a thick inner
race, v^hile others would malae a veiy thin one.
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-'O'^f
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This, then r^ould te very intpracticalDle.
The only practical method left, and the one to
solve all difficulties is to "base the coefficient
of friction on the radius of the shaft. It will
be said that this will give a larger coefficient
of friction than really exists, which is true, hut,
if all hall hearing manufacturers hased their
results on this, comparative values would he
obtained which would he just as good as the
actual coefficients fo r determining the merits of
one hearing over another. Another advantage of
this raet^iod of determining the coefficr'ent of
friction is that it can he compared directly to
hearings of the roller and hahhit type. Therefore,
in tlae investigation, all coefficients of friction
will he hased on the radius of the shaft.
In CO nduc t ingf -th is the si s t he f o 11 ow ing . me th od
of procudure was adopted:
IS.
A zero load rim was made first. This was
made with all of the levers removed, therefore, the
only load on the hearings is that produced h.y the
weight of the sliaft, which amounts ftf ahout 10#
per hearing. The torque necessary to drive under
no load was found to 0/16 pounds for speeds as
high as 2500 R.P.M. Under these condit:ons the
coefficient of friction is constant and the
horse power to drive varies directly with the spe.ed.
The radial load test was made neict. Starting
with as low a speed as possible and keeping it con -
stant, the hearings were loaded hy increments of 100
pounds, up to 2060 pounds, the torque heing noted
at each nev load.
Knowing the toruqe, speed and load on the
hearing the horse -ower to drive and coefficient of
driction may he calculated.
In testing out the hearings under asial loads
•«i.r' e ?■-■!«! ■■^.:
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â– Jj ,3
10
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16.
it was foimd tliat the two bearings supporting
the shaft could not M loaded Sixially as
originally intended, because the hearing housing
would not slide freely in the main support. This
necessitated the following procedure:
The two, center bearings were removed, and the
machine run at different speeds under no load the
torque being noted at each speed. The two center
bearings were then replaced, and axial loads applied
.to them at different speeds, (starting vrith a zero
load, and increasing to 800 pounds by increments
of 100 - rounds.) T'le torque was noted at each
changed of speed or load, from which was subtracted
the torque produced by the two outer bearings.
This then gave the true torque duetothe axial load
on the bearings, from which may be calculated the
coefficient of friction and horse power to drive.
It is not thw scope of this investigation to
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deteimine the Id est shape of the races, and the number
and size of "balls, for the least frictional losses,
hut a â– ^ord may he said ahout these factors. It has
"been fotind that the frictional resistance was least
for halls rolling hetween straight line sections,
or perfectly flat sar faces, giving two points of
contact. Increasing the points of contact to three
and foTir produced higher frictional resistance,
without materially affecting the cariying capacity.
Curving the race resulted in an important increase
in carrying- capacity, v/ith a harely measurahle
increase in friction. The spacing of the halls
"by a separator has proven to he more satisfactory,
allowing them to rest one against the other.
Striheck developed from his experiments the
following equation for the carrying capacity of an
annular or radial hearing:
2
L r End in which
â– i sr^t
• 9.t9Jb
16,
I = load capacity in potuids
d r ball diameter in eights of an
inch, e.g., \ inch diameter hall,
d = 4
n = nimiher of halls
k r a constant dependent upon the material
the shape of the hall - supporting
surface and the speed.
An attempt was made at getting the coefficient
of friction and horse power to drive unSer a com-
hination of radial and axial loads, hut the results
obtained v;ere inconsistent with the results obtained
when the loads were applied individually. In most
cases no change was noted between combination loads
and radial loads alowe. An explanation of this
arises from the fact that the ball bearing did not
slide freely in the housing as soon as a radial load
was applied. It was therefore necessary to redesign
the two center housings, as shownrin figure 7. Time
did not permit the making of the housings and there-
fore presentable data was not obtained for combination.
1^.
Z t: J.
nGURE 7
20.
loads.
If no\7 we refer to the various curves we can
study the action of the tiearings. Referring to
the radial load curve (horse power vs. speed) we
see that the horse power to drive varies directly
with the speed and increases with the load on the
hearing. Since these are straigltt lines the equa-
tion of them will he in the form of y « m x •♦• "B,
hut h is zero for all of these, therefore the
'equation vail reduce y = ra 3: or H.P. = m*R.P.M.
â– dierern is the slope of x he line and has the
folloT7ing values:
Radial Load
Radial load
in pounds. m
in
pounds, m
200 .0000SV5
1200
.0000545
400 .0CC040
1400
.0000605
600 .0000450
1600
.0000672
800 .0000465
1800
.0000740
1000 .000050
2000
.0000835
For intermediate loads we can get ra hy inter-
polation.
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u'lO'A) OV-i.t)
^P,"'.'
o or; '"! e
»t:
i-'O r
Z(.
Referring to the axiBl load cin-ve (horse power
vs. epeed) we see that the curves are straight lines
np to 1600 R.P.M. Thus? the same equation vrill hold
here, as was used for radial loads, H.P. - in R.P.M. in
which m has the following value as determined from
the cu.rves:
Axial load in nounds n
200 .00002
400 .0000280
600 .0000S4S
800 .00004075
For internediate loads get m hy interpolation.
If we tiirn to the coeffic:!ent of friction
curves we see that the two curves liave practically
the sane fomi, the only difference heing that one
is dravm closer to the y axis thsm the oth^r. They
show that the coefficient of friction is less for
asial loads, varying from to 1000 pounds, than for
this range in radial loads. Beginning with 1000
poujids and up the coefficient of friction is the
same for "both types of loads.
snton) sv ' - o.r .L'
APPMDIZ
zz.
Sample O?:lculg-tions
Radial I OP a
Let C - coefficient of friction
T - total torque (torque of 4 'bearincrs ) in inch
H - Radius of shaft in inches Pounds
I - lOvid on bearing in pounds
H.P. - horse power to drive one hearing
R - radius of dTnEmometer arm - Sl-g-"
i\ - R. P. ::.
Y/ - net scale reading in pounds
Calculations made for 1260# load and 1000 R.P.M.
P r, ^ ■- «46 ^ gl»5 z .00324
4 ^ ** 4 X .8858 X 1260 ==
H.P. -
2 ff R IT W - 2 tr X SI. 5 X 1000 x .46 = .0576
35,000 X 4 12 X 33,000 x 4 '""^
iUcipl Load
IJotations same as ahove.
Calcula,tions made for 400# load and 1010 R. P. M.
_JL_ = .12 X 51.5 Q .00533
- 2 r b E X .8858 x 400 " '
_ 2 TT R IT V7 . 2 tr X 31.5 x 1010 x .15 t .0304
^-' ^* ' 33,000 X 2 " 12 X 33,000 x2 ==
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.14
2.205
.00498
.0123
600
.28
.14
.14
2.205
.00414
.0123
700
.50
.14
.16
2.52
.00406
.0140
800
.32
.14
.18
2.835
.00400
.0158
680 R.
P. M.
L
Scale
Reading
Scale
W
t
C
H P
20
.18
.02
.315
.0034
100
22
.18
.04
.630
.00711
.0068
200
24
.18
.Q6
.945
.00533
.0104
300
26
.18
..08
1.260
.00474
.0136
400
28
.18
.10
1.575
.00445
.01700
500
30
.18
.12
1.890
.00426
,0204
600
32
.18
.14
2.205
.00413
.0233
700
34
.18
.16
2.520
.0040 6
.6276
800
36
.18
.18
2.835
.00483
.0306
37.
1010 R. P. M.
I
Scale
Reading
Scale
w
T
C
HP
20
10
.02
.315
.00506
100
24
10
.06
.945
.01065
.0152
200
26
10
.08
1.260
.00711
.0202
500
28
10-
.10
1.57 5
.00597
.0253
400
30
10
.12
1.890
.00533
.0304
500
30
10
.12
1.890
.00427
.0304
600
32
10
.14
8.205
,00^0.4
.0354
700
32
10
.14
2.205
,00457
.0354
800
34
10
.16
2.520
.00356
.0404
1380 R. P. M.
I
Scale
Reading
Scale
W
T
C
HP
.22
.20
.06
.315
,00666
100
.26
.20
.06
.945
.01065 ,
,0247
200
.28
.20
.08
1.26
.00711
.03300
300
..-^o
.20
.10
1.575
.00597
,0418
400
.30
.20
.10
1.575
.00501
,0412
500
.32
.20
.12
1.89
.00427
.0495
600
.32
.20
.12
1.89
.00356
.049 5
700
.34
.20
.14
2.20 5
.00356
.oa77
800
.36
.20
.16
2.52
.00356
.066
30.
1650 R. P. M.
I
Scale
Readin
Scale
I
W
t
C
H P
.22
.20
.02
.515
.
00825
IOC
,26
.00
.06
.945
.01065
.0247
200
.28
.20
.08
1.26
,00711
.05300
300
.50
.20
.10
1.575
.00597
.0412
400
.50
.20
.10
1.575
.00501
.0412
500
.52
.20
.12
1.89
.00427
.0495
600
.52
.20
.12
1.89
.00556
.0495
700
.54
.20
.14
2.205
,00556
.0577
800
.26
.20
.16
2.52
.00556
.066
2000 -
2080 R
. P. M.
I
Scale
Readin,
Seal e
W
t
C
H P
24
22
.02
.315
.0104
100
26
22
.04
.650
.00711
.0208
200
28
22
.06
.945
.00553
.0316
500
50
22
.08
1.260
. 00414
.0416
400
30
22
.Q8
1.260
.00556
.0416
^nn
3 P.
P.P.
.in
1 .i^75
.00356
.0520
600
58
22
.10
1.575
.00098
.0520
700
34
22
.12
1.890
.00305
.0624
800
56
22
.14
2.205
.00311
.728