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Jacobson2003-StribeckLecture.pdf
Tribology International 36 (2003) 781–789
www.elsevier.com/locate/triboint
The Stribeck memorial lecture
Bo Jacobson ∗
Machine Elements Division, Department of Mechanical Engineering, Lund University, Box 118, Lund 221 00, Sweden
Abstract
Stribeck’s publications ‘Kugellager für beliebige Belastungen’ and ‘Die wesentlichen Eigenschaften der Gleit- und Rollenlager’
were remarkable for their time. Many of the findings were revolutionary and are still considered quite modern.
The method used by Stribeck to calculate load distribution between the balls within a ball bearing is still used today, otherwise,
a full finite element analysis would be required.
His measurements and analysis of bearing friction is also astonishingly accurate. Most of the ball-bearing types studied by
Stribeck are not in use any more, but the deep-groove ball bearings he investigated were similar enough to be compared with
modern deep-groove ball bearings. Stribeck measured the friction coefficient 0.0015 and SKF gives the friction coefficient 0.0015
for deep-groove ball bearings in the catalogue from 1989. For the first time, in the SKF catalogue from 1989, the life calculation
method took into account contamination also. Stribeck had stated already 100 years ago that cleanliness was very important for
bearing life and function.
Stribeck’s investigation of journal bearing friction, as a function of load and speed, was also extremely important, as he showed
the possibility of finding a point of minimum friction for lubricated applications. He also showed that the friction for sliding bearings
started at high friction at low speeds, decreased to a minimum friction when metal to metal contact stopped, and then increased
again at higher speeds, which is the well known ‘Stribeck curve’.
 2003 Elsevier Ltd. All rights reserved.
Keywords: Friction; Load; Ball bearings; Roller bearings; Journal bearings
1. Introduction
It is now 101 and 100 years, respectively, since Richard Stribeck published his famous papers, ‘Kugellager
für beliebige Belastungen’ (1901), and ‘Die wesentlichen
Eigenshaften der Gleit- und Rollenlager’ (1902) in Zeitschrift des Vereines deutscher Ingenieure [2,3]. The first
paper treated the load-carrying capacity of ball bearings
brilliantly and the second paper resolved the longstanding dispute over bearing friction characteristics as a
function of load, speed and lubrication.
Stribeck [1] was born in Stuttgart in Germany on 7
July 1861. He received his basic education in his hometown where he finished his studies at the Technische
Hochschule in Stuttgart. Later, he gained practical
experience in various places, including Königsberg. At
the age of 27, in 1888, he was back in his hometown,
Stuttgart, and was appointed as Professor of Machine
∗
Fax: +46-46222-8504.
E-mail address: [email protected] (B. Jacobson).
0301-679X/$ - see front matter  2003 Elsevier Ltd. All rights reserved.
doi:10.1016/S0301-679X(03)00094-X
Construction at the Building School. Two years later, in
1890, he moved to the Technische Hochschule in Darmstadt. After three years, on 1 April 1893, he moved to
the Technische Hochschule in Dresden. As Professor für
Maschineningenieurswesen, he gave his inaugural
address in Dresden on 6 May 1893 on the subject ‘Progress in the production of steam and in the exploitation
of its energy in steam engines during the last 20 years’.
Seven weeks later, on 1 July 1893, he went on a sabbatical leave to the United States of America.
Later, when he was back in Dresden, in the academic
year 1896–1897, he was appointed to the governing
body of the newly founded Machines Laboratory I
(Strengths). He did not stay there long and moved to
Berlin in March 1898. There, he became head of the
metallurgical division of the Centralstelle für wissenschaftlich-technische Untersuchungen in Neubabelsberg, Berlin.
During the years 1889–1893, when Stribeck worked
in Stuttgart and in Darmstadt, he published some 10
scientific papers. This large early output was mainly on
giving solutions to the problems of boilers and steam
782
B. Jacobson / Tribology International 36 (2003) 781–789
turbines for marine applications. Only in the last paper
in this group, Stribeck dealt with dynamic loading on
the main and cross-head bearings of steam engines. That
was thus the first time he became involved with tribology
and tribological problems. When he was in Dresden,
Stribeck studied gears, with particular emphasis on
experimental studies of worm gear drives. He experimentally determined the limiting loading for correctly
working worm gear drives as a function of load (F) and
speed (n). He found that a hyperbola described the function well and that F·n = constant described the combination of load and speed, which could be allowed for
any specific drive lubricated with a certain specific oil.
In 1897, he published a paper on the consequences of
wear of gears.
As mentioned above, Stribeck moved in 1898 to
Berlin to work as one of the directors of the newly created Centralstelle für wissenschaftlich-technische Untersuchungen in Neubabelsberg, and to head the metallurgical division. There, he undertook and published the
findings from his most basic tribological studies between
1898 and 1902. They were in two different areas: the
load-carrying capacity of ball bearings and the friction
characteristics of plain bearings and radial rolling
element bearings. Stribeck applied sound physical principles to his experimental investigations, and used the
theoretical background information available at that
time.
In his very well known study on the load-carrying
capacity of ball bearings Stribeck [2] gave scientific
basis for the development of a large industry. He used
the Hertzian contact theory, which by then had been
known for 20 years, to connect the loads on the different
balls to the external load. He then compared the loads
on single ball contacts of different geometry, which were
needed to give visible plastic deformations. He used
three balls in contact, see Fig. 1, a steel plate sandwiched
between two balls, and more conforming contacts of the
type found in deep-groove ball bearings. Stribeck’s studies, which were commissioned by the Deutsche Waffenund Munionsfabriken, Berlin, were epoch-making in
their effect on the ball-bearing industry. Its major impact
arose from the analysis of the carefully conducted
experiments, see Fig. 2, and the sound appreciation of
Hertzian contact theory, together with the presentation
of practical formulae in a form suitable for the bearing
designer. Stribeck’s approach to the question of loadcarrying capacity of ball bearings was about seven years
in advance of similar considerations in gear design.
2. Load-carrying capacity for ball bearings
Stribeck’s paper on ball bearings ‘Kugellager für
beliebige Belastungen’ was published in Zeitschrift des
Vereines deutcher Ingenieure on Saturday 19 January
Fig. 1.
Test set-up for ball deformation measurements.
1901, as a ‘Mitteilung aus der Zentralstelle für wissenschaftlich- techniche Untersuchungen’. Stribeck stated
there that his results were applicable not only to ball
bearings, but also to all other types of concentrated contacts of Hertzian type.
He used Hertz’ equations for deformation, contact
dimension and contact pressure, and applied it to ballbearing geometry. By comparing the permanent deformation left after the unloading, he found that contact
pressures could be much higher than the yield strength
of the steel, and the elastic deformations were almost
halved due to the hydrostatic stress components in the
contact centre. He made experiments with three balls in
series and a flat or concave plate between two balls to
find the allowable load before plastic deformation
B. Jacobson / Tribology International 36 (2003) 781–789
783
ing the maximum contact pressure 4.6 GPa for selfaligning ball bearings, 4.2 GPa for other ball bearings
and 4 GPa for roller bearings. By measuring the load and
ball diameter combination, which just started the plastic
deformation, Stribeck found that P / d 2 = k was constant
for each geometry type (ball–ball, ball–flat, ball–
concave), and that for each steel hardness, the allowable
load was proportional to the square of the ball diameter.
For the materials he investigated, different geometry
gave different k-values between 2 and 10 kg per oneeighth of an inch depending on the groove forms in
the rings.
3. Friction for ball bearings
One week later, on 26 January 1901, Stribeck published the second half of the paper ‘Ball bearings for
varying loads’ (Kugellager für beliebige Belastungen),
and there the friction in ball bearings was analysed. He
elegantly split the power loss in each contact point into
one rolling resistance component and one spinning
resistance component, see Fig. 3. By summing up the
power loss components, it was obvious that the power
loss was proportional to the ratio of ring diameter to ball
diameters, or for full complement bearings proportional
to the number of balls. The bearings should have a small
number of large balls to get low power loss.
Stribeck needed to know the load distribution between
the balls, and assumed then that only the contact points
Fig. 2. Contact deformation shown as a function of load for three
types of contact.
occurred, see Fig. 2. The geometry balls on balls had
sizes 3/80, 1/20, 5/80, 3/40, 7/80, 10 and 1 1/80. For flat
between balls, the sizes were 3/80, 1/20, 5/80, 7/80 and
1 1/80.
By using linear elasticity and measuring the elastic
compression deformations of diameter 16 mm cylinders
with 32 mm length, he found that the modulus of elasticity varied slightly between un-hardened, oil hardened
and water hardened materials. The difference between
the highest and the lowest value was 1.25%, so he used
a mean value in his analysis (E = 2,120,000 kg/cm2).
Stribeck found, just as predicted by Hertz’ theory, that
the elastic deformation was largest for a ball on a ball
contact, smaller for a ball on a flat contact and still
smaller for concave contacts. His limit of detection of
plastic deformation was 0.00025·d, which is 2.5 times
as large as the plastic deformation used by ISO before
the ISO 76:1987 standard for static load-carrying
capacity for rolling element bearings was adopted.
Today, the static load-carrying capacity for rolling
element bearings is defined by an elastic calculation giv-
Fig. 3.
Rolling and spinning contacts between ball and race.
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B. Jacobson / Tribology International 36 (2003) 781–789
were deformed, and the rings had no bending deformations. For a bearing with zero radial play, the load
per ball was proportional to cosg 3/2 where the angle g
was measured from the ball with the maximum load.
Summing up the load components, he found that for 10,
15 and 20 balls in the bearing, the total load-carrying
capacity divided with the maximum load on one ball was
the number of balls divided with 4.37. To be on the safe
side, he gave the famous equation:
P0 5 5·P / z
(1)
where P0 = maximum load on one ball; P = maximum
load on one ball; and z = number of balls.
For friction calculations, he also found that the sum
of the radial forces around the circumference of the bearing was 1.2 times the external load.
4. Experiments with ball bearings
Stribeck designed and manufactured a friction torque
measurement rig, which automatically compensated for
the weight of the different moving parts, see Fig. 4. He
was thereby able to measure the small friction with high
accuracy. He found that for small amounts of lubricant
in the bearings, the friction was independent of the temperature. His experiments were run at 65, 100, 130, 190,
380, 580, 780 and 1150 rpm, starting with the lowest
speed. When decreasing the speed from 1150 rpm, Stri-
Fig. 4. Friction measurement set-up with compensating counterweight.
beck noticed that the friction was lower than during the
speed increase despite almost constant temperature. He
assumed that the friction decrease was due to running in
of the surfaces, as this phenomenon was much stronger
for rough surfaces than for polished surfaces, where the
friction was not changed by running in. Stribeck ran tests
with five different types of contact geometry for ball
bearings and found that a deep-groove ball bearing with
open osculation had the lowest coefficient of friction,
0.0015, of the tested types, see Fig. 5. For that bearing
the largest allowable ball load was 11·d2 kg where d was
the ball diameter in eighths of an inch. That shows the
material development the last 100 years. The maximum
load to avoid plastic deformation then was 3100 N on a
diameter 17 mm ball. Today, the ball load is 10,000 N
for infinite life. Stribeck also ran endurance tests and
detected that also small variations in material hardness
had a large influence on life. As soon as a plastically
deformed rolling track was visible, the bearing life was
short. For good steel hardness, he found the maximum
allowable ball load to be
P 5 10·d2 kg (d in inches)
(2)
P 5 100·d2kg (d in cm)
(3)
when the ring groove radius was 2d / 3.
Such good results were only possible when balls and
rings were made of good hardened material, and dust
and contamination, e.g. forming sand or surface parts
from the moulding, were removed from the bearing
lubricant, just like for sliding bearings.
Fig. 5. Ball bearings with grooved rings similar to modern deepgroove ball bearings.
B. Jacobson / Tribology International 36 (2003) 781–789
5. Bearing lubrication
One and a half year later, Stribeck published his paper
on friction and lubrication for journal bearings and rolling bearings, ‘Die wesentlichen Eigenshaften der Gleitund Rollenlager’, starting on Saturday 6 September
1902, No. 36, of Zeitschrift des Vereines deutscher Ingenieure.
He wrote the paper in order to show which bearing
type was the best performer under different running conditions, as no hard data were available for journal bearings. No information was available earlier on the different lubrication regimes and their friction coefficients; it
was impossible to know which bearing type to choose
for different applications. During the investigation, Stribeck not only compared hydrodynamic bearings and ball
bearings, but also studied roller bearings.
6. Journal bearings
Stribeck had found that not only roughly manufactured bearing surfaces needed to run-in for some time.
How fast the running in process could be, was mainly
a function of the bearing material. Fastest were some
white metals, slightly slower were bearing bronzes and
very slow was cast iron. Steel was practically impossible
to run-in. The load possible to use during the running in
process was also a strong function of the material. White
metals could stand the highest running in load, and steel
the lowest. For white metals with a yield point of 20
MPa, mean pressures P / (L·d) = 9 MPa were possible to
carry. White metals also gave the least wear of the steel
shaft, and were therefore considered the best bearing
materials.
Stribeck also mentioned that a bearing, which had
been run-in at a low load, had to run-in again if the load
was increased. That is still true, but still not very well
known in industry.
7. The lubricant
Stribeck saw that the lubricant had a very strong
influence on the friction, but also on the allowable load
and the safety of the bearing function. If a bearing was
run-in with one oil, another oil could start the running
in anew or fail the bearing. The best oil found, and used
for the tests, was ‘Gasmotorenöl’ delivered from ‘Gasmotorenfabrik Deutz’. The same oil had also been used
in the earlier ball-bearing tests. The viscosity at 20 °C
was 0.28 Ns/m2, at 40 °C, it was 0.075 Ns/m2, and at
100 °C, the viscosity was 0.012 Ns/m2. The flame point
was 180 °C. By comparing the viscosity variation for
different temperatures, Stribeck found that the thin Velocite Spindle Oil changed its viscosity in the same way
785
between 20 and 50 °C as a heavy mineral oil between
80 and 110 °C, see Fig. 6.
8. Cast iron Sellers-bearing with ring-lubrication
Stribeck’s test bearings were different from most
modern journal bearings, in that the bearing width (230
mm) was much larger than the diameter (70 mm), see
Fig. 7. This made the bearing sensitive for shaft bending
and misalignment. To avoid temperature transients,
causing temperatures at different points in the bearing
to differ by a few degrees, measurements were made
slowly. Steady state temperatures were reached after 2–
3 h of running. That also showed Stribeck that most
machines in transient use never would reach the steady
state temperature.
Stribeck’s experiments clearly indicated that the
higher the speed is, the higher is also the bearing load
when the minimum coefficient of friction is experienced.
When the bearing speed approached zero, all tested loads
between 0.25 and 20 kg/cm2 gave the same coefficient
of friction 0.14, see Fig. 8. At low speeds, a temperature
increase resulted in an increased friction, which was
explained as increased metal contact in the bearing. That
metal contact also made the friction vary a lot from
experiment to experiment. At high speeds, when the oil
film was thick and without metal to metal contact, the
friction was much more repeatable. At increasing speed
above the minimum friction point, the friction increased
linearly with the speed, but at higher speeds, the increase
was lower due to the increased oil film thickness. Stribeck even registered decreasing friction at increasing
speed, when the bearing temperature was allowed to
increase at increasing speed and thereby lowered the
lubricant viscosity.
9. White-metal bearings
The Sellers-bearings were unable to run-in by themselves, but the white-metal bearings were expected to
easily run-in without any specific run-in procedure. Different low hardness alloys containing lead, tin and antimony were tested. Their yield points were low, hardly
above 20 MPa. The bearing diameter was 70 mm and
the bearing length was here 137 mm. The mean pressure
for such long bearings was normally below 2.5 MPa and
reached only exceptionally 5 MPa pressure. To reach
such high pressures in the test set-up, Stribeck halved
the bearing length after he had finished the full-length
experiments. He then found that the short bearing
worked as well as the long one for high speeds, and
better than the long one for low speeds. At low film
thickness, the long bearing seems to have had problems
with edge loading. Stribeck loaded the part of the bear-
786
B. Jacobson / Tribology International 36 (2003) 781–789
Fig. 6.
Oil viscosity (°E) as a function of temperature.
Fig. 7.
Sellers-bearing with ring-lubrication.
ing surface containing the lubrication grooves, so the oil
film was thinner than if the grooves had been in the
unloaded part.
10. The running-in
To get a smooth surface fast for the white-metal surface, running in was started at a low load (1.15 MPa)
and a low speed (64 rpm). The friction decreased quickly
from 0.020 and stabilised at 0.0028 after 18 h. The load
was then increased and the running in restarted at 1.9
MPa pressure. The friction stabilised at 0.0022 after 16
h. When the load was increased later to 2.6 MPa, the
coefficient of friction was strongly increased, and still
after 6 h, it was 0.0044, double the value for lower pressures. After 16 h of running, the bearing was still not runin. The load was then increased to 3.6 MPa for 10 h,
but the bearing still did not run-in correctly. When the
load then was decreased to2.6 MPa, the bearing worked
very well with the coefficient of friction 0.0021. The
bearing material (magnoliametalle) worked well and
became smoother by the running. The use of a low viscosity oil for the running in process made the surfaces
so smooth, that when the bearing later was loaded to 5.1
MPa using a higher viscosity oil, no further running in
was taking place. Stribeck also tried to use water as
lubricant, but that gave wear and particle production.
B. Jacobson / Tribology International 36 (2003) 781–789
Fig. 8.
787
Journal bearing friction as a function of rotational speed for different mean pressures. ‘Stribeck curves’.
11. The run-in bearing lubricated with gas motor
oil (gasmotorenöl)
When the bearing was so well run-in that no metallic
contact occurred for the running conditions tested, Stribeck measured the coefficient of friction as a function
of temperature for loads from 0.07 to 7.5 MPa and
speeds from 190 to 1100 rpm. The minimum friction
was around 0.005 for the highest load and temperature.
Stribeck compared the change in viscosity and temperature with the change in friction. He then saw that the
friction was not proportional to the viscosity, but the oil
film thickness also changed and influenced the friction.
The starting friction at zero speed was much higher for
the white-metal bearings (0.21–0.24) than for the cast
iron bearing (0.14), but for increasing speed, the friction
in the white-metal bearing decreased much faster, and
the minimum friction was lower than for the cast iron
bearing. Even lower friction was found in a smaller bearing with 7.5 MPa load.
A comparison between the white-metal bearing and
the cast iron bearing showed their different behaviour.
The largest difference was in the degree of running in.
The polished area in the white-metal bearing was much
788
B. Jacobson / Tribology International 36 (2003) 781–789
larger than the polished area in the cast iron bearing.
The conformity between shaft and bearing was thus different for the two bearings after running in. For the same
bearing pressure, the bearing performance was very
similar for the two bearing materials studied.
12. Roller bearings
Stribeck also investigated a number of roller bearings
with different types of long cylindrical rollers to compare their friction and load-carrying capacity with ball
bearings and hydrodynamic journal bearings. The five
bearing types he studied were Laschenrollenlager, bearings with loose tube-formed rollers mounted on small
shafts, Kynoch’s roller bearings, Hyatt bearings, and
bearings from Mossberg & Granville Mfg. Co.
The Laschenrollenlager had loose rollers coupled
together 4 by 4 using scissors-like coupling mechanisms
at the roller ends. The roller length was about as long
as the length along the circumference for one group of
four rollers. This caused self-locking by axial motion
and turning of the roller group like a self-locking drawer.
As Stribeck concluded, ‘Die Konstruktion erwies sich
unbrauchbar’ (the design was useless).
The other four bearing types were tested extensively,
both regarding load-carrying capacity and friction/
power loss.
The first bearing type tested had 20 loose rollers with
diameter 10 mm and length 90 mm coupled together by
two washers having 20 holes around the circumference,
one at each side of the bearing. The rollers were
‘extremely accurately manufactured with a diameter
variation of only ±0.04 mm’, which is at least 100 times
less accurate than modern bearings.
The bearing worked well and had a coefficient of friction at high loads of 0.0039, which can be compared
with Stribeck’s ball-bearing friction 0.0015. The bearings failed by pitting, probably caused by over-rolling
of large numbers of wear particles. The load-carrying
capacity of the bearing type was
P 5 11·L·d·z / 5 kg
(4)
where z is the number of rollers; L is the roller length
in cm; and d is the roller diameter in cm.
This can be compared to Stribeck’s ball-bearing tests,
which gave
P 5 100·d2·z / 5 kg
(5)
To give the same load-carrying capacity for a roller
bearing and a ball bearing, the roller length had to be
9.1 times longer than the ball diameter if the roller and
ball diameters were the same.
Kynoch’s roller bearing had a similar geometry, but
the long tube-formed rollers were cut into 5–6 pieces
giving larger flexibility to accommodate manufacturing
errors. The friction was about the same, but the loadcarrying capacity was only 60% of the capacity of the
previous bearing type.
The Hyatt-roller bearing had also about the same friction but a load-carrying capacity of 85% of the second
bearing type. The big problem with the Hyatt-roller bearing was its noise level. Already at 600 rpm ‘ist das
Geräusch selbst in einer Schlosserwerkstätte recht störend’ (the bearing noise is really disturbing even in a drop
forging shop).
The bearings from Mossberg & Granville Mfg. Co.
had cages, which surrounded and closed in the rollers
with a small gap around most of their circumference.
That gave a large hydrodynamic resistance to motion, so
its power loss gave very high working temperature at
high speed and high friction. The cage did guide the
rollers very well though, which gave as a result noise
that was the lowest among the roller bearings.
Stribeck summed the results up in his Fig. 33, see Fig.
9, where it is obvious that roller bearings had much (2–
10 times) higher friction than ball bearings, but that the
starting friction was lower than for sliding bearings. The
starting friction, and the steady state high-speed friction,
was the same for most of the roller bearing types. The
load-carrying capacity for roller bearings was
P 5 k·L·d·z / 5
(6)
where 6#k#11 for the different bearing types tested.
When he compared cast iron journal bearings with
roller bearings, only the best roller bearings could reach
the high load-carrying capacity of a journal bearing.
13. Conclusions
Stribeck’s publications ‘Kugellager für beliebige
Belastungen’ and ‘Die wesentlichen Eigenschaften der
Gleit- und Rollenlager’ were remarkable for their time.
Many of the findings were revolutionary and are still
considered quite modern.
The method used by Stribeck to calculate load distribution between the balls within a ball bearing is still
used today, otherwise, a full finite element analysis
would be required.
His measurements and analysis of bearing friction is
also astonishingly accurate. Most of the ball-bearing
types studied by Stribeck are not in use any more, but the
deep-groove ball bearings he investigated were similar
enough to be compared with modern deep-groove ball
bearings. Stribeck measured the friction coefficient
0.0015 and SKF gives the friction coefficient 0.0015 for
deep-groove ball bearings in the catalogue from 1989.
For the first time, in the SKF catalogue from 1989, the
life calculation method that took into account contamination also. Stribeck had stated already 100 years ago
B. Jacobson / Tribology International 36 (2003) 781–789
Fig. 9.
789
Comparison of friction for different bearing types.
that cleanliness was very important for bearing life
and function.
Stribeck’s investigation of journal bearing friction, as
a function of load and speed, was also extremely
important, as he showed the possibility to find a point
of minimum friction for lubricated applications. He also
showed that the friction for sliding bearings started at
high friction at low speeds, decreased to a minimum friction when metal to metal contact stopped, and then
increased again at higher speeds, the well known ‘Stribeck curve’.
References
[1] Dowson D. History of tribology. London: Longman Group
Limited, 1979.
[2] Stribeck R. Kugellager für beliebige Belastungen. Zeitschrift des
Vereines deutscher Ingenieure 1901;45(3):73–9 (pt I) & 45(4):
118–125 (pt II).
[3] Stribeck R. Die wesentlischen Eigenschaften der Gleit- und Rollenlager. Zeitschrift des Vereines deutscher Ingenieure 1902;
46(37):1341-1348 (pt I) & 46(38):1432-1438 (pt II) & 46(39)
1463–1470 (pt III).
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