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 Construction at the Building School. Two years later, in

1890, he moved to the Technische Hochschule in Darm-
It is now 101 and 100 years, respectively, since Rich- stadt. After three years, on 1 April 1893, he moved to
ard Stribeck published his famous papers, ‘Kugellager the Technische Hochschule in Dresden. As Professor für
für beliebige Belastungen’ (1901), and ‘Die wesentlichen Maschineningenieurswesen, he gave his inaugural
Eigenshaften der Gleit- und Rollenlager’ (1902) in Zeit- address in Dresden on 6 May 1893 on the subject ‘Pro-
schrift des Vereines deutscher Ingenieure [2,3]. The first gress in the production of steam and in the exploitation
paper treated the load-carrying capacity of ball bearings of its energy in steam engines during the last 20 years’.
brilliantly and the second paper resolved the longstand- Seven weeks later, on 1 July 1893, he went on a sabbati-
ing dispute over bearing friction characteristics as a cal leave to the United States of America.
function of load, speed and lubrication. Later, when he was back in Dresden, in the academic
Stribeck [1] was born in Stuttgart in Germany on 7 year 1896–1897, he was appointed to the governing
July 1861. He received his basic education in his home- body of the newly founded Machines Laboratory I
town where he finished his studies at the Technische (Strengths). He did not stay there long and moved to
Hochschule in Stuttgart. Later, he gained practical Berlin in March 1898. There, he became head of the
experience in various places, including Königsberg. At metallurgical division of the Centralstelle für wissen-
the age of 27, in 1888, he was back in his hometown, schaftlich-technische Untersuchungen in Neubab-
Stuttgart, and was appointed as Professor of Machine elsberg, Berlin.
During the years 1889–1893, when Stribeck worked
in Stuttgart and in Darmstadt, he published some 10
∗
Fax: +46-46222-8504. scientific papers. This large early output was mainly on
E-mail address: bo.jacobson@mel.lth.se (B. Jacobson). giving solutions to the problems of boilers and steam

0301-679X/$ - see front matter  2003 Elsevier Ltd. All rights reserved.
doi:10.1016/S0301-679X(03)00094-X
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 exper-
imentally 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 func-
tion well and that F·n = constant described the combi-
nation 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 cre-
ated Centralstelle für wissenschaftlich-technische Unter-
suchungen in Neubabelsberg, and to head the metallurgi-
cal 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 prin-
ciples 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 stud-
ies, which were commissioned by the Deutsche Waffen- Fig. 1. Test set-up for ball deformation measurements.
und Munionsfabriken, Berlin, were epoch-making in
their effect on the ball-bearing industry. Its major impact 1901, as a ‘Mitteilung aus der Zentralstelle für wissen-
arose from the analysis of the carefully conducted schaftlich- techniche Untersuchungen’. Stribeck stated
experiments, see Fig. 2, and the sound appreciation of there that his results were applicable not only to ball
Hertzian contact theory, together with the presentation bearings, but also to all other types of concentrated con-
of practical formulae in a form suitable for the bearing tacts of Hertzian type.
designer. Stribeck’s approach to the question of load- He used Hertz’ equations for deformation, contact
carrying capacity of ball bearings was about seven years dimension and contact pressure, and applied it to ball-
in advance of similar considerations in gear design. bearing geometry. By comparing the permanent defor-
mation left after the unloading, he found that contact
pressures could be much higher than the yield strength
2. Load-carrying capacity for ball bearings of the steel, and the elastic deformations were almost
halved due to the hydrostatic stress components in the
Stribeck’s paper on ball bearings ‘Kugellager für contact centre. He made experiments with three balls in
beliebige Belastungen’ was published in Zeitschrift des series and a flat or concave plate between two balls to
Vereines deutcher Ingenieure on Saturday 19 January 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 self-

aligning 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 one-
eighth of an inch depending on the groove forms in
the rings.

3. Friction for ball bearings

One week later, on 26 January 1901, Stribeck pub-

lished 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 3/8⬙, 1/2⬙, 5/8⬙, 3/4⬙, 7/8⬙, 1⬙ and 1 1/8⬙. For flat
between balls, the sizes were 3/8⬙, 1/2⬙, 5/8⬙, 7/8⬙ and
1 1/8⬙.
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 elas-
ticity 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.
784 B. Jacobson / Tribology International 36 (2003) 781–789

were deformed, and the rings had no bending defor- beck noticed that the friction was lower than during the
mations. For a bearing with zero radial play, the load speed increase despite almost constant temperature. He
per ball was proportional to cosg 3/2 where the angle g assumed that the friction decrease was due to running in
was measured from the ball with the maximum load. of the surfaces, as this phenomenon was much stronger
Summing up the load components, he found that for 10, for rough surfaces than for polished surfaces, where the
15 and 20 balls in the bearing, the total load-carrying friction was not changed by running in. Stribeck ran tests
capacity divided with the maximum load on one ball was with five different types of contact geometry for ball
the number of balls divided with 4.37. To be on the safe bearings and found that a deep-groove ball bearing with
side, he gave the famous equation: open osculation had the lowest coefficient of friction,
0.0015, of the tested types, see Fig. 5. For that bearing
P0 ⫽ 5·P / z (1)
the largest allowable ball load was 11·d2 kg where d was
where P0 = maximum load on one ball; P = maximum the ball diameter in eighths of an inch. That shows the
load on one ball; and z = number of balls. material development the last 100 years. The maximum
For friction calculations, he also found that the sum load to avoid plastic deformation then was 3100 N on a
of the radial forces around the circumference of the bear- diameter 17 mm ball. Today, the ball load is 10,000 N
ing was 1.2 times the external load. 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
4. Experiments with ball bearings deformed rolling track was visible, the bearing life was
short. For good steel hardness, he found the maximum
Stribeck designed and manufactured a friction torque allowable ball load to be
measurement rig, which automatically compensated for
the weight of the different moving parts, see Fig. 4. He P ⫽ 10·d2 kg (d in inches) (2)
was thereby able to measure the small friction with high
P ⫽ 100·d2kg (d in cm) (3)
accuracy. He found that for small amounts of lubricant
in the bearings, the friction was independent of the tem- when the ring groove radius was 2d / 3.
perature. His experiments were run at 65, 100, 130, 190, Such good results were only possible when balls and
380, 580, 780 and 1150 rpm, starting with the lowest rings were made of good hardened material, and dust
speed. When decreasing the speed from 1150 rpm, Stri- and contamination, e.g. forming sand or surface parts
from the moulding, were removed from the bearing
lubricant, just like for sliding bearings.

Fig. 4. Friction measurement set-up with compensating counter-

weight. Fig. 5. Ball bearings with grooved rings similar to modern deep-
groove ball bearings.
 B. Jacobson / Tribology International 36 (2003) 781–789 785

5. Bearing lubrication between 20 and 50 °C as a heavy mineral oil between

80 and 110 °C, see Fig. 6.
One and a half year later, Stribeck published his paper
on friction and lubrication for journal bearings and rol-
ling bearings, ‘Die wesentlichen Eigenshaften der Gleit- 8. Cast iron Sellers-bearing with ring-lubrication
und Rollenlager’, starting on Saturday 6 September
1902, No. 36, of Zeitschrift des Vereines deutscher Inge- Stribeck’s test bearings were different from most
nieure. modern journal bearings, in that the bearing width (230
He wrote the paper in order to show which bearing mm) was much larger than the diameter (70 mm), see
type was the best performer under different running con- Fig. 7. This made the bearing sensitive for shaft bending
ditions, as no hard data were available for journal bear- and misalignment. To avoid temperature transients,
ings. No information was available earlier on the differ- causing temperatures at different points in the bearing
ent lubrication regimes and their friction coefficients; it to differ by a few degrees, measurements were made
was impossible to know which bearing type to choose slowly. Steady state temperatures were reached after 2–
for different applications. During the investigation, Stri- 3 h of running. That also showed Stribeck that most
beck not only compared hydrodynamic bearings and ball machines in transient use never would reach the steady
bearings, but also studied roller bearings. state temperature.
Stribeck’s experiments clearly indicated that the
higher the speed is, the higher is also the bearing load
6. Journal bearings when the minimum coefficient of friction is experienced.
When the bearing speed approached zero, all tested loads
Stribeck had found that not only roughly manufac- between 0.25 and 20 kg/cm2 gave the same coefficient
tured bearing surfaces needed to run-in for some time. of friction 0.14, see Fig. 8. At low speeds, a temperature
How fast the running in process could be, was mainly increase resulted in an increased friction, which was
a function of the bearing material. Fastest were some explained as increased metal contact in the bearing. That
white metals, slightly slower were bearing bronzes and metal contact also made the friction vary a lot from
very slow was cast iron. Steel was practically impossible experiment to experiment. At high speeds, when the oil
to run-in. The load possible to use during the running in film was thick and without metal to metal contact, the
process was also a strong function of the material. White friction was much more repeatable. At increasing speed
metals could stand the highest running in load, and steel above the minimum friction point, the friction increased
the lowest. For white metals with a yield point of 20 linearly with the speed, but at higher speeds, the increase
MPa, mean pressures P / (L·d) = 9 MPa were possible to was lower due to the increased oil film thickness. Stri-
carry. White metals also gave the least wear of the steel beck even registered decreasing friction at increasing
shaft, and were therefore considered the best bearing speed, when the bearing temperature was allowed to
materials. increase at increasing speed and thereby lowered the
Stribeck also mentioned that a bearing, which had lubricant viscosity.
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. 9. White-metal bearings

The Sellers-bearings were unable to run-in by them-

7. The lubricant selves, but the white-metal bearings were expected to
easily run-in without any specific run-in procedure. Dif-
Stribeck saw that the lubricant had a very strong ferent low hardness alloys containing lead, tin and anti-
influence on the friction, but also on the allowable load mony were tested. Their yield points were low, hardly
and the safety of the bearing function. If a bearing was above 20 MPa. The bearing diameter was 70 mm and
run-in with one oil, another oil could start the running the bearing length was here 137 mm. The mean pressure
in anew or fail the bearing. The best oil found, and used for such long bearings was normally below 2.5 MPa and
for the tests, was ‘Gasmotorenöl’ delivered from ‘Gas- reached only exceptionally 5 MPa pressure. To reach
motorenfabrik Deutz’. The same oil had also been used such high pressures in the test set-up, Stribeck halved
in the earlier ball-bearing tests. The viscosity at 20 °C the bearing length after he had finished the full-length
was 0.28 Ns/m2, at 40 °C, it was 0.075 Ns/m2, and at experiments. He then found that the short bearing
100 °C, the viscosity was 0.012 Ns/m2. The flame point worked as well as the long one for high speeds, and
was 180 °C. By comparing the viscosity variation for better than the long one for low speeds. At low film
different temperatures, Stribeck found that the thin Vel- thickness, the long bearing seems to have had problems
ocite Spindle Oil changed its viscosity in the same way 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 coefficient of friction was strongly increased, and still
film was thinner than if the grooves had been in the after 6 h, it was 0.0044, double the value for lower press-
unloaded part. ures. After 16 h of running, the bearing was still not run-
in. The load was then increased to 3.6 MPa for 10 h,
but the bearing still did not run-in correctly. When the
10. The running-in load then was decreased to2.6 MPa, the bearing worked
very well with the coefficient of friction 0.0021. The
To get a smooth surface fast for the white-metal sur- bearing material (magnoliametalle) worked well and
face, running in was started at a low load (1.15 MPa) became smoother by the running. The use of a low vis-
and a low speed (64 rpm). The friction decreased quickly cosity oil for the running in process made the surfaces
from 0.020 and stabilised at 0.0028 after 18 h. The load so smooth, that when the bearing later was loaded to 5.1
was then increased and the running in restarted at 1.9 MPa using a higher viscosity oil, no further running in
MPa pressure. The friction stabilised at 0.0022 after 16 was taking place. Stribeck also tried to use water as
h. When the load was increased later to 2.6 MPa, the lubricant, but that gave wear and particle production.
 B. Jacobson / Tribology International 36 (2003) 781–789 787

Fig. 8. Journal bearing friction as a function of rotational speed for different mean pressures. ‘Stribeck curves’.

11. The run-in bearing lubricated with gas motor film thickness also changed and influenced the friction.
oil (gasmotorenöl) The starting friction at zero speed was much higher for
the white-metal bearings (0.21–0.24) than for the cast
When the bearing was so well run-in that no metallic iron bearing (0.14), but for increasing speed, the friction
contact occurred for the running conditions tested, Stri- in the white-metal bearing decreased much faster, and
beck measured the coefficient of friction as a function the minimum friction was lower than for the cast iron
of temperature for loads from 0.07 to 7.5 MPa and bearing. Even lower friction was found in a smaller bear-
speeds from 190 to 1100 rpm. The minimum friction ing with 7.5 MPa load.
was around 0.005 for the highest load and temperature. A comparison between the white-metal bearing and
Stribeck compared the change in viscosity and tempera- the cast iron bearing showed their different behaviour.
ture with the change in friction. He then saw that the The largest difference was in the degree of running in.
friction was not proportional to the viscosity, but the oil 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. errors. The friction was about the same, but the load-
The conformity between shaft and bearing was thus dif- carrying capacity was only 60% of the capacity of the
ferent for the two bearings after running in. For the same previous bearing type.
bearing pressure, the bearing performance was very The Hyatt-roller bearing had also about the same fric-
similar for the two bearing materials studied. tion but a load-carrying capacity of 85% of the second
bearing type. The big problem with the Hyatt-roller bear-
ing was its noise level. Already at 600 rpm ‘ist das
12. Roller bearings Geräusch selbst in einer Schlosserwerkstätte recht stör-
end’ (the bearing noise is really disturbing even in a drop
Stribeck also investigated a number of roller bearings forging shop).
with different types of long cylindrical rollers to com- The bearings from Mossberg & Granville Mfg. Co.
pare their friction and load-carrying capacity with ball had cages, which surrounded and closed in the rollers
bearings and hydrodynamic journal bearings. The five with a small gap around most of their circumference.
bearing types he studied were Laschenrollenlager, bear- That gave a large hydrodynamic resistance to motion, so
ings with loose tube-formed rollers mounted on small its power loss gave very high working temperature at
shafts, Kynoch’s roller bearings, Hyatt bearings, and high speed and high friction. The cage did guide the
bearings from Mossberg & Granville Mfg. Co. rollers very well though, which gave as a result noise
The Laschenrollenlager had loose rollers coupled that was the lowest among the roller bearings.
together 4 by 4 using scissors-like coupling mechanisms Stribeck summed the results up in his Fig. 33, see Fig.
at the roller ends. The roller length was about as long 9, where it is obvious that roller bearings had much (2–
as the length along the circumference for one group of 10 times) higher friction than ball bearings, but that the
four rollers. This caused self-locking by axial motion starting friction was lower than for sliding bearings. The
and turning of the roller group like a self-locking drawer. starting friction, and the steady state high-speed friction,
As Stribeck concluded, ‘Die Konstruktion erwies sich was the same for most of the roller bearing types. The
unbrauchbar’ (the design was useless). load-carrying capacity for roller bearings was
The other four bearing types were tested extensively,
both regarding load-carrying capacity and friction/ P ⫽ k·L·d·z / 5 (6)
power loss. where 6ⱕkⱕ11 for the different bearing types tested.
The first bearing type tested had 20 loose rollers with When he compared cast iron journal bearings with
diameter 10 mm and length 90 mm coupled together by roller bearings, only the best roller bearings could reach
two washers having 20 holes around the circumference, the high load-carrying capacity of a journal bearing.
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. 13. Conclusions
The bearing worked well and had a coefficient of fric-
tion at high loads of 0.0039, which can be compared Stribeck’s publications ‘Kugellager für beliebige
with Stribeck’s ball-bearing friction 0.0015. The bear- Belastungen’ and ‘Die wesentlichen Eigenschaften der
ings failed by pitting, probably caused by over-rolling Gleit- und Rollenlager’ were remarkable for their time.
of large numbers of wear particles. The load-carrying Many of the findings were revolutionary and are still
capacity of the bearing type was considered quite modern.
The method used by Stribeck to calculate load distri-
P ⫽ 11·L·d·z / 5 kg (4)
bution between the balls within a ball bearing is still
where z is the number of rollers; L is the roller length used today, otherwise, a full finite element analysis
in cm; and d is the roller diameter in cm. would be required.
This can be compared to Stribeck’s ball-bearing tests, His measurements and analysis of bearing friction is
which gave also astonishingly accurate. Most of the ball-bearing
types studied by Stribeck are not in use any more, but the
P ⫽ 100·d2·z / 5 kg (5)
deep-groove ball bearings he investigated were similar
To give the same load-carrying capacity for a roller enough to be compared with modern deep-groove ball
bearing and a ball bearing, the roller length had to be bearings. Stribeck measured the friction coefficient
9.1 times longer than the ball diameter if the roller and 0.0015 and SKF gives the friction coefficient 0.0015 for
ball diameters were the same. deep-groove ball bearings in the catalogue from 1989.
Kynoch’s roller bearing had a similar geometry, but For the first time, in the SKF catalogue from 1989, the
the long tube-formed rollers were cut into 5–6 pieces life calculation method that took into account contami-
giving larger flexibility to accommodate manufacturing nation also. Stribeck had stated already 100 years ago
 B. Jacobson / Tribology International 36 (2003) 781–789 789

Fig. 9. Comparison of friction for different bearing types.

that cleanliness was very important for bearing life References

and function.
Stribeck’s investigation of journal bearing friction, as
a function of load and speed, was also extremely [1] Dowson D. History of tribology. London: Longman Group
important, as he showed the possibility to find a point Limited, 1979.
[2] Stribeck R. Kugellager für beliebige Belastungen. Zeitschrift des
of minimum friction for lubricated applications. He also Vereines deutscher Ingenieure 1901;45(3):73–9 (pt I) & 45(4):
showed that the friction for sliding bearings started at 118–125 (pt II).
high friction at low speeds, decreased to a minimum fric- [3] Stribeck R. Die wesentlischen Eigenschaften der Gleit- und Rol-
tion when metal to metal contact stopped, and then lenlager. Zeitschrift des Vereines deutscher Ingenieure 1902;
increased again at higher speeds, the well known ‘Stri- 46(37):1341-1348 (pt I) & 46(38):1432-1438 (pt II) & 46(39)
beck curve’. 1463–1470 (pt III).