ISO/TR 1281-1:2008
(Main)Rolling bearings — Explanatory notes on ISO 281 — Part 1: Basic dynamic load rating and basic rating life
Rolling bearings — Explanatory notes on ISO 281 — Part 1: Basic dynamic load rating and basic rating life
ISO/TR 1281-1:2008 gives supplementary background information regarding the derivation of mathematical expressions and factors given in ISO 281:2007.
Wälzlager - Erläuternde Anmerkungen zur ISO 281 - Teil 1: Dynamische Tragzahlen und nominelle Lebensdauer
Roulements — Notes explicatives sur l'ISO 281 — Partie 1: Charges dynamiques de base et durée nominale de base
L'ISO/TR 1281-1:2008 donne un certain nombre d'informations sur la manière dont ont été définis les expressions mathématiques et les facteurs donnés dans l'ISO 281:2007.
Kotalni ležaji - Pojasnilo k standardu ISO 281 - 1. del: Imenska dinamična obremenitev in imenska doba trajanja
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TECHNICAL ISO/TR
REPORT 1281-1
First edition
2008-12-01
Rolling bearings — Explanatory notes
on ISO 281 —
Part 1:
Basic dynamic load rating and basic
rating life
Roulements — Notes explicatives sur l'ISO 281 —
Partie 1: Charges dynamiques de base et durée nominale de base
Reference number
ISO/TR 1281-1:2008(E)
©
ISO 2008
---------------------- Page: 1 ----------------------
ISO/TR 1281-1:2008(E)
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ii © ISO 2008 – All rights reserved
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ISO/TR 1281-1:2008(E)
Contents Page
Foreword. iv
Introduction . v
1 Scope . 1
2 Normative references . 1
3 Symbols . 1
4 Basic dynamic load rating . 3
4.1 Basic dynamic radial load rating, C , for radial ball bearings . 4
r
4.2 Basic dynamic axial load rating, C , for single row thrust ball bearings. 7
a
4.3 Basic dynamic axial load rating, C , for thrust ball bearings with two or more rows of
a
balls. 9
4.4 Basic dynamic radial load rating, C , for radial roller bearings. 10
r
4.5 Basic dynamic axial load rating, C , for single row thrust roller bearings. 12
a
4.6 Basic dynamic axial load rating, C , for thrust roller bearings with two or more rows of
a
rollers . 13
5 Dynamic equivalent load. 15
5.1 Expressions for dynamic equivalent load. 15
5.2 Factors X, Y, and e. 27
6 Basic rating life . 38
7 Life adjustment factor for reliability. 39
Bibliography . 40
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ISO/TR 1281-1:2008(E)
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies
(ISO member bodies). The work of preparing International Standards is normally carried out through ISO
technical committees. Each member body interested in a subject for which a technical committee has been
established has the right to be represented on that committee. International organizations, governmental and
non-governmental, in liaison with ISO, also take part in the work. ISO collaborates closely with the
International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.
The main task of technical committees is to prepare International Standards. Draft International Standards
adopted by the technical committees are circulated to the member bodies for voting. Publication as an
International Standard requires approval by at least 75 % of the member bodies casting a vote.
In exceptional circumstances, when a technical committee has collected data of a different kind from that
which is normally published as an International Standard (“state of the art”, for example), it may decide by a
simple majority vote of its participating members to publish a Technical Report. A Technical Report is entirely
informative in nature and does not have to be reviewed until the data it provides are considered to be no
longer valid or useful.
Attention is drawn to the possibility that some of the elements of this document may be the subject of patent
rights. ISO shall not be held responsible for identifying any or all such patent rights.
ISO/TR 1281-1 was prepared by Technical Committee ISO/TC 4, Rolling bearings, Subcommittee SC 8, Load
ratings and life.
This first edition of ISO/TR 1281-1, together with the first edition of ISO/TR 1281-2, cancels and replaces the
first edition of ISO/TR 8646:1985, which has been technically revised.
ISO/TR 1281 consists of the following parts, under the general title Rolling bearings — Explanatory notes on
ISO 281:
⎯ Part 1: Basic dynamic load rating and basic rating life
⎯ Part 2: Modified rating life calculation, based on a systems approach of fatigue stresses
iv © ISO 2008 – All rights reserved
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ISO/TR 1281-1:2008(E)
Introduction
ISO/R281:1962
A first discussion on an international level of the question of standardizing calculation methods for load ratings
of rolling bearings took place at the 1934 conference of the International Federation of the National
Standardizing Associations (ISA). When ISA held its last conference in 1939, no progress had been made.
However, in its 1945 report on the state of rolling bearing standardization, the ISA 4 Secretariat included
proposals for definition of concepts fundamental to load rating and life calculation standards. This report was
distributed in 1949 as document ISO/TC 4 (Secretariat-1)1, and the definitions it contained are in essence
those given in ISO 281:2007 for the concepts “life” and “basic dynamic load rating” (now divided into “basic
dynamic radial load rating” and “basic dynamic axial load rating”).
In 1946, on the initiative of the Anti-Friction Bearing Manufacturers Association (AFBMA), New York,
discussions of load rating and life calculation standards started between industries in the USA and Sweden.
Chiefly on the basis of the results appearing in Reference [1], an AFBMA standard, Method of evaluating load
ratings of annular ball bearings, was worked out and published in 1949. On the same basis, the member body
for Sweden presented, in February 1950, a first proposal to ISO, “Load rating of ball bearings” [doc.
ISO/TC 4/SC 1 (Sweden-1)1].
In view of the results of both further research and a modification to the AFBMA standard in 1950, as well as
interest in roller bearing rating standards, in 1951, the member body for Sweden submitted a modified
proposal for rating of ball bearings [doc. ISO/TC 4/SC 1 (Sweden-6)20] as well as a proposal for rating of
roller bearings [doc. ISO/TC 4/SC 1 (Sweden-7)21].
Load rating and life calculation methods were then studied by ISO/TC 4, ISO/TC 4/SC 1 and ISO/TC 4/WG 3
at 11 different meetings from 1951 to 1959. Reference [2] was then of considerable use, serving as a major
basis for the sections regarding roller bearing rating.
The framework for the Recommendation was settled at a TC 4/WG 3 meeting in 1956. At the time,
deliberations on the draft for revision of AFBMA standards were concluded in the USA and ASA B3 approved
the revised standard. It was proposed to the meeting by the USA and discussed in detail, together with the
Secretariat's proposal. At the meeting, a WG 3 proposal was prepared which adopted many parts of the USA
proposal.
In 1957, a Draft Proposal (document TC 4 N145) based on the WG proposal was issued. At the WG 3
meeting the next year, this Draft Proposal was investigated in detail, and at the subsequent TC 4 meeting, the
adoption of TC 4 N145, with some minor amendments, was concluded. Then, Draft ISO Recommendation
No. 278 (as TC 4 N188) was issued in 1959, and ISO/R281 accepted by ISO Council in 1962.
ISO 281/1:1977
In 1964, the member body for Sweden suggested that, in view of the development of imposed bearing steels,
the time had come to review ISO/R281 and submitted a proposal [ISO/TC 4/WG 3 (Sweden-1)9]. However, at
this time, WG 3 was not in favour of a revision.
In 1969, on the other hand, TC 4 followed a suggestion by the member body for Japan (doc. TC 4 N627) and
reconstituted its WG 3, giving it the task of revising ISO/R281. The AFBMA load rating working group had at
this time started revision work. The member body for the USA submitted the Draft AFBMA standard, Load
ratings and fatigue life for ball bearings [ISO/TC 4/WG 3 (USA-1)11], for consideration in 1970 and Load
ratings and fatigue life for roller bearings [ISO/TC 4/WG 3 (USA-3)19] in 1971.
In 1972, TC 4/WG 3 was reorganized and became TC 4/SC 8. This proposal was investigated in detail at five
meetings from 1971 to 1974. The third and final Draft Proposal (doc. TC 4/SC 8 N23), with some amendments,
was circulated as a Draft International Standard in 1976 and became ISO 281-1:1977.
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ISO/TR 1281-1:2008(E)
The major part of ISO 281-1:1977 constituted a re-publication of ISO/R281, the substance of which had been
only very slightly modified. However, based mainly on American investigations during the 1960s, a new clause
was added, dealing with adjustment of rating life for reliability other than 90 % and for material and operating
conditions.
Furthermore, supplementary background information regarding the derivation of mathematical expressions
and factors given in ISO 281-1:1977 was published, first as ISO 281-2, Explanatory notes, in 1979; however,
TC 4/SC 8 and TC 4 later decided to publish it as ISO/TR 8646:1985.
vi © ISO 2008 – All rights reserved
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TECHNICAL REPORT ISO/TR 1281-1:2008(E)
Rolling bearings — Explanatory notes on ISO 281 —
Part 1:
Basic dynamic load rating and basic rating life
1 Scope
This part of ISO/TR 1281 gives supplementary background information regarding the derivation of
mathematical expressions and factors given in ISO 281:2007.
2 Normative references
The following referenced documents are indispensable for the application of this document. For dated
references, only the edition cited applies. For undated references, the latest edition of the referenced
document (including any amendments) applies.
ISO 281:2007, Rolling bearings — Dynamic load ratings and rating life
3 Symbols
Clause
A constant of proportionality 7
A constant of proportionality determined experimentally 4
1
B constant of proportionality determined experimentally 4
1
C basic dynamic radial load rating of a rotating ring 4, 5
1
C basic dynamic radial load rating of a stationary ring 4, 5
2
C basic dynamic axial load rating for thrust ball or roller bearing 4, 6
a
C basic dynamic axial load rating of the rotating ring of an entire thrust ball or roller 4
a1
bearing
C basic dynamic axial load rating of the stationary ring of an entire thrust ball or roller 4
a2
bearing
C basic dynamic axial load rating as a row k of an entire thrust ball or roller bearing 4
ak
C basic dynamic axial load rating as a row k of the rotating ring of thrust ball or roller 4
a1k
bearing
C basic dynamic axial load rating as a row k of the stationary ring of thrust ball or roller 4
a2k
bearing
C basic dynamic load rating for outer ring 5
e
C basic dynamic load rating for inner ring 5
i
C basic dynamic radial load rating for radial ball or roller bearing 4, 5, 6
r
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ISO/TR 1281-1:2008(E)
D pitch diameter of ball or roller set 4
pw
D ball diameter 4, 5
w
D mean roller diameter 4
we
E modulus of elasticity 4
o
F axial load 5
a
F radial load 4, 5
r
J factor relating mean equivalent load on a rotating ring to Q 4, 5
1 max
J factor relating mean equivalent load on a stationary ring to Q 4, 5
2 max
J axial load integral 5
a
J radial load integral 4, 5
r
L bearing life 7
L basic rating life 6, 7
10
L effective contact length of roller 4
we
L L per row k 4
wek we
N number of stress applications to a point on the raceway 4
P dynamic equivalent axial load for thrust bearing 5, 6
a
P dynamic equivalent radial load for radial bearing 5, 6
r
P dynamic equivalent radial load for the rotating ring 5
r1
P dynamic equivalent radial load for the stationary ring 5
r2
Q normal force between a rolling element and the raceways 4, 6
Q rolling element load for the basic dynamic load rating of the bearing 4, 6
C
Q rolling element load for the basic dynamic load rating of a ring rotating relative to the 4, 5
C
1
applied load
Q rolling element load for the basic dynamic load rating of a ring stationary relative to the 4, 5
C
2
applied load
Q maximum rolling element load 4, 5
max
S probability of survival, reliability 4, 7
V volume representative of the stress concentration 4
V rotation factor 5
f
X radial load factor for radial bearing 5
X radial load factor for thrust bearing 5
a
Y axial load factor for radial bearing 5
Y axial load factor for thrust bearing 5
a
Z number of balls or rollers per row 4, 5
Z number of balls or rollers per row k 4
k
a semimajor axis of the projected contact ellipse 4
a life adjustment factor for reliability 7
1
b semiminor axis of the projected contact ellipse 4
c exponent determined experimentally 4, 6
c compression constant 5
c
2 © ISO 2008 – All rights reserved
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ISO/TR 1281-1:2008(E)
e measure of life scatter, i.e. Weibull slope determined experimentally 4, 5, 6, 7
f factor which depends on the geometry of the bearing components, the accuracy to 4
c
which the various components are made, and the material
h exponent determined experimentally 4, 6
i number of rows of balls or rollers 4
l circumference of the raceway 4
r cross-sectional raceway groove radius 5
r cross-sectional raceway groove radius of outer ring or housing washer 4
e
r cross-sectional raceway groove radius of inner ring or shaft washer 4
i
t auxiliary parameter 4
v J (0,5)/J (0,5) 5
2 1
z depth of the maximum orthogonal subsurface shear stress 4
o
α nominal contact angle 4, 5
α ′ actual contact angle 5
γ D cos α/D for ball bearings with α ≠ 90° 4
w pw
D /D for ball bearings with α = 90°
w pw
D cos α/D for roller bearings with α ≠ 90°
we pw
D /D for roller bearings with α = 90°
we pw
ε parameter indicating the width of the loaded zone in the bearing 5
η reduction factor 4, 5
λ reduction factor 4
µ factor introduced by Hertz 4
ν factor introduced by Hertz, or adjustment factor for exponent variation 4
σ maximum contact stress 4
max
Σρ curvature sum 4
τ maximum orthogonal subsurface shear stress 4
o
ϕ one half of the loaded arc 5
o
4 Basic dynamic load rating
The background to basic dynamic load ratings of rolling bearings according to ISO 281 appears in
References [1] and [2].
The expressions for calculation of basic dynamic load ratings of rolling bearings develop from a power
correlation that can be written as follows:
ce
τ NV
1
o
ln ∝ (1)
h
S
z
o
where
S is the probability of survival;
τ is the maximum orthogonal subsurface shear stress;
o
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ISO/TR 1281-1:2008(E)
N is the number of stress applications to a point on the raceway;
V is the volume representative of the stress concentration;
z is the depth of the maximum orthogonal subsurface shear stress;
o
c, h are experimentally determined exponents;
e is the measure of life scatter, i.e. the Weibull slope determined experimentally.
For “point” contact conditions (ball bearings) it is assumed that the volume, V, representative of the stress
concentration in Correlation (1) is proportional to the major axis of the projected contact ellipse, 2a, the
circumference of the raceway, l, and the depth, z , of the maximum orthogonal subsurface shear stress, τ
o o:
Va∝ zl (2)
o
Substituting Correlation (2) into Correlation (1):
ce
1 τ Nal
o
ln ∝ (3)
h−1
S
z
o
“Line” contact was considered in References [1] and [2] to be approached under conditions where the major
axis of the calculated Hertz contact ellipse is 1,5 times the effective roller contact length:
21a,=5L (4)
we
2
In addition, b/a should be small enough to permit the introduction of the limit value of ab as b/a approaches 0:
23Q
2
ab = (5)
π∑E ρ
o
(for variable definitions, see 4.1).
4.1 Basic dynamic radial load rating, C , for radial ball bearings
r
From the theory of Hertz, the maximum orthogonal subsurface shear stress, τ , and the depth, z , can be
o o
expressed in terms of a radial load F , i.e. a maximum rolling element load, Q , or a maximum contact
r max
stress, σ , and dimensions for the contact area between a rolling element and the raceways. The
max
relationships are:
τσ= T
omax
z = ζ b
o
1/2
(2t −1)
T =
2(tt +1)
1
ζ =
1/2
(1tt+−)(2 1)
1/3
⎛⎞
3 Q
a = µ
⎜⎟
E ∑ ρ
⎝⎠o
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ISO/TR 1281-1:2008(E)
1/3
⎛⎞
3 Q
bv=
⎜⎟
E ∑ ρ
o
⎝⎠
where
σ is the maximum contact stress;
max
t is the auxiliary parameter;
a is the semimajor axis of the projected contact ellipse;
b is the semiminor axis of the projected contact ellipse;
Q is the normal force between a rolling element and the raceways;
E is the modulus of elasticity;
o
Σρ is the curvature sum;
µ, v are factors introduced by Hertz.
Consequently, for a given rolling bearing, τ , a, l and z can be expressed in terms of bearing geometry, load
o o
and revolutions. Correlation (3) is changed to an equation by inserting a constant of proportionality. Inserting a
6
specific number of revolutions (e.g. 10 ) and a specific reliability (e.g. 0,9), the equation is solved for a rolling
element load for basic dynamic load rating which is designated to point contact rolling bearings introducing a
constant of proportionality, A :
1
04, 1
(1,59ch+−1,41 5,82)/(c−h+2)
⎛⎞
1,3 2r (1∓ γ )
QA= ×
⎜⎟
C 1
(2c+h−2)/(ch−+2) 3e/ (ch−+2) 3e/(ch−+2)
2rD−
40,5 (1± γ)
⎝⎠w
(6)
3/(ch−+2)
γ
⎛⎞
(2ch+−5)/(c−h+2) −3e/(c−h+2)
DZ
⎜⎟ w
cosα
⎝⎠
where
Q is the rolling element load for the basic dynamic load rating of the bearing;
C
D is the ball diameter;
w
γ is D cos α/D ;
w pw
in which
D is the pitch diameter of the ball set,
pw
α is the nominal contact angle;
Z is the number of balls per row.
The basic dynamic radial load rating, C , of a rotating ring is given by:
1
J
r
CQ==Z cos α 0,407Q Z cos α (7)
1CC
11
J
1
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ISO/TR 1281-1:2008(E)
The basic dynamic radial load rating, C , of a stationary ring is given by:
2
J
r
CQ==Z cos α 0,389Q Z cos α (8)
2CC
22
J
2
where
Q is the rolling element load for the basic dynamic load rating of a ring rotating relative to the
C
1
applied load;
Q is the rolling element load for the basic dynamic load rating of a ring stationary relative to
C
2
the applied load;
J = J (0,5) is the radial load integral (see Table 3);
r r
J = J (0,5) is the factor relating mean equivalent load on a rotating ring to Q (see Table 3);
1 1 max
J = J (0,5) is the factor relating mean equivalent load on a stationary ring to Q (see Table 3).
2 2 max
The relationship between C for an entire radial ball bearing, and C and C , is expressed in terms of the
r 1 2
product law of probability as:
−3/(ch−+2)
(2ch−+)/3
⎡⎤
⎛⎞
C
1
⎢⎥
CC=+1 (9)
⎜⎟
r1
⎢⎥C
⎝⎠2
⎣⎦
Substituting Equations (6), (7) and (8) into Equation (9), the basic dynamic radial load rating, C , for an entire
r
ball bearing is expressed as:
04, 1
(1,59ch+−1,41 5,82)/(c−h+2)
⎡⎤
13, 2r (1− γ)
i 3/(ch−+2)
C,=041 A γ ×
r1⎢⎥
(2ch+ −2)/(ch−+2) 3e/(c−+h 2) 3e/(ch−+2)
2rD−
40,5 (1+ γ)
⎣⎦iw
−−3/(ch+2)
� �
(2ch−+)/3
04, 1
(1,59ch++1,41 3e− 5,82)/(c−h+2)
� ⎧ ⎫ �
⎡⎤
⎛⎞
� r 2rD− ⎛⎞1−γ �
⎪⎪
i ew
�11+ ,04 � × (10)
⎨⎬⎢⎥⎜⎟
⎜⎟
� �
rr21−+D γ
⎢⎥ ⎝⎠
ei⎝⎠w
�⎪⎪⎣⎦ �
⎩⎭
� �
� �
(ch−−1)/(2ch−+) (ch−−3e+2)/(2c−h+) (25c+h−)/(2c−h+)
(ciZos α) D
w
where
A is the experimentally determined proportionality constant;
1
r is the cross-sectional raceway groove radius of the inner ring;
i
r is the cross-sectional raceway groove radius of the outer ring;
e
i is the number of rows of balls.
Here, the contact angle, α, the number of rolling elements (balls), Z, and the diameter, D , depend on bearing
w
design. On the other hand, the ratios of raceway groove radii, r and r , to a half-diameter of a rolling element
i e
(ball), D /2 and γ = D cosα/D , are not dimensional, therefore it is convenient in practice that the value for
w w pw
the initial terms on the right-hand side of Equation (10) to be designated as a factor, f :
c
(ch−−1)/(2c−+h) (ch−−3e+2)/(2c−h+) (25c+h−)/(2c−h+)
Cf= (ci os α) Z D (11)
rc w
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ISO/TR 1281-1:2008(E)
With radial ball bearings, the faults in bearings resulting from manufacturing need to be taken into
consideration, and a reduction factor, λ, is introduced to reduce the value for a basic dynamic radial load
rating for radial ball bearings from its theoretical value. It is convenient to include λ in the factor, f . The value
c
of λ is determined experimentally.
Consequently, the factor f is given by:
c
04, 1
(1,59ch+−1,41 5,82)/(c−h+2)
⎛⎞
13, 2r (1− γ)
3/(ch−+2)
i
f,=041λ A γ ×
⎜⎟
c1
(2c+h−2)/(ch−+2) 3e/(c−+h 2) 3e/(ch−+2)
2rD−
40,5 (1+ γ)
⎝⎠iw
−−3/(ch+2)
�� (12)
(2ch−+)/3
04, 1
��(1,59ch++1,41 3e−5,82)/(c−h+2)
⎧⎫
⎡⎤
��⎛⎞2rD−⎛⎞
⎪⎪r 1− γ
i ew
��
11+ ,04
⎢⎥
⎨⎬⎜⎟ ⎜⎟
��
rr21−+D γ
��
⎢⎥⎝⎠
⎪⎪ei⎝⎠w
⎣⎦
��
⎩⎭
��
Based on References [1] and [2], the following values were assigned to the experimental constants in the load
rating equations:
e = 10/9
c = 31/3
h = 7/3
Substituting the numerical values into Equation (11) gives the following, however, a sufficient number of test
results are only available for small balls, i.e. up to a diameter of about 25 mm, and these show that the load
1,8
rating may be taken as being proportional to D . In the case of larger balls, the load rating appears to
w
1,4
increase even more slowly in relation to the ball diameter, and D can be assumed where D > 25,4 mm:
w
w
07, 2/3 1,8
Cf= (ci osα) Z D for D u 25,4 mm (13)
rc w w
0,7 2/3 1,4
C,= 3 647f (i cos α) Z D for D > 25,4 mm (14)
rc w w
04, 1
03,,139
⎛⎞
2r γγ(1− )
i
f,=×0089A 0,41λ
⎜⎟
c1
1/3
2rD−
(1+ γ )
⎝⎠iw
−3/10
(15)
��10/3
04, 1
��17, 2
⎧⎫
��⎡⎤
⎛⎞ r⎛⎞2rD−
⎪⎪1− γ
i ew
��
11+ ,04
⎢⎥
⎨⎜⎜⎟ ⎟⎬
��
12+−γ rrD
��
⎝⎠⎢⎥ei w
⎪⎪⎝⎠
⎣⎦
��
⎩⎭
��
Values of f in ISO 281:2007, Table 2, are calculated by substituting raceway groove radii and reduction
c
factors given in Table 1 into Equation (15).
The value for 0,089A is 98,066 5 to calculate C in newtons.
1 r
4.2 Basic dynamic axial load rating, C , for single row thrust ball bearings
a
4.2.1 Thrust ball bearings with contact angle α ≠ 90°
As in 4.1, for thrust ball bearings with contact angle α ≠ 90°:
(ch−−1)/(2ch−+) (ch−−3e+2)/(2c−h+) (2c+h−5)/(2c−h+)
Cf= (cosαα) tanZ D (16)
ac w
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ISO/TR 1281-1:2008(E)
For most thrust ball bearings, the theoretical value of a basic dynamic axial load rating has to be reduced on
the basis of unequal distribution of load among the rolling elements in addition to the reduction factor, λ, which
is introduced in to radial ball bearing load ratings. This reduction factor is designated as η.
Consequently, the factor f is given by:
c
04, 1
(1,59ch+−1,41 5,82)/(c−h+2)
⎛⎞
13, 2r (1− γ)
i 3/(ch−+2)
fA= λη γ ×
⎜⎟
c1
(2ch+ −2)/(c−+h 2) 3e/(c−+h 2) 3e/(c−+h 2)
2rD−
40,5 (1+ γ)
⎝⎠iw
−−3/(ch+2)
(17)
��
(2ch−+)/3
04, 1
��(1,59ch++1,41 3e− 5,82)/(c−h+2)
⎧⎫
⎡⎤
��⎛⎞
⎪ r 2rD− ⎛⎞1−γ ⎪
i ew
��
1+
⎨⎜⎢⎥⎟ ⎬
⎜⎟
��
rr21−+D γ
��
⎢⎥ ⎝⎠
ei⎝⎠w
⎪⎣⎦ ⎪
��
⎩ ⎭
��
Similarly, to take the effect of ball size into account, substitute experimental constants e = 10/9, c = 31/3, and
h = 7/3 into Equations (16) and 17) to give:
07, 2/3 1,8
Cf= (cosαα) tanZ D for 25,D u 4mm (18)
ac w w
07,/2 3 1,4
C,= 3 647f (cosαα) tanZ D for 25,D > 4mm (19)
ac w w
04, 1
03,,139
⎛⎞
2r γγ(1− )
i
f,=×0089A λη
⎜⎟
c1
1/3
2rD−
(1+ γ )
i w
⎝⎠
−3/10
(20)
��
10/3
04, 1
��17, 2
⎧⎫
⎡⎤
��⎛⎞2rD−
⎪⎪r ⎛⎞1− γ
i ew
��
1+
⎢⎥
⎨⎜ ⎟ ⎬
⎜⎟
��
rr21−+D γ
��
⎢⎥ ⎝⎠
⎪⎪ei⎝⎠w
⎣⎦
��
⎩⎭
��
The value for 0,089A is 98,066 5 to calculate C in newtons. Values of f in ISO 281:2007, Table 4, rightmost
1 a c
column, are calculated by substituting raceway groove radii and reduction factors given in Table 1 into
Equation (20).
4.2.2 Thrust ball bearings with contact angle α = 90°
As in 4.1, for thrust ball bearings with contact angle α = 90°:
(ch−−3e+2)/(c−h+2)
(2ch+−5)/(c−h+2)
Cf= Z D (21)
ac w
04, 1
⎛⎞2r
13,
i 3/(ch−+2)
fA=×λη γ
⎜⎟
c1
(2ch+ −2)/(c−+h 2) 3e/(c−+h 2)
2rD−
40,5
⎝⎠iw
−−3/(ch+2)
(22)
��(2ch−+)/3
04, 1
��⎧⎫
��⎡⎤
r⎛⎞2rD−
⎪⎪
i ew
��
1+
⎢⎥
⎨⎜ ⎟ ⎬
��
rr2 −D
��
⎢⎥ei w
⎪⎪⎝⎠
⎣⎦
��
⎩⎭
��
in which γ = D /D .
w pw
Similarly, to take the effect of ball size into account, substitute experimental constants e = 10/9, c = 31/3, and
h = 7/3 into Equations (21) and (22), to give:
2/3 1,8
Cf= Z D for D u 25,4 mm (23)
ac w w
8 © ISO 2008 – All rights reserved
---------------------- Page: 14 ----------------------
ISO/TR 1281-1:2008(E)
2/3 1,4
Cf= 3,647ZD for D > 25,4 mm (24)
ac w w
−31/ 0
��10/3
04, 1
04, 1
��⎧⎫
��⎡⎤
⎛⎞2rr⎛ 2rD−⎞
⎪⎪
03,
ii ew
��
fA=+0,089 λη γ 1 (25)
⎢⎥
c1⎜⎟ ⎨⎜ ⎟⎬
��
22rD−−r r D
��
iw ⎢⎥e i w
⎝⎠ ⎪⎪⎝ ⎠
⎣⎦
��
⎩⎭
��
The value for 0,089A is 98,066 5 to calculate C in newtons. Values of f in ISO 281:2007, Table 4, second
1 a c
column from left, are calculated by substituting raceway groove radii and reduction factors which are given in
Table 1 into Equation (25).
4.3 Basic dynamic axial load rating, C , for thrust ball bearings with two or more rows of
a
balls
According to the product law of probability, relationships between the basic axial load rating of an entire thrust
ball bearing and of both the rotating and stationary rings are given as:
−−3/(ch+2)
−−(2ch+)/3 −(ch− +2)/3
⎡⎤
CC=+C (26)
ak
a1kka2
⎣⎦
CQ= sinαZ ⎫
a1kC k
1 ⎪
(27)
⎬
CQ= sinαZ
a2kC k
⎪
2 ⎭
−−3/(ch+2)
−−(2ch+)/3 −(ch− +2)/3
⎡⎤
CC=+C (28)
a
a1 a2
⎣⎦
n
⎫
CQ=⎪sinα Z
a1Ck
∑
1
⎪
k=1 ⎪
(29)
⎬
n
⎪
CQ= sinα Z
a2Ck∑ ⎪
2
⎪
k =1 ⎭
where
C is the basic dynamic axial load rating as a row k of an entire thrust ball bearing;
ak
C is the basic dynamic axial load rating as a row k of the rotating ring of an entire thrust ball bearing;
a1k
C is the basic dynamic axial load rating as a row k of the stationary ring of an entire thrust ball
a2k
bearing;
C is the basic dynamic axial load rating of an entire thrust ball bearing;
a
C is the basic dynamic axial load rating of the rotating ring of an entire thrust ball bearing;
a1
C is the basic dynamic axial load rating of the stationary ring of an entire thrust ball bearing;
a2
Z is the number of balls per row k.
k
© ISO 2008 – All rights reserved 9
---------------------- Page: 15 ----------------------
ISO/TR 1281-1:2008(E)
Substituting Equations (26), (27), and (29) into Equation (28), and rearranging, gives:
−−3/(ch+2)
−−(2ch+)/3 −(ch− +2)/3
⎡⎤
nn
⎛⎞ ⎛ ⎞
⎢⎥
⎜⎟QZsinαα+⎜Q sinZ⎟
Ck∑∑C k
12
n⎢⎥
⎜⎟ ⎜ ⎟
⎝⎠kk==11⎝ ⎠
⎢⎥
CZ=
a k
∑
−−(2ch+)/3
⎢⎥
n
⎛⎞
k=1
⎢⎥
⎜⎟
Z
∑ k
⎢⎥
⎜⎟
⎝⎠k =1
⎣⎦
−−3/(ch+2)
� −−(2ch+)/3�
−−3/(ch+2)
� �
⎧⎫
−−(2ch+)/3 −(ch− +2)/3
�⎪⎪⎡⎤ �
QZsinαα+Q sin
...
SLOVENSKI STANDARD
SIST-TP ISO/TR 1281-1:2009
01-junij-2009
.RWDOQLOHåDML3RMDVQLORNVWDQGDUGX,62GHO ,PHQVNDGLQDPLþQD
REUHPHQLWHYLQLPHQVNDGREDWUDMDQMD
Rolling bearings - Explanatory notes on ISO 281 - Part 1: Basic dynamic load rating and
basic rating life
Wälzlager - Erläuternde Anmerkungen zur ISO 281 - Teil 1: Dynamische Tragzahlen
und nominelle Lebensdauer
Roulements - Notes explicatives sur l'ISO 281 - Partie 1: Charges dynamiques de base
et durée nominale de base
Ta slovenski standard je istoveten z: ISO/TR 1281-1:2008
ICS:
21.100.20 Kotalni ležaji Rolling bearings
SIST-TP ISO/TR 1281-1:2009 en,fr
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.
---------------------- Page: 1 ----------------------
SIST-TP ISO/TR 1281-1:2009
---------------------- Page: 2 ----------------------
SIST-TP ISO/TR 1281-1:2009
TECHNICAL ISO/TR
REPORT 1281-1
First edition
2008-12-01
Rolling bearings — Explanatory notes
on ISO 281 —
Part 1:
Basic dynamic load rating and basic
rating life
Roulements — Notes explicatives sur l'ISO 281 —
Partie 1: Charges dynamiques de base et durée nominale de base
Reference number
ISO/TR 1281-1:2008(E)
©
ISO 2008
---------------------- Page: 3 ----------------------
SIST-TP ISO/TR 1281-1:2009
ISO/TR 1281-1:2008(E)
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electronic or mechanical, including photocopying and microfilm, without permission in writing from either ISO at the address below or
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Published in Switzerland
ii © ISO 2008 – All rights reserved
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SIST-TP ISO/TR 1281-1:2009
ISO/TR 1281-1:2008(E)
Contents Page
Foreword. iv
Introduction . v
1 Scope . 1
2 Normative references . 1
3 Symbols . 1
4 Basic dynamic load rating . 3
4.1 Basic dynamic radial load rating, C , for radial ball bearings . 4
r
4.2 Basic dynamic axial load rating, C , for single row thrust ball bearings. 7
a
4.3 Basic dynamic axial load rating, C , for thrust ball bearings with two or more rows of
a
balls. 9
4.4 Basic dynamic radial load rating, C , for radial roller bearings. 10
r
4.5 Basic dynamic axial load rating, C , for single row thrust roller bearings. 12
a
4.6 Basic dynamic axial load rating, C , for thrust roller bearings with two or more rows of
a
rollers . 13
5 Dynamic equivalent load. 15
5.1 Expressions for dynamic equivalent load. 15
5.2 Factors X, Y, and e. 27
6 Basic rating life . 38
7 Life adjustment factor for reliability. 39
Bibliography . 40
© ISO 2008 – All rights reserved iii
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SIST-TP ISO/TR 1281-1:2009
ISO/TR 1281-1:2008(E)
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies
(ISO member bodies). The work of preparing International Standards is normally carried out through ISO
technical committees. Each member body interested in a subject for which a technical committee has been
established has the right to be represented on that committee. International organizations, governmental and
non-governmental, in liaison with ISO, also take part in the work. ISO collaborates closely with the
International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.
The main task of technical committees is to prepare International Standards. Draft International Standards
adopted by the technical committees are circulated to the member bodies for voting. Publication as an
International Standard requires approval by at least 75 % of the member bodies casting a vote.
In exceptional circumstances, when a technical committee has collected data of a different kind from that
which is normally published as an International Standard (“state of the art”, for example), it may decide by a
simple majority vote of its participating members to publish a Technical Report. A Technical Report is entirely
informative in nature and does not have to be reviewed until the data it provides are considered to be no
longer valid or useful.
Attention is drawn to the possibility that some of the elements of this document may be the subject of patent
rights. ISO shall not be held responsible for identifying any or all such patent rights.
ISO/TR 1281-1 was prepared by Technical Committee ISO/TC 4, Rolling bearings, Subcommittee SC 8, Load
ratings and life.
This first edition of ISO/TR 1281-1, together with the first edition of ISO/TR 1281-2, cancels and replaces the
first edition of ISO/TR 8646:1985, which has been technically revised.
ISO/TR 1281 consists of the following parts, under the general title Rolling bearings — Explanatory notes on
ISO 281:
⎯ Part 1: Basic dynamic load rating and basic rating life
⎯ Part 2: Modified rating life calculation, based on a systems approach of fatigue stresses
iv © ISO 2008 – All rights reserved
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SIST-TP ISO/TR 1281-1:2009
ISO/TR 1281-1:2008(E)
Introduction
ISO/R281:1962
A first discussion on an international level of the question of standardizing calculation methods for load ratings
of rolling bearings took place at the 1934 conference of the International Federation of the National
Standardizing Associations (ISA). When ISA held its last conference in 1939, no progress had been made.
However, in its 1945 report on the state of rolling bearing standardization, the ISA 4 Secretariat included
proposals for definition of concepts fundamental to load rating and life calculation standards. This report was
distributed in 1949 as document ISO/TC 4 (Secretariat-1)1, and the definitions it contained are in essence
those given in ISO 281:2007 for the concepts “life” and “basic dynamic load rating” (now divided into “basic
dynamic radial load rating” and “basic dynamic axial load rating”).
In 1946, on the initiative of the Anti-Friction Bearing Manufacturers Association (AFBMA), New York,
discussions of load rating and life calculation standards started between industries in the USA and Sweden.
Chiefly on the basis of the results appearing in Reference [1], an AFBMA standard, Method of evaluating load
ratings of annular ball bearings, was worked out and published in 1949. On the same basis, the member body
for Sweden presented, in February 1950, a first proposal to ISO, “Load rating of ball bearings” [doc.
ISO/TC 4/SC 1 (Sweden-1)1].
In view of the results of both further research and a modification to the AFBMA standard in 1950, as well as
interest in roller bearing rating standards, in 1951, the member body for Sweden submitted a modified
proposal for rating of ball bearings [doc. ISO/TC 4/SC 1 (Sweden-6)20] as well as a proposal for rating of
roller bearings [doc. ISO/TC 4/SC 1 (Sweden-7)21].
Load rating and life calculation methods were then studied by ISO/TC 4, ISO/TC 4/SC 1 and ISO/TC 4/WG 3
at 11 different meetings from 1951 to 1959. Reference [2] was then of considerable use, serving as a major
basis for the sections regarding roller bearing rating.
The framework for the Recommendation was settled at a TC 4/WG 3 meeting in 1956. At the time,
deliberations on the draft for revision of AFBMA standards were concluded in the USA and ASA B3 approved
the revised standard. It was proposed to the meeting by the USA and discussed in detail, together with the
Secretariat's proposal. At the meeting, a WG 3 proposal was prepared which adopted many parts of the USA
proposal.
In 1957, a Draft Proposal (document TC 4 N145) based on the WG proposal was issued. At the WG 3
meeting the next year, this Draft Proposal was investigated in detail, and at the subsequent TC 4 meeting, the
adoption of TC 4 N145, with some minor amendments, was concluded. Then, Draft ISO Recommendation
No. 278 (as TC 4 N188) was issued in 1959, and ISO/R281 accepted by ISO Council in 1962.
ISO 281/1:1977
In 1964, the member body for Sweden suggested that, in view of the development of imposed bearing steels,
the time had come to review ISO/R281 and submitted a proposal [ISO/TC 4/WG 3 (Sweden-1)9]. However, at
this time, WG 3 was not in favour of a revision.
In 1969, on the other hand, TC 4 followed a suggestion by the member body for Japan (doc. TC 4 N627) and
reconstituted its WG 3, giving it the task of revising ISO/R281. The AFBMA load rating working group had at
this time started revision work. The member body for the USA submitted the Draft AFBMA standard, Load
ratings and fatigue life for ball bearings [ISO/TC 4/WG 3 (USA-1)11], for consideration in 1970 and Load
ratings and fatigue life for roller bearings [ISO/TC 4/WG 3 (USA-3)19] in 1971.
In 1972, TC 4/WG 3 was reorganized and became TC 4/SC 8. This proposal was investigated in detail at five
meetings from 1971 to 1974. The third and final Draft Proposal (doc. TC 4/SC 8 N23), with some amendments,
was circulated as a Draft International Standard in 1976 and became ISO 281-1:1977.
© ISO 2008 – All rights reserved v
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SIST-TP ISO/TR 1281-1:2009
ISO/TR 1281-1:2008(E)
The major part of ISO 281-1:1977 constituted a re-publication of ISO/R281, the substance of which had been
only very slightly modified. However, based mainly on American investigations during the 1960s, a new clause
was added, dealing with adjustment of rating life for reliability other than 90 % and for material and operating
conditions.
Furthermore, supplementary background information regarding the derivation of mathematical expressions
and factors given in ISO 281-1:1977 was published, first as ISO 281-2, Explanatory notes, in 1979; however,
TC 4/SC 8 and TC 4 later decided to publish it as ISO/TR 8646:1985.
vi © ISO 2008 – All rights reserved
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SIST-TP ISO/TR 1281-1:2009
TECHNICAL REPORT ISO/TR 1281-1:2008(E)
Rolling bearings — Explanatory notes on ISO 281 —
Part 1:
Basic dynamic load rating and basic rating life
1 Scope
This part of ISO/TR 1281 gives supplementary background information regarding the derivation of
mathematical expressions and factors given in ISO 281:2007.
2 Normative references
The following referenced documents are indispensable for the application of this document. For dated
references, only the edition cited applies. For undated references, the latest edition of the referenced
document (including any amendments) applies.
ISO 281:2007, Rolling bearings — Dynamic load ratings and rating life
3 Symbols
Clause
A constant of proportionality 7
A constant of proportionality determined experimentally 4
1
B constant of proportionality determined experimentally 4
1
C basic dynamic radial load rating of a rotating ring 4, 5
1
C basic dynamic radial load rating of a stationary ring 4, 5
2
C basic dynamic axial load rating for thrust ball or roller bearing 4, 6
a
C basic dynamic axial load rating of the rotating ring of an entire thrust ball or roller 4
a1
bearing
C basic dynamic axial load rating of the stationary ring of an entire thrust ball or roller 4
a2
bearing
C basic dynamic axial load rating as a row k of an entire thrust ball or roller bearing 4
ak
C basic dynamic axial load rating as a row k of the rotating ring of thrust ball or roller 4
a1k
bearing
C basic dynamic axial load rating as a row k of the stationary ring of thrust ball or roller 4
a2k
bearing
C basic dynamic load rating for outer ring 5
e
C basic dynamic load rating for inner ring 5
i
C basic dynamic radial load rating for radial ball or roller bearing 4, 5, 6
r
© ISO 2008 – All rights reserved 1
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SIST-TP ISO/TR 1281-1:2009
ISO/TR 1281-1:2008(E)
D pitch diameter of ball or roller set 4
pw
D ball diameter 4, 5
w
D mean roller diameter 4
we
E modulus of elasticity 4
o
F axial load 5
a
F radial load 4, 5
r
J factor relating mean equivalent load on a rotating ring to Q 4, 5
1 max
J factor relating mean equivalent load on a stationary ring to Q 4, 5
2 max
J axial load integral 5
a
J radial load integral 4, 5
r
L bearing life 7
L basic rating life 6, 7
10
L effective contact length of roller 4
we
L L per row k 4
wek we
N number of stress applications to a point on the raceway 4
P dynamic equivalent axial load for thrust bearing 5, 6
a
P dynamic equivalent radial load for radial bearing 5, 6
r
P dynamic equivalent radial load for the rotating ring 5
r1
P dynamic equivalent radial load for the stationary ring 5
r2
Q normal force between a rolling element and the raceways 4, 6
Q rolling element load for the basic dynamic load rating of the bearing 4, 6
C
Q rolling element load for the basic dynamic load rating of a ring rotating relative to the 4, 5
C
1
applied load
Q rolling element load for the basic dynamic load rating of a ring stationary relative to the 4, 5
C
2
applied load
Q maximum rolling element load 4, 5
max
S probability of survival, reliability 4, 7
V volume representative of the stress concentration 4
V rotation factor 5
f
X radial load factor for radial bearing 5
X radial load factor for thrust bearing 5
a
Y axial load factor for radial bearing 5
Y axial load factor for thrust bearing 5
a
Z number of balls or rollers per row 4, 5
Z number of balls or rollers per row k 4
k
a semimajor axis of the projected contact ellipse 4
a life adjustment factor for reliability 7
1
b semiminor axis of the projected contact ellipse 4
c exponent determined experimentally 4, 6
c compression constant 5
c
2 © ISO 2008 – All rights reserved
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SIST-TP ISO/TR 1281-1:2009
ISO/TR 1281-1:2008(E)
e measure of life scatter, i.e. Weibull slope determined experimentally 4, 5, 6, 7
f factor which depends on the geometry of the bearing components, the accuracy to 4
c
which the various components are made, and the material
h exponent determined experimentally 4, 6
i number of rows of balls or rollers 4
l circumference of the raceway 4
r cross-sectional raceway groove radius 5
r cross-sectional raceway groove radius of outer ring or housing washer 4
e
r cross-sectional raceway groove radius of inner ring or shaft washer 4
i
t auxiliary parameter 4
v J (0,5)/J (0,5) 5
2 1
z depth of the maximum orthogonal subsurface shear stress 4
o
α nominal contact angle 4, 5
α ′ actual contact angle 5
γ D cos α/D for ball bearings with α ≠ 90° 4
w pw
D /D for ball bearings with α = 90°
w pw
D cos α/D for roller bearings with α ≠ 90°
we pw
D /D for roller bearings with α = 90°
we pw
ε parameter indicating the width of the loaded zone in the bearing 5
η reduction factor 4, 5
λ reduction factor 4
µ factor introduced by Hertz 4
ν factor introduced by Hertz, or adjustment factor for exponent variation 4
σ maximum contact stress 4
max
Σρ curvature sum 4
τ maximum orthogonal subsurface shear stress 4
o
ϕ one half of the loaded arc 5
o
4 Basic dynamic load rating
The background to basic dynamic load ratings of rolling bearings according to ISO 281 appears in
References [1] and [2].
The expressions for calculation of basic dynamic load ratings of rolling bearings develop from a power
correlation that can be written as follows:
ce
τ NV
1
o
ln ∝ (1)
h
S
z
o
where
S is the probability of survival;
τ is the maximum orthogonal subsurface shear stress;
o
© ISO 2008 – All rights reserved 3
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SIST-TP ISO/TR 1281-1:2009
ISO/TR 1281-1:2008(E)
N is the number of stress applications to a point on the raceway;
V is the volume representative of the stress concentration;
z is the depth of the maximum orthogonal subsurface shear stress;
o
c, h are experimentally determined exponents;
e is the measure of life scatter, i.e. the Weibull slope determined experimentally.
For “point” contact conditions (ball bearings) it is assumed that the volume, V, representative of the stress
concentration in Correlation (1) is proportional to the major axis of the projected contact ellipse, 2a, the
circumference of the raceway, l, and the depth, z , of the maximum orthogonal subsurface shear stress, τ
o o:
Va∝ zl (2)
o
Substituting Correlation (2) into Correlation (1):
ce
1 τ Nal
o
ln ∝ (3)
h−1
S
z
o
“Line” contact was considered in References [1] and [2] to be approached under conditions where the major
axis of the calculated Hertz contact ellipse is 1,5 times the effective roller contact length:
21a,=5L (4)
we
2
In addition, b/a should be small enough to permit the introduction of the limit value of ab as b/a approaches 0:
23Q
2
ab = (5)
π∑E ρ
o
(for variable definitions, see 4.1).
4.1 Basic dynamic radial load rating, C , for radial ball bearings
r
From the theory of Hertz, the maximum orthogonal subsurface shear stress, τ , and the depth, z , can be
o o
expressed in terms of a radial load F , i.e. a maximum rolling element load, Q , or a maximum contact
r max
stress, σ , and dimensions for the contact area between a rolling element and the raceways. The
max
relationships are:
τσ= T
omax
z = ζ b
o
1/2
(2t −1)
T =
2(tt +1)
1
ζ =
1/2
(1tt+−)(2 1)
1/3
⎛⎞
3 Q
a = µ
⎜⎟
E ∑ ρ
⎝⎠o
4 © ISO 2008 – All rights reserved
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SIST-TP ISO/TR 1281-1:2009
ISO/TR 1281-1:2008(E)
1/3
⎛⎞
3 Q
bv=
⎜⎟
E ∑ ρ
o
⎝⎠
where
σ is the maximum contact stress;
max
t is the auxiliary parameter;
a is the semimajor axis of the projected contact ellipse;
b is the semiminor axis of the projected contact ellipse;
Q is the normal force between a rolling element and the raceways;
E is the modulus of elasticity;
o
Σρ is the curvature sum;
µ, v are factors introduced by Hertz.
Consequently, for a given rolling bearing, τ , a, l and z can be expressed in terms of bearing geometry, load
o o
and revolutions. Correlation (3) is changed to an equation by inserting a constant of proportionality. Inserting a
6
specific number of revolutions (e.g. 10 ) and a specific reliability (e.g. 0,9), the equation is solved for a rolling
element load for basic dynamic load rating which is designated to point contact rolling bearings introducing a
constant of proportionality, A :
1
04, 1
(1,59ch+−1,41 5,82)/(c−h+2)
⎛⎞
1,3 2r (1∓ γ )
QA= ×
⎜⎟
C 1
(2c+h−2)/(ch−+2) 3e/ (ch−+2) 3e/(ch−+2)
2rD−
40,5 (1± γ)
⎝⎠w
(6)
3/(ch−+2)
γ
⎛⎞
(2ch+−5)/(c−h+2) −3e/(c−h+2)
DZ
⎜⎟ w
cosα
⎝⎠
where
Q is the rolling element load for the basic dynamic load rating of the bearing;
C
D is the ball diameter;
w
γ is D cos α/D ;
w pw
in which
D is the pitch diameter of the ball set,
pw
α is the nominal contact angle;
Z is the number of balls per row.
The basic dynamic radial load rating, C , of a rotating ring is given by:
1
J
r
CQ==Z cos α 0,407Q Z cos α (7)
1CC
11
J
1
© ISO 2008 – All rights reserved 5
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SIST-TP ISO/TR 1281-1:2009
ISO/TR 1281-1:2008(E)
The basic dynamic radial load rating, C , of a stationary ring is given by:
2
J
r
CQ==Z cos α 0,389Q Z cos α (8)
2CC
22
J
2
where
Q is the rolling element load for the basic dynamic load rating of a ring rotating relative to the
C
1
applied load;
Q is the rolling element load for the basic dynamic load rating of a ring stationary relative to
C
2
the applied load;
J = J (0,5) is the radial load integral (see Table 3);
r r
J = J (0,5) is the factor relating mean equivalent load on a rotating ring to Q (see Table 3);
1 1 max
J = J (0,5) is the factor relating mean equivalent load on a stationary ring to Q (see Table 3).
2 2 max
The relationship between C for an entire radial ball bearing, and C and C , is expressed in terms of the
r 1 2
product law of probability as:
−3/(ch−+2)
(2ch−+)/3
⎡⎤
⎛⎞
C
1
⎢⎥
CC=+1 (9)
⎜⎟
r1
⎢⎥C
⎝⎠2
⎣⎦
Substituting Equations (6), (7) and (8) into Equation (9), the basic dynamic radial load rating, C , for an entire
r
ball bearing is expressed as:
04, 1
(1,59ch+−1,41 5,82)/(c−h+2)
⎡⎤
13, 2r (1− γ)
i 3/(ch−+2)
C,=041 A γ ×
r1⎢⎥
(2ch+ −2)/(ch−+2) 3e/(c−+h 2) 3e/(ch−+2)
2rD−
40,5 (1+ γ)
⎣⎦iw
−−3/(ch+2)
� �
(2ch−+)/3
04, 1
(1,59ch++1,41 3e− 5,82)/(c−h+2)
� ⎧ ⎫ �
⎡⎤
⎛⎞
� r 2rD− ⎛⎞1−γ �
⎪⎪
i ew
�11+ ,04 � × (10)
⎨⎬⎢⎥⎜⎟
⎜⎟
� �
rr21−+D γ
⎢⎥ ⎝⎠
ei⎝⎠w
�⎪⎪⎣⎦ �
⎩⎭
� �
� �
(ch−−1)/(2ch−+) (ch−−3e+2)/(2c−h+) (25c+h−)/(2c−h+)
(ciZos α) D
w
where
A is the experimentally determined proportionality constant;
1
r is the cross-sectional raceway groove radius of the inner ring;
i
r is the cross-sectional raceway groove radius of the outer ring;
e
i is the number of rows of balls.
Here, the contact angle, α, the number of rolling elements (balls), Z, and the diameter, D , depend on bearing
w
design. On the other hand, the ratios of raceway groove radii, r and r , to a half-diameter of a rolling element
i e
(ball), D /2 and γ = D cosα/D , are not dimensional, therefore it is convenient in practice that the value for
w w pw
the initial terms on the right-hand side of Equation (10) to be designated as a factor, f :
c
(ch−−1)/(2c−+h) (ch−−3e+2)/(2c−h+) (25c+h−)/(2c−h+)
Cf= (ci os α) Z D (11)
rc w
6 © ISO 2008 – All rights reserved
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SIST-TP ISO/TR 1281-1:2009
ISO/TR 1281-1:2008(E)
With radial ball bearings, the faults in bearings resulting from manufacturing need to be taken into
consideration, and a reduction factor, λ, is introduced to reduce the value for a basic dynamic radial load
rating for radial ball bearings from its theoretical value. It is convenient to include λ in the factor, f . The value
c
of λ is determined experimentally.
Consequently, the factor f is given by:
c
04, 1
(1,59ch+−1,41 5,82)/(c−h+2)
⎛⎞
13, 2r (1− γ)
3/(ch−+2)
i
f,=041λ A γ ×
⎜⎟
c1
(2c+h−2)/(ch−+2) 3e/(c−+h 2) 3e/(ch−+2)
2rD−
40,5 (1+ γ)
⎝⎠iw
−−3/(ch+2)
�� (12)
(2ch−+)/3
04, 1
��(1,59ch++1,41 3e−5,82)/(c−h+2)
⎧⎫
⎡⎤
��⎛⎞2rD−⎛⎞
⎪⎪r 1− γ
i ew
��
11+ ,04
⎢⎥
⎨⎬⎜⎟ ⎜⎟
��
rr21−+D γ
��
⎢⎥⎝⎠
⎪⎪ei⎝⎠w
⎣⎦
��
⎩⎭
��
Based on References [1] and [2], the following values were assigned to the experimental constants in the load
rating equations:
e = 10/9
c = 31/3
h = 7/3
Substituting the numerical values into Equation (11) gives the following, however, a sufficient number of test
results are only available for small balls, i.e. up to a diameter of about 25 mm, and these show that the load
1,8
rating may be taken as being proportional to D . In the case of larger balls, the load rating appears to
w
1,4
increase even more slowly in relation to the ball diameter, and D can be assumed where D > 25,4 mm:
w
w
07, 2/3 1,8
Cf= (ci osα) Z D for D u 25,4 mm (13)
rc w w
0,7 2/3 1,4
C,= 3 647f (i cos α) Z D for D > 25,4 mm (14)
rc w w
04, 1
03,,139
⎛⎞
2r γγ(1− )
i
f,=×0089A 0,41λ
⎜⎟
c1
1/3
2rD−
(1+ γ )
⎝⎠iw
−3/10
(15)
��10/3
04, 1
��17, 2
⎧⎫
��⎡⎤
⎛⎞ r⎛⎞2rD−
⎪⎪1− γ
i ew
��
11+ ,04
⎢⎥
⎨⎜⎜⎟ ⎟⎬
��
12+−γ rrD
��
⎝⎠⎢⎥ei w
⎪⎪⎝⎠
⎣⎦
��
⎩⎭
��
Values of f in ISO 281:2007, Table 2, are calculated by substituting raceway groove radii and reduction
c
factors given in Table 1 into Equation (15).
The value for 0,089A is 98,066 5 to calculate C in newtons.
1 r
4.2 Basic dynamic axial load rating, C , for single row thrust ball bearings
a
4.2.1 Thrust ball bearings with contact angle α ≠ 90°
As in 4.1, for thrust ball bearings with contact angle α ≠ 90°:
(ch−−1)/(2ch−+) (ch−−3e+2)/(2c−h+) (2c+h−5)/(2c−h+)
Cf= (cosαα) tanZ D (16)
ac w
© ISO 2008 – All rights reserved 7
---------------------- Page: 15 ----------------------
SIST-TP ISO/TR 1281-1:2009
ISO/TR 1281-1:2008(E)
For most thrust ball bearings, the theoretical value of a basic dynamic axial load rating has to be reduced on
the basis of unequal distribution of load among the rolling elements in addition to the reduction factor, λ, which
is introduced in to radial ball bearing load ratings. This reduction factor is designated as η.
Consequently, the factor f is given by:
c
04, 1
(1,59ch+−1,41 5,82)/(c−h+2)
⎛⎞
13, 2r (1− γ)
i 3/(ch−+2)
fA= λη γ ×
⎜⎟
c1
(2ch+ −2)/(c−+h 2) 3e/(c−+h 2) 3e/(c−+h 2)
2rD−
40,5 (1+ γ)
⎝⎠iw
−−3/(ch+2)
(17)
��
(2ch−+)/3
04, 1
��(1,59ch++1,41 3e− 5,82)/(c−h+2)
⎧⎫
⎡⎤
��⎛⎞
⎪ r 2rD− ⎛⎞1−γ ⎪
i ew
��
1+
⎨⎜⎢⎥⎟ ⎬
⎜⎟
��
rr21−+D γ
��
⎢⎥ ⎝⎠
ei⎝⎠w
⎪⎣⎦ ⎪
��
⎩ ⎭
��
Similarly, to take the effect of ball size into account, substitute experimental constants e = 10/9, c = 31/3, and
h = 7/3 into Equations (16) and 17) to give:
07, 2/3 1,8
Cf= (cosαα) tanZ D for 25,D u 4mm (18)
ac w w
07,/2 3 1,4
C,= 3 647f (cosαα) tanZ D for 25,D > 4mm (19)
ac w w
04, 1
03,,139
⎛⎞
2r γγ(1− )
i
f,=×0089A λη
⎜⎟
c1
1/3
2rD−
(1+ γ )
i w
⎝⎠
−3/10
(20)
��
10/3
04, 1
��17, 2
⎧⎫
⎡⎤
��⎛⎞2rD−
⎪⎪r ⎛⎞1− γ
i ew
��
1+
⎢⎥
⎨⎜ ⎟ ⎬
⎜⎟
��
rr21−+D γ
��
⎢⎥ ⎝⎠
⎪⎪ei⎝⎠w
⎣⎦
��
⎩⎭
��
The value for 0,089A is 98,066 5 to calculate C in newtons. Values of f in ISO 281:2007, Table 4, rightmost
1 a c
column, are calculated by substituting raceway groove radii and reduction factors given in Table 1 into
Equation (20).
4.2.2 Thrust ball bearings with contact angle α = 90°
As in 4.1, for thrust ball bearings with contact angle α = 90°:
(ch−−3e+2)/(c−h+2)
(2ch+−5)/(c−h+2)
Cf= Z D (21)
ac w
04, 1
⎛⎞2r
13,
i 3/(ch−+2)
fA=×λη γ
⎜⎟
c1
(2ch+ −2)/(c−+h 2) 3e/(c−+h 2)
2rD−
40,5
⎝⎠iw
−−3/(ch+2)
(22)
��(2ch−+)/3
04, 1
��⎧⎫
��⎡⎤
r⎛⎞2rD−
⎪⎪
i ew
��
1+
⎢⎥
⎨⎜ ⎟ ⎬
��
rr2 −D
��
⎢⎥ei w
⎪⎪⎝⎠
⎣⎦
��
⎩⎭
��
in which γ = D /D .
w pw
Similarly, to take the effect of ball size into account, substitute experimental constants e = 10/9, c = 31/3, and
h = 7/3 into Equations (21) and (22), to give:
2/3 1,8
Cf= Z D for D u 25,4 mm (23)
ac w w
8 © ISO 2008 – All rights reserved
---------------------- Page: 16 ----------------------
SIST-TP ISO/TR 1281-1:2009
ISO/TR 1281-1:2008(E)
2/3 1,4
Cf= 3,647ZD for D > 25,4 mm (24)
ac w w
−31/ 0
��10/3
04, 1
04, 1
��⎧⎫
��⎡⎤
⎛⎞2rr⎛ 2rD−⎞
⎪⎪
03,
ii ew
��
fA=+0,089 λη γ 1 (25)
⎢⎥
c1⎜⎟ ⎨⎜ ⎟⎬
��
22rD−−r r D
��
iw ⎢⎥e i w
⎝⎠ ⎪⎪⎝ ⎠
⎣⎦
��
⎩⎭
��
The value for 0,089A is 98,066 5 to calculate C in newtons. Values of f in ISO 281:2007, Table 4, second
1 a c
column from left, are calculated by substituting raceway groove radii and reduction factors which are given in
Table 1 into Equation (25).
4.3 Basic dynamic axial load rating, C , for thrust ball bearings with two or more rows of
a
balls
According to the product law of probability, relationships between the basic axial load rating of an entire thrust
ball bearing and of both the rotating and stationary rings are given as:
−−3/(ch+2)
−−(2ch+)/3 −(ch−
...
RAPPORT ISO/TR
TECHNIQUE 1281-1
Première édition
2008-12-01
Roulements — Notes explicatives
sur l'ISO 281 —
Partie 1:
Charges dynamiques de base et durée
nominale de base
Rolling bearings — Explanatory notes on ISO 281 —
Part 1: Basic dynamic load rating and basic rating life
Numéro de référence
ISO/TR 1281-1:2008(F)
©
ISO 2008
---------------------- Page: 1 ----------------------
ISO/TR 1281-1:2008(F)
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Publié en Suisse
ii © ISO 2008 – Tous droits réservés
---------------------- Page: 2 ----------------------
ISO/TR 1281-1:2008(F)
Sommaire Page
Avant-propos. iv
Introduction . v
1 Domaine d'application. 1
2 Références normatives . 1
3 Symboles . 1
4 Charge dynamique de base . 3
4.1 Charge radiale dynamique de base, C , des roulements radiaux à billes. 4
r
4.2 Charge axiale dynamique de base, C , des butées à billes à une rangée . 8
a
4.3 Charge axiale dynamique de base, C , des butées à billes à deux ou plusieurs rangées. 9
a
4.4 Charge radiale dynamique de base, C , des roulements (radiaux) à rouleaux. 10
r
4.5 Charge axiale dynamique de base, C , des butées à rouleaux à une rangée. 12
a
4.6 Charge axiale dynamique de base, C , des butées à deux ou plusieurs rangées de
a
rouleaux . 13
5 Charge dynamique équivalente. 15
5.1 Expressions de la charge dynamique équivalente . 15
5.2 Facteurs X, Y et e. 27
6 Durée nominale . 39
7 Facteur de réduction de la durée en fonction de la fiabilité. 40
Bibliographie . 42
© ISO 2008 – Tous droits réservés iii
---------------------- Page: 3 ----------------------
ISO/TR 1281-1:2008(F)
Avant-propos
L'ISO (Organisation internationale de normalisation) est une fédération mondiale d'organismes nationaux de
normalisation (comités membres de l'ISO). L'élaboration des Normes internationales est en général confiée
aux comités techniques de l'ISO. Chaque comité membre intéressé par une étude a le droit de faire partie du
comité technique créé à cet effet. Les organisations internationales, gouvernementales et non
gouvernementales, en liaison avec l'ISO participent également aux travaux. L'ISO collabore étroitement avec
la Commission électrotechnique internationale (CEI) en ce qui concerne la normalisation électrotechnique.
Les Normes internationales sont rédigées conformément aux règles données dans les Directives ISO/CEI,
Partie 2.
La tâche principale des comités techniques est d'élaborer les Normes internationales. Les projets de Normes
internationales adoptés par les comités techniques sont soumis aux comités membres pour vote. Leur
publication comme Normes internationales requiert l'approbation de 75 % au moins des comités membres
votants.
Exceptionnellement, lorsqu'un comité technique a réuni des données de nature différente de celles qui sont
normalement publiées comme Normes internationales (ceci pouvant comprendre des informations sur l'état
de la technique par exemple), il peut décider, à la majorité simple de ses membres, de publier un Rapport
technique. Les Rapports techniques sont de nature purement informative et ne doivent pas nécessairement
être révisés avant que les données fournies ne soient plus jugées valables ou utiles.
L'attention est appelée sur le fait que certains des éléments du présent document peuvent faire l'objet de
droits de propriété intellectuelle ou de droits analogues. L'ISO ne saurait être tenue pour responsable de ne
pas avoir identifié de tels droits de propriété et averti de leur existence.
L'ISO/TR 1281-1 a été élaboré par le comité technique ISO/TC 4, Roulements, sous-comité SC 8, Charges de
base et durée.
Cette première édition de l'ISO/TR 1281-1, conjointement avec la première édition de l'ISO/TR 1281-2, annule
et remplace la première édition de l'ISO/TR 8646:1985, qui a fait l'objet d'une révision technique.
L'ISO/TR 1281 comprend les parties suivantes, présentées sous le titre général Roulements — Notes
explicatives sur l'ISO 281:
⎯ Partie 1: Charges dynamiques de base et durée nominale de base
⎯ Partie 2: Calcul modifié de la durée nominale de base fondé sur une approche système du travail de
fatigue
iv © ISO 2008 – Tous droits réservés
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ISO/TR 1281-1:2008(F)
Introduction
ISO/R281:1992
Une première discussion de niveau international portant sur la normalisation des méthodes de calcul des
charges de base des roulements eut lieu en 1934 lors de la conférence de la Fédération Internationale des
Associations Nationales de Normalisation (ISA). Lorsque l'ISA tint sa dernière réunion en 1939, aucun
progrès n'était encore intervenu. Pourtant, dans son rapport de 1945 sur l'état de la normalisation dans le
domaine des roulements, le Secrétariat de l'ISA 4 incluait des propositions de définitions de concepts
fondamentaux pour les normes de calcul de charges de base et de durée. Ce rapport fut diffusé en 1949 sous
la référence ISO/TC 4 (Secrétariat-1)1, les définitions qu'il contenait étant en substance celles que reprend
l'ISO 281:2007 sous les termes de «durée» et de «charge dynamique de base» (cette dernière étant
maintenant séparée en «charge dynamique radiale de base» et «charge dynamique axiale de base».
Les discussions sur les normes de calcul de durée et de charges de base reprirent en 1946 entre les
spécialistes américains et suédois à l'initiative de l'AFBMA — Anti-Friction Bearing Manufacturers Association
(New York). Une norme AFBMA, publiée en 1949, intitulée «Method of evaluating load ratings of annular ball
bearings» fut élaborée sur la base principalement des résultats des recherches scientifiques effectuées par
G. Lundberg et A. Palmgren et parues en 1947 (voir la Référence [1]). Partant de la même source, le Comité
membre suédois soumit en février 1950 une première proposition à l'ISO [doc. ISO/TC 4/SC 1 (Suède-1)1]
intitulée «Charges de base des roulements à billes».
Compte tenu des recherches nouvelles, de la révision de la norme AFBMA en 1950 et également de l'intérêt
pour les normes de calcul des roulements à rouleaux, le Comité membre suédois présenta, en 1951, une
proposition modifiée de calcul des roulements à billes [doc. ISO/TC 4/SC 1 (Suède-6)20] puis une proposition
de calcul des roulements à rouleaux [doc. ISO/TC 4/SC 1 (Suède-7)21].
Ces méthodes de calcul furent étudiées par l'ISO/TC 4, le TC 4/SC 1 et le TC 4/GT 3 lors de 11 réunions
différentes s'étalant entre 1951 et 1959. Vint s'ajouter à ces documents une étude Lundberg-Palmgren
publiée en 1952 (voir la Référence [2]) qui eut un retentissement considérable sur l'élaboration des chapitres
relatifs au calcul des roulements à rouleaux.
Le cadre de la recommandation fut arrêté définitivement par l'ISOTC 4/GT 3 lors de sa réunion de 1956. À la
même époque, les États-Unis avaient fini de réviser les normes AFBMA et le Comité ASA B3 avait approuvé
le document révisé. Cette norme fut présentée en réunion par les États-Unis et discutée en détail en même
temps que la proposition du Secrétariat. Lors de la réunion, le GT 3 prépara une proposition qui reprenait de
nombreuses parties de la proposition américaine.
L'avant-projet issu de cette proposition de groupe de travail (doc. TC 4 N145) parut en 1957. Il fut examiné en
détail par le GT 3 lors de sa réunion de l'année suivante et adopté à la réunion suivante du TC 4 avec
quelques amendements mineurs. Le projet de Recommandation ISO N° 278 fut publié en 1959 (TC 4 N188)
et accepté en 1962 par le Conseil de l'ISO sous la référence Recommandation ISO/R281.
ISO 281-1:1977
En 1964, le Comité membre suédois suggéra, au vu de l'amélioration des aciers pour roulements, qu'il était
temps de réviser l'ISO/R281. La Suède soumit une proposition ISO/TC 4/GT 3 (Suède-1)9, mais à l'époque le
GT 3 ne se déclara pas en faveur d'une telle révision.
En 1969, cependant, le TC 4 suivit la suggestion du Comité membre du Japon (doc. TC 4 N627) de
reconstituer le GT 3 et de lui donner pour tâche de réviser l'ISO/R281. Le groupe AFBMA de calcul des
charges de base avait également à l'époque reprit les travaux pour réviser la norme et, en 1970, le Comité
membre américain soumit un projet de norme AFBMA intitulé «Load ratings and fatigue life for ball bearings»
[doc. ISO/TC 4/GT 3 (USA-1)11]), suivi en 1971 d'un autre projet «Load ratings and fatigue life for roller
bearings» [doc. ISO/TC 4/GT 3 (USA-3)19].
© ISO 2008 – Tous droits réservés v
---------------------- Page: 5 ----------------------
ISO/TR 1281-1:2008(F)
En 1972, le statut du GT 3 fut modifié et le groupe devint le sous-comité TC 4/SC 8. Le projet fut examiné en
e
détail au cours de cinq réunions s'étalant entre 1971 et 1974 et le projet final (3 avant-projet TC 4/SC 8 N23)
fut diffusé avec quelques modifications sous forme de projet de Norme internationale en 1976. En 1977,
l'ISO 281-1:1977 était acceptée par le Conseil de l'ISO.
La majeure partie de cette Norme internationale constitue une réédition de la Recommandation ISO/R281
dont le fond n'est que très peu modifié. Un nouvel article a cependant été ajouté, résultat de recherches
américaines des années 1960 et qui traite de la correction à apporter à la durée si la fiabilité est supérieure à
90 % ou pour tenir compte des matériaux et des conditions de fonctionnement.
Des informations complémentaires relatives à la manière dont sont déterminés les expressions et facteurs de
l'ISO 281-1:1977 devaient être publiées sous la référence ISO 281-2, Notes explicatives, mais en 1979, le
TC 4/SC 8 et le TC 4 décidèrent de les publier sous la forme d'un Rapport technique, l'ISO/TR 8646:1985.
vi © ISO 2008 – Tous droits réservés
---------------------- Page: 6 ----------------------
RAPPORT TECHNIQUE ISO/TR 1281-1:2008(F)
Roulements — Notes explicatives sur l'ISO 281 —
Partie 1:
Charges dynamiques de base et durée nominale de base
1 Domaine d'application
La présente partie de l'ISO/TR 1281 donne un certain nombre d'informations sur la manière dont ont été
définis les expressions mathématiques et les facteurs donnés dans l'ISO 281:2007.
2 Références normatives
Les documents de référence suivants sont indispensables pour l'application du présent document. Pour les
références datées, seule l'édition citée s'applique. Pour les références non datées, la dernière édition du
document de référence s'applique (y compris les éventuels amendements).
ISO 281:2007, Roulements — Charges dynamiques de base et durée nominale
3 Symboles
Article
A constante de proportionnalité 7
A constante de proportionnalité déterminée expérimentalement 4
1
B constante de proportionnalité déterminée expérimentalement 4
1
C charge radiale dynamique de base d'une bague tournante 4, 5
1
C charge radiale dynamique de base d'une bague fixe 4, 5
2
C charge axiale dynamique de base d'une butée à bille ou à rouleaux 4, 6
a
C charge axiale dynamique de base de la rondelle tournante d'une butée à billes ou à 4
a1
rouleaux
C charge axiale dynamique de base de la rondelle fixe d'une butée à billes ou à 4
a2
rouleaux
C charge axiale dynamique de base de la rangée k d'une butée à billes ou à rouleaux 4
ak
C charge axiale dynamique de base de la rangée k de la rondelle tournante d'une 4
a1k
butée à billes ou à rouleaux
C charge axiale dynamique de base de la rangée k de la rondelle fixe d'une butée à 4
a2k
billes ou à rouleaux
C charge dynamique de base d'une bague extérieure 5
e
C charge dynamique de base d'une bague intérieure 5
i
C charge radiale dynamique de base d'un roulement (radial) à billes ou à rouleaux 4, 5, 6
r
© ISO 2008 – Tous droits réservés 1
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ISO/TR 1281-1:2008(F)
D diamètre primitif 4
pw
D diamètre de bille 4, 5
w
D diamètre moyen de rouleau 4
we
E module d'élasticité 4
o
F charge axiale 5
a
F charge radiale 4, 5
r
J facteur rapportant à Q la charge moyenne équivalente sur une bague tournante 4, 5
1 max
(par rapport à la charge appliquée)
J facteur rapportant à Q la charge moyenne équivalente sur une bague fixe (par 4, 5
2 max
rapport à la charge appliquée)
J intégrale de charge axiale 5
a
J intégrale de charge radiale 4, 5
r
L durée du roulement 7
L durée nominale 6, 7
10
L longueur effective de contact du rouleau 4
we
L L pour la rangée k 4
wek we
N nombre d'application pour la contrainte en un point du chemin de roulement 4
P charge axiale dynamique équivalente d'une butée 5, 6
a
P charge radiale dynamique équivalente d'un roulement (radial) 5, 6
r
P charge radiale dynamique équivalente de la bague tournante 5
r1
P charge radiale dynamique équivalente de la bague fixe 5
r2
Q force normale entre un élément roulant et les chemins de roulement 4, 6
Q charge sur l'élément roulant correspondant à la charge dynamique de base du 4, 6
C
roulement
Q charge sur l'élément roulant correspondant à la charge dynamique de base d'une 4, 5
C
1
bague tournante (par rapport à la charge appliquée)
Q charge sur l'élément roulant correspondant à la charge dynamique de base d'une 4, 5
C
2
bague fixe (par rapport à la charge appliquée)
Q charge maximale sur l'élément roulant 4,5
max
S probabilité de survie, fiabilité 4, 7
V volume représentatif de la concentration des contraintes 4
V facteur de rotation 5
f
X facteur de charge radiale pour roulement (radial) 5
X facteur de charge radiale pour butée 5
a
Y facteur de charge axiale pour roulement (radial) 5
Y facteur de charge axiale pour butée 5
a
Z nombre de billes ou de rouleaux par rangée 4, 5
Z nombre de billes ou de rouleaux par rangée k 4
k
a demi grand axe de l'ellipse de contact projetée 4
a facteur de réduction de la durée en fonction de la fiabilité 7
1
b demi petit axe de l'ellipse de contact projetée 4
2 © ISO 2008 – Tous droits réservés
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ISO/TR 1281-1:2008(F)
c exposant déterminé expérimentalement 4, 6
c constante de compression 5
c
e mesure de dispersion de durée, c'est-à-dire pente de la courbe de Weibull, 4, 5, 6, 7
déterminée expérimentalement
f facteur qui dépend de la géométrie des éléments du roulement, de leur précision 4
c
d'exécution et des matériaux
h exposant déterminé expérimentalement 4, 6
i nombre de rangées de billes ou de rouleaux 4
l circonférence du chemin du roulement 4
r rayon de courbure (transversal) d'un chemin de roulement 5
r rayon de courbure (transversal) d'un chemin de roulement de bague extérieure ou 4
e
de rondelle-logement
r rayon de courbure (transversal) d'un chemin de roulement de bague intérieure ou de 4
i
rondelle-arbre
t paramètre auxiliaire 4
v
J (0,5)/J (0,5) 5
2 1
z profondeur de la contrainte maximale de cisaillement orthogonale sous la surface 4
o
α angle nominal de contact 4, 5
α ′ angle réel de contact 5
γ D cos α/D pour roulements à billes avec α ≠ 90° 4
wpw
DD/ pour roulements à billes avec α = 90°
wpw
D cos α/D pour roulements à rouleaux avec α ≠ 90°
we pw
DD/ pour roulements à rouleaux avec α = 90°
we pw
ε paramètre caractéristique de la grandeur de la zone chargée dans le roulement 5
η facteur de réduction 4, 5
λ facteur de réduction 4
µ facteur introduit par Hertz 4
ν facteur introduit par Hertz, ou facteur de réduction de la variation de l'exposant 4
σ contrainte maximale de contact 4
max
Σρ somme des courbures 4
τ contrainte maximale de cisaillement orthogonale sous la surface 4
o
ϕ moitié de l'arc chargé 5
o
4 Charge dynamique de base
Les calculs de charges dynamiques de base de l'ISO 281 sur les roulements sont fondés sur les
Références [1] et [2].
Les formules de calcul des charges dynamiques de base des roulements dérivent de la relation suivante:
ce
1 τ NV
o
ln ∝ (1)
h
S
z
o
© ISO 2008 – Tous droits réservés 3
---------------------- Page: 9 ----------------------
ISO/TR 1281-1:2008(F)
où
S est la probabilité de survie;
τ est la composante orthogonale de la contrainte maximale de cisaillement sous la surface;
o
N est le nombre d'applications de la contrainte en un point donné du chemin de roulement;
V est le volume représentatif de la concentration des contraintes;
z est la profondeur de la composante orthogonale de la contrainte maximale de cisaillement sous la
o
surface;
c, h sont des exposants déterminés expérimentalement;
e est la mesure de la dispersion de la durée, c'est-à-dire la pente de la courbe de Weibull, déterminée
expérimentalement.
Dans les conditions de contact «ponctuel» (roulements à billes), on prend comme hypothèse que le volume, V,
représentatif de la concentration des contraintes dans la Relation (1) est proportionnel au grand axe de
l'ellipse de contact projetée, 2a, à la circonférence du chemin de roulement, l, et à la profondeur, z , de la
o
composante orthogonale de la contrainte maximale de cisaillement sous la surface, τ .
o
Va∝ zl (2)
o
D'où, si l'on introduit (2) dans la Relation (1):
ce
τ Nal
1
o
ln ∝ (3)
h−1
S
z
o
Lundberg et Palmgren (voir Références [1] et [2]) ont considéré qu'on pouvait admettre un contact «linéaire»
lorsque le grand axe de l'ellipse de contact calculée (ellipse de Hertz) était de 1,5 fois la longueur effective de
contact du rouleau:
21aL=,5 (4)
we
2
Il convient, en outre, que b/a soit suffisamment petit pour permettre d'introduire la valeur-limite de ab pour b/a
tendant vers 0:
23Q
2
ab = (5)
π∑E ρ
o
(pour les notations, se reporter à 4.1).
4.1 Charge radiale dynamique de base, C , des roulements radiaux à billes
r
D'après la théorie de Hertz, la composante orthogonale de la contrainte maximale de cisaillement sous la
surface, τ , et sa profondeur, z , peuvent se rattacher à une charge radiale, F , c'est-à-dire une charge
o o r
maximale sur l'élément roulant, Q , ou une contrainte maximale de contact, σ , et aux dimensions de la
max max
zone de contact entre un élément roulant et les chemins de roulement. Les relations correspondantes
s'expriment comme suit:
τσ= T
omax
z = ζ b
o
4 © ISO 2008 – Tous droits réservés
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ISO/TR 1281-1:2008(F)
1/2
(2t −1)
T =
2(tt +1)
1
ζ =
1/2
(1tt+−)(2 1)
1/3
⎛⎞
3 Q
a = µ
⎜⎟
E ∑ ρ
o
⎝⎠
1/3
⎛⎞
3 Q
bv=
⎜⎟
E ∑ ρ
o
⎝⎠
où
σ est la contrainte maximale de contact;
max
t est le paramètre auxiliaire;
a est le demi grand axe de l'ellipse de contact projetée;
b est le demi petit axe de l'ellipse de contact projetée;
Q est la force normale entre l'élément roulant et les chemins de roulement;
E est le module d'élasticité;
o
Σρ est la somme des courbures;
µ, v sont des facteurs introduits par Hertz.
En conséquence, pour un roulement donné, τ , a, l et z peuvent s'exprimer en fonction de la géométrie du
o o
roulement, de la charge et du nombre de tours. La Relation (3) devient une équation si l'on y introduit une
6
constante de proportionnalité. En supposant un nombre déterminé de tours (par exemple 10 ) et une fiabilité
également déterminée (par exemple 0,9), l'équation peut être résolue pour une charge sur l'élément roulant
correspondant à la charge dynamique de base sur le roulement. Pour un contact ponctuel et en désignant par
A la constante de proportionnalité, cette charge s'exprime par
1
0,41
(1,59ch+−1,41 5,82)/(c−h+2)
⎛⎞
1,3 2r (1∓ γ )
QA= ×
⎜⎟
C 1
(2ch+ −2)/(c−+h 2) 3e /(c−+h 2) 3e/(c−+h 2)
2rD−
40,5 w (1± γ)
⎝⎠
(6)
3/(ch−+2)
⎛⎞γ
(2ch+−5)/(c−h+2) −3e/(c−h+2)
DZ
⎜⎟ w
cosα
⎝⎠
où
Q est la charge sur l'élément roulant correspondant à la charge dynamique de base du roulement;
C
D est le diamètre de bille;
w
γ est D cos α/D
w pw
où
D est le diamètre primitif,
pw
α est l'angle nominal de contact,
Z est le nombre de billes par rangée.
© ISO 2008 – Tous droits réservés 5
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ISO/TR 1281-1:2008(F)
La charge radiale dynamique de base, C , d'une bague tournante s'obtient comme suit:
1
J
r
CQ==Z cos α 0,407Q Z cos α (7)
1CC
11
J
1
La charge radiale dynamique de base, C , d'une bague fixe s'obtient comme suit:
2
J
r
CQ==Z cos α 0,389Q Z cos α (8)
2CC
22
J
2
où
Q est la charge sur l'élément roulant correspondant à la charge dynamique de base d'une
C
1
bague tournante par rapport à la charge appliquée;
Q est la charge sur l'élément roulant correspondant à la charge dynamique de base d'une
C
2
bague fixe par rapport à la charge appliquée;
J = J (0,5) est l'intégrale de la charge radiale (voir Tableau 3);
r r
J = J (0,5) est la facteur rapportant à Q la charge moyenne équivalente sur une bague tournante
1 1 max
par rapport à la charge appliquée (voir Tableau 3);
J = J (0,5) est la facteur rapportant à Q la charge moyenne équivalente sur une bague fixe par
2 2 max
rapport à la charge appliquée (voir Tableau 3).
La relation entre C , pour un roulement (radial) à billes complet, et C et C s'exprime selon la loi du produit
r 1 2
des probabilités:
−3/(ch−+2)
(2ch−+)/3
⎡⎤
⎛⎞C
1
⎢⎥
CC=+1 (9)
⎜⎟
r1
⎢⎥
C
2
⎝⎠
⎣⎦
Si l'on remplace les termes de l'Équation (9) par leurs valeurs données dans les Équations (6), (7) et (8), la
charge radiale dynamique de base, C , d'un roulement à billes complet s'exprime de la façon suivante:
r
04, 1
(1,59ch+−1,41 5,82)/(c−h+2)
⎡⎤2r
13, (1− γ)
i 3/(ch−+2)
C,=041 A γ ×
⎢⎥
r1
(2ch+ −2)/(ch−+2) 3e/(c−+h 2) 3e/(ch−+2)
2rD−
40,5 (1+ γ)
⎣⎦iw
−−3/(ch+2)
��(2ch−+)/3
04, 1
��(1,59ch++1,41 3e− 5,82)/(c−h+2)
⎧ ⎫
⎡⎤
��r⎛⎞2rD− ⎛⎞
⎪⎪1−γ
i ew
��11+ ,04 × (10)
⎢⎥
⎨⎬⎜⎟ ⎜⎟
��
rr21−+D γ
⎢⎥ ⎝⎠
⎪⎪ei⎝⎠w
��⎣⎦
⎩⎭
��
��
(ch−−1)/(2ch−+) (ch−−3e+2)/(2c−h+) (25c+h−)/(2c−h+)
(ciZos α) D
w
où
A est la constante de proportionnalité déterminée expérimentalement;
1
r est le rayon de courbure du chemin de roulement de la bague inférieure (en section transversale);
i
r est le rayon de courbure du chemin de roulement de la bague extérieure (en section transversale);
e
i est le nombre de rangées de billes.
6 © ISO 2008 – Tous droits réservés
---------------------- Page: 12 ----------------------
ISO/TR 1281-1:2008(F)
Dans le cas considéré, l'angle de contact, α, le nombre d'éléments roulants (billes), Z, et le diamètre, D ,
w
dépendent de la conception du roulement. Par ailleurs, le rapport des rayons de courbure, r et r , au demi-
i e
diamètre de l'élément roulant (bille), D /2 et λ = D cosα/D , sont des grandeurs sans dimension. Il est donc
w w pw
commode, dans la pratique, de remplacer les premiers termes du membre de droite de l'Équation (10) par un
facteur f :
c
(ch−−1)/(c−+h 2) (ch−−3e+2)/(c−h+2) (2c+h−5)/(c−h+2)
Cf= (ci os α) Z D (11)
rc w
Dans le cas de roulements (radiaux) à billes, il faut considérer les défauts pouvant résulter de la fabrication et
introduire un facteur de réduction, λ, qui réduit la valeur théorique de la charge radiale dynamique de base du
roulement; il est pratique également d'inclure le facteur λ dans le facteur f , la valeur de ce facteur λ étant
c
déterminée expérimentalement.
Le facteur f devient ainsi:
c
04, 1
(1,59ch+−1,41 5,82)/(c−h+2)
⎛⎞
13, 2r (1− γ)
3/(ch−+2)
i
f,=041λ A γ ×
⎜⎟
c1
(2c+h−2)/(ch−+2) 3e/(c−+h 2) 3e/(ch−+2)
2rD−
40,5 (1+ γ)
⎝⎠iw
(12)
−−3/(ch+2)
��(2ch−+)/3
04, 1
��(1,59ch++1,41 3e−5,82)/(c−h+2)
⎧⎫
⎡⎤
��r⎛⎞2rD−⎛⎞
⎪⎪1−γ
i ew
��11+ ,04
⎢⎥
⎨⎬⎜⎟ ⎜⎟
��
rr21−+D γ
⎢⎥ ⎝⎠
⎪⎪ei⎝⎠w
��⎣⎦
⎩⎭
��
��
À la suite des premiers travaux expérimentaux de Lundberg et Palmgren sur les roulements à billes (voir
Références [1] et [2]), les valeurs suivantes ont été attribuées aux constantes expérimentales des équations
de calcul de charge:
e = 10/9
c = 31/3
h = 7/3
Si l'on remplace ces termes par leur valeur numérique dans l'Équation (11), on obtient ce qui suit. Toutefois,
les résultats d'essai disponibles concernent essentiellement les petites billes, c'est-à-dire jusqu'à un diamètre
d'environ 25 mm, et les résultats montrent que la charge de base peut dans ce cas être prise comme
1,8
proportionnelle à D . Dans le cas de billes plus grosses, la charge de base semble augmenter bien plus
w
1,4
lentement avec le diamètre de bille et on peut admettre la proportionnalité à D lorsque D > 25,4 mm:
w w
07, 2/3 1,8
Cf= (ci osα) Z D pour D u 25,4 mm (13)
rc w w
0,7 2/3 1,4
C,= 3 647f (i cos α) Z D pour D > 25,4 mm (14)
rc w w
04, 1
03,,139
⎛⎞
2r γγ(1− )
i
f,=×0 089A 0,41λ
⎜⎟
c1
1/3
2rD−
(1+ γ )
⎝⎠iw
(15)
−3/10
��10/3
04, 1
��17, 2
⎧⎫
⎡⎤
��⎛⎞ r⎛⎞2rD−
⎪⎪1−γ
i e w
��11+ ,04
⎢⎥
⎨⎬⎜⎟ ⎜⎟
��
12+−γ rrD
⎝⎠⎢⎥ewi
⎪⎪⎝⎠
��⎣⎦
⎩⎭
��
��
© ISO 2008 – Tous droits réservés 7
---------------------- Page: 13 ----------------------
ISO/TR 1281-1:2008(F)
Les valeurs de f de l'ISO 281:2007, Tableau 2, sont calculées d'après l'Équation (15) où l'on a remplacé les
c
rayons de courbure et les facteurs de réduction par les valeurs données dans le Tableau 1.
La valeur de 0,089A pour calculer C en newtons est 98,066 5.
1 r
4.2 Charge axiale dynamique de base, C , des butées à billes à une rangée
a
4.2.1 Butées à billes à angle de contact α ≠ 90°
De même qu'en 4.1, pour les butées à billes à angle de contact α ≠ 90:
(ch−−1)/(2ch−+) (ch−−3e+2)/(2c−h+) (2c+h−5)/(2c−h+)
Cf= (cosαα) tanZ D (16)
ac w
Dans la plupart des cas, la valeur théorique de la charge axiale dynamique de base doit être réduite pour tenir
compte de la répartition non uniforme de la charge entre les éléments roulants, et ce en sus du facteur de
réduction, λ, déjà prévu pour les roulements (radiaux) à billes. Ce facteur de réduction supplémentaire est ici
désigné par η.
Le facteur f s'écrit en conséquence:
c
04, 1
(1,59ch+−1,41 5,82)/(c−h+2)
⎛⎞2r
13, (1− γ)
i 3/(ch−+2)
fA= λη γ ×
⎜⎟
c1
(2ch+ −2)/(c−+h 2) 3e/(c−+h 2) 3e/(c−+h 2)
2rD−
40,5 (1+ γ)
⎝⎠iw
(17)
−−3/(ch+2)
��(2ch−+)/3
04, 1
��(1,59ch++1,41 3e− 5,82)/(c−h+2)
⎧⎫
⎡⎤
��r⎛⎞2rD− ⎛⎞
⎪ 1− γ ⎪
i ew
��1+
⎢⎥
⎨⎜ ⎟ ⎜⎟ ⎬
��
rr21−+D γ
⎢⎥ei w ⎝⎠
⎪⎝⎠ ⎪
��⎣⎦
⎩ ⎭
��
��
En remplaçant dans les Équations (16) et (17) les constantes expérimentales par leurs valeurs e = 10/9,
c = 31/3 et h = 7/3 et en tenant compte à nouveau de l'effet de la dimension des billes, on obtient:
0,7 2/3 1,8
Cf= (cosαα) tanZ D pour D u 25,4 mm (18)
ac w w
0,7 2 / 3 1,4
Cf= 3,647 (cosαα) tanZD pour D > 25,4 mm (19)
ac w w
04, 1
03,,139
⎛⎞
2r γγ(1− )
i
f,=×0 089A λη
⎜⎟
c1
1/3
2rD−
(1+ γ )
⎝⎠iw
(20)
−3/10
��10/3
04, 1
��⎧⎫17, 2
⎡⎤
��r⎛⎞2rD−⎛ 1− γ⎞
⎪⎪
i ew
��
1+
⎢⎥
⎨⎜ ⎟ ⎜⎟ ⎬
��
rr21−+D γ
⎢⎥ei w ⎝⎠
⎪⎪⎝⎠
��⎣⎦
⎩⎭
��
��
La valeur de 0,089A pour calculer C en newtons est de 98,066 5. Les valeurs de f dans l'ISO 281:2007,
1 a c
Tableau 4, colonne de droite, sont calculées d'après l'Équation (20) où l'on a remplacé les rayons de courbure
et le facteur de réduction par leurs valeurs données dans le Tab
...
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