ISO/TR 13989-2:2000

Calculation of scuffing load capacity of cylindrical, bevel and hypoid gears - Part 2: Integral temperature method

Calcul de la capacité de charge au grippage des engrenages cylindriques, coniques et hypoïdes — Partie 2: Méthode de la température intégrale

Izračun nosilnosti glede na toplotno razjedanje zobnih bokov valjastih, stožčastih in hipoidnih zobnikov - 2. del: Metoda povprečne temperature

General Information

• Status: Withdrawn
• Publication Date: 22-Mar-2000
• Withdrawal Date: 22-Mar-2000
• ICS: 21.200 - Gears
• Technical Committee: ISO/TC 60/SC 2 - Gear capacity calculation
• Drafting Committee: ISO/TC 60/SC 2/WG 6 - Gear calculations
• Current Stage: 9599 - Withdrawal of International Standard
• Start Date: 07-Jan-2014
• Completion Date: 18-Dec-2013
• Ref Project: SIST ISO/TR 13989-2:2002 - Calculation of scuffing load capacity of cylindrical, bevel and hypoid gears -- Part 2: Integral temperature method

Relations

• Revised: ISO/TS 6336-21:2017 - Calculation of load capacity of spur and helical gears - Part 21: Calculation of scuffing load capacity (also applicable to bevel and hypoid gears) - Integral temperature method
• Effective Date: 15-Dec-2017

Standards Content (Sample)

TECHNICAL ISO/TR
REPORT 13989-2
First edition
2000-03-15
Calculation of scuffing load capacity of
cylindrical, bevel and hypoid gears —
Part 2:
Integral temperature method
Calcul de la capacité de charge au grippage des engrenages cylindriques,
coniques et hypoïdes —
Partie 2: Méthode de la température intégrale
Reference number
©
ISO 2000
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ii © ISO 2000 – All rights reserved

Contents Page
Foreword.v
Introduction.vi
1 Scope .1
2 Normative references .1
3 Terms, definitions, symbols and units .1
3.1 Terms and definitions .1
3.2 Symbols and units.1
4 Field of application .6
4.1 Scuffing damage.6
4.2 Integral temperature criterion.7
5 Influence factors .7
5.1 Mean coefficient of friction� .7
mC
5.2 Run-in factor X .10
E
5.3 Thermal flash factor X .10
M
5.4 Pressure angle factor X .11
��
6 Calculation.12
6.1 Cylindrical gears.12
6.1.1 Scuffing safety factor S .12
intS
6.1.2 Permissible integral temperature� .12
intP
6.1.3 Integral temperature� .12
int
6.1.4 Flash temperature at pinion tooth tip� .13
flaE
6.1.5 Bulk temperature� .13
M
6.1.6 Mean coefficient of friction� .14
mC
6.1.7 Run-in factor X .14
E
6.1.8 Thermal flash factor X .14
M
6.1.9 Pressure angle factor X .14
��
6.1.10 Geometry factor at tip of pinion X .14
BE
6.1.11 Approach factor X .14
Q
6.1.12 Tip relief factor X .15
Ca
6.1.13 Contact ratio factor: X .16
ε
6.2 Bevel gears.19
6.2.1 Scuffing safety factor S .20
intS
6.2.2 Permissible integral temperature� .20
intP
6.2.3 Integral temperature� .20
int
6.2.4 Flash temperature at pinion tooth tip� .20
flaE
6.2.5 Bulk temperature� .20
M
6.2.6 Mean coefficient of friction� .20
mC
6.2.7 Run-in factor X .21
E
6.2.8 Thermal flash factor X .21
M
6.2.9 Pressure angle factor X .21
��
6.2.10 Geometry factor at tip of pinion X .21
BE
6.2.11 Approach factor X .21
Q
6.2.12 Tip relief factor X .21
Ca
6.2.13 Contact ratio factor X .22
ε
6.3 Hypoid gears .22
6.3.1 Scuffing safety factor S .22
intS
6.3.2 Permissible integral temperature� .22
intP
6.3.3 Integral temperature� .22
int
6.3.4 Bulk temperature� .22
M
6.3.5 Mean coefficient of friction� .23
mC
6.3.6 Run-in factor X .23
E
6.3.7 Geometry factor X .23
G
6.3.8 Approach factor X .24
Q
6.3.9 Tip relief factor X .25
Ca
6.3.10 Contact ratio factor X .25
ε
6.3.11 Calculation of virtual crossed axes helical gears .25
6.4 Scuffing integral temperature.29
6.4.1 Scuffing integral temperature� .29
intS
6.4.2 Relative welding factor X .33
WrelT
Annex A (informative) Examples.34
Annex B (informative) Contact-time-dependent scuffing temperature.44
Bibliography .48
iv © ISO 2000 – All rights reserved

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.
The main task of technical committees is to prepare International Standards, but in exceptional circumstances a
technical committee may propose the publication of a Technical Report of one of the following types:
� type 1, when the required support cannot be obtained for the publication of an International Standard, despite
repeated efforts;
� type 2, when the subject is still under technical development or where for any other reason there is the future
but not immediate possibility of an agreement on an International Standard;
� type 3, 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).
Technical Reports of types 1 and 2 are subject to review within three years of publication, to decide whether they
can be transformed into International Standards. Technical Reports of type 3 do not necessarily have to be
reviewed until the data they provide are considered to be no longer valid or useful.
Technical Reports are drafted in accordance with the rules given in the ISO/IEC Directives, Part 3.
Attention is drawn to the possibility that some of the elements of this part of ISO/TR 13989 may be the subject of
patent rights. ISO shall not be held responsible for identifying any or all such patent rights.
ISO/TR 13989-2, which is a Technical Report of type 2, was prepared by Technical Committee ISO/TC 60, Gears,
Subcommittee SC 2, Gear capacity calculation.
This document is being issued in the Technical Report (type 2) series of publications (according to
subclause G.3.2.2 of Part 1 of the ISO/IEC Directives, 1995) as a “prospective standard for provisional application”
in the field of scuffing load capacity of gears because there is an urgent need for guidance on how standards in this
field should be used to meet an identified need. In 1975, two methods to evaluate the risk of scuffing were
documented to be studied by ISO/TC 60. It was agreed that after a period of experience one method shall be selected.
Since the subject is still under technical development and there is a future possibility of an agreement on an
International Standard, the publication of a type 2 Technical Report was proposed.
This document is not to be regarded as an “International Standard”. It is proposed for provisional application so that
information and experience of its use in practice may be gathered. Comments on the content of this document
should be sent to the ISO Central Secretariat.
A review of this Technical Report (type 2) will be carried out not later than three years after its publication with the
options of: extension for another three years; conversion into an International Standard; or withdrawal.
ISO/TR 13989 consists of the following parts, under the general title Calculation of scuffing load capacity of
cylindrical, bevel and hypoid gears:
� Part 1: Flash temperature method
� Part 2: Integral temperature method
Annexes A and B of this part of ISO/TR 13989 are for information only.
Introduction
This part of ISO/TR 13989 describes the surface damage "warm scuffing" for cylindrical (spur and helical), bevel
and hypoid gears for generally used gear materials and different heat treatments. "Warm scuffing" is characterized
by typical scuffing and scoring marks, which can lead to increasing power loss, dynamic load, noise and wear. For
"cold scuffing", in general associated with low temperature and low speed, under approximately 4 m/s, and
through-hardened, heavily loaded gears, the equations are not suitable.
There is a particularly severe form of gear tooth surface damage in which seizure or welding together of areas of
tooth surfaces occurs, due to absence or breakdown of a lubricant film between the contacting tooth flanks of
mating gears, caused by high temperature and high pressure. This form of damage is termed "scuffing" and most
relevant when surface velocities are high. Scuffing may also occur for relatively low sliding velocities when tooth
surface pressures are high enough, either generally or, because of uneven surface geometry and loading, in
discrete areas.
Risk of scuffing damage varies with the properties of gear materials, the lubricant used, the surface roughness of
tooth flanks, the sliding velocities and the load. Excessive aeration or the presence of contaminants in the lubricant
such as metal particles in suspension, also increase the risk of scuffing damage. Consequences of the scuffing of
high speed gears include a tendency to high levels of dynamic loading due to increase of vibration, which usually
leads to further damage by scuffing, pitting or tooth breakage.
High surface temperatures due to high surface pressures and sliding velocities can initiate the breakdown of
lubricant films. On the basis of this hypothesis two approaches to relate temperature to lubricant film breakdown
are presented:
� the flash temperature method (presented in ISO/TR 13989-1), based on contact temperatures which vary
along the path of contact;
� the integral temperature method (presented in this part of ISO/TR 13989), based on the weighted average of
the contact temperatures along the path of contact.
The integral temperature method is based on the assumption that scuffing is likely to occur when the mean value of
the contact temperature (integral temperature) is equal to or exceeds a corresponding critical value. The risk of
scuffing of an actual gear unit can be predicted by comparing the integral temperature with the critical value,
derived from a gear test for scuffing resistance of lubricants. The calculation method takes account of all significant
influence parameters, i.e. the lubricant (mineral oil with and without EP-additives, synthetic oils), the surface
roughness, the sliding velocities, the load, etc.
In order to ensure that all types of scuffing and comparable forms of surface damage due to the complex
relationships between hydrodynamical, thermodynamical and chemical phenomena are dealt with, further methods
of assessment may be necessary. The development of such methods is the objective of ongoing research.
vi © ISO 2000 – All rights reserved

TECHNICAL REPORT ISO/TR 13989-2:2000(E)
Calculation of scuffing load capacity of cylindrical, bevel and
hypoid gears —
Part 2:
Integral temperature method
1 Scope
This part of ISO/TR 13989 specifies the integral temperature method for calculating the scuffing load capacity of
cylindrical, bevel and hypoid gears.
2 Normative references
The following normative documents contain provisions which, through reference in this text, constitute provisions of
this part of ISO/TR 13989. For dated references, subsequent amendments to, or revisions of, any of these
publications do not apply. However, parties to agreements based on this part of ISO/TR 13989 are encouraged to
investigate the possibility of applying the most recent editions of the normative documents indicated below. For
undated references, the latest edition of the normative document referred to applies. Members of ISO and IEC
maintain registers of currently valid International Standards.
ISO 53:1998, Cylindrical gears for general and heavy engineering — Standard basic rack tooth profile.
ISO 1122-1:1998, Vocabulary of gear terms — Part 1: Definitions related to geometry.
ISO 1328-1:1995, Cylindrical gears — ISO system of accuracy — Part 1: Definitions and allowable values of
deviations relevant to corresponding flanks of gear teeth.
ISO 6336-1:1996, Calculation of load capacity of spur and helical gears — Part 1: Basic principles, introduction and
general influence factors.
1)
ISO 10300-1:— , Calculation of load capacity of bevel gears — Part 1: Introduction and general influence factors.
3 Terms, definitions, symbols and units
3.1 Terms and definitions
For the purposes of this part of ISO/TR 13989, the terms and definitions given in ISO 1122-1 apply.
3.2 Symbols and units
The symbols used in this part of ISO/TR 13989 are given in Table 1.
1) To be published.
Table 1 — Symbols and units
Symbol Description Unit Reference
a centre distance mm —
a
virtual centre distance of virtual cylindrical gear mm ISO 10300-1
v
b face width, smaller value of pinion or wheel mm —
b
effective facewidth for scuffing mm Eq. (46)
eB
c specific heat capacity per unit volume N/(mm ·K) —
v
single stiffness N/(mm·µm) ISO 6336-1
c�
c mesh stiffness N/(mm·µm) ISO 6336-1
�
d referencecirclediameter mm —
d effective tip diameter mm —
Na
d tip diameter mm Eq. (69)
a
d base diameter mm Eq. (70)
b
d
diameter at mid-facewidth mm —
m
d reference circle of virtual crossed axes helical gear mm Eq. (68)
s
d reference diameter of virtual cylindrical gear mm ISO 10300-1
v
d tip diameter of virtual cylindrical gear mm ISO 10300-1
va
d base diameter of virtual cylindrical gear mm ISO 10300-1
vb
g recess path of contact of pinion, wheel mm Eqs. (90), (91)
an1,2
g approach path of contact of pinion, wheel mm Eqs. (90), (91)
fn1,2
g* sliding factor — Eq. (62)
h addendum at mid-facewidth of hypoid gear mm —
am
m module mm —
m normal module of hypoid gear at mid-facewidth mm —
mn
m normal module of virtual crossed axes helical gear mm Eq. (73)
sn
n number of meshing gears ——
p
p normal base pitch mm Eq. (74)
en
u gear ratio ——
u gear ratio of virtual cylindrical gear — ISO 10300-1
v
v reference line velocity m/s —
v tangential velocity of pinion, wheel of hypoid gear m/s Eqs. (77), (78)
t1,2
v
maximum sliding velocity at tip of pinion m/s Eq. (83)
g�1
v sliding velocity at pitch point m/s Eq. (82)
gs
v sliding velocity m/s Eqs. (84), (85)
g1,2
v sliding velocity m/s Eq. (87)
g�1
2 © ISO 2000 – All rights reserved

Table 1 (continued)
Symbol Description Unit Reference
v sliding velocity m/s Eq. (88)
g�1
v tangential speed at reference cone at mid-facewidth of m/s —
mt
bevel gear
v sums of tangential speeds at pitch point m/s Eqs. (2), (47), (81)
ΣC
tangential speed m/s Eq. (79)
v
Σs
v tangential speed m/s Eq. (80)
Σh
w specific tooth load, scuffing N/mm Eq. (4)
Bt
z number of teeth ——
z number of teeth of virtual cylindrical gear — ISO 10300-1
v
1/2
B thermal contact coefficient N/(mm·s ·K) Eq. (12)
M
C ,C ,C weighting factors ——
1 2 2H
C nominal tip reliefµm —
a
C
effective tip reliefµm Eqs. (37), (38), (49)
eff
E module of elasticity (Young's modulus) N/mm —
F
nominal tangential load at reference cone at mid-facewidth N —
mt
F normal tooth load N Eq. (51)
n
F nominal tangential load at reference circle N —
t
K application factor — ISO 6336-1,
A
ISO 10300-1
K dynamic factor — ISO 6336-1,
v
ISO 10300-1
K = K transverse load factor (scuffing) — 6.2.4, ISO 6336-1,
B� H�
ISO 10300-1
K = K face load factor (scuffing) — ISO 6336-1
B� H�
ISO 10300-1, 6.2.4,
Eqs. (52), (53)
K helical load factor (scuffing) — Eq. (5), 6.2.4, 6.3.5
B�
K bearing factor — 6.3.3
B�be
K transverse load factor — ISO 6336-1,
H�
ISO 10300-1
K face load factor — ISO 6336-1,
H�
ISO 10300-1
K bearing factor — ISO 10300-1
H�be
L contact parameter — Eq. (55)
Ra
arithmetic mean roughnessµm Eq. (6)
S scuffing safety factor — Eq. (14)
intS
S minimum required scuffing safety factor ——
Smin
Table 1 (continued)
Symbol Description Unit Reference
T torque of the pinion Nm —
T scuffing torque of test pinion Nm Eq. (96)
1T
X geometry factor at pinion tooth tip — Eq. (22)
BE
X
run-in factor — Eq. (8)
E
X tip relief factor — Eq. (32)
Ca
X geometry factor of hypoid gears — Eq. (54)
G
X lubricant factor — 5.1
L
X thermal flash factor — Eq. (9)
M
X approach factor — Eqs. (25), (26), (27)
Q
X roughness factor — Eq. (7)
R
X lubrication factor — 6.1.5.3
S
X welding factor of executed gear — Table 3
W
X welding factor of test gear — 6.4.2
WT
X relative welding factor — Eq. (102)
WrelT
X contact factor — Eq. (21)
mp
X pressure angle factor — Eqs. (13), (48)
��
X contact ratio factor — Eqs. (39) to (44)
ε
pressure angle °—
�
normal pressure angle at mid-facewidth of hypoid gear °—
�
mn
� normal pressure angle °—
n
normal pressure angle of crossed axes helical gear ° Eq. (64)
�
sn
� transverse pressure angle of crossed axes helical gear ° Eq. (66)
st
transverse pressure angle °—
�
t
� ´ transverse working pressure angle °—
t
transverse pressure angle of virtual cylindrical gear ° ISO 10300-1
�
vt
arbitrary angle ° Figure 2
�
y
helix angle °—
�
helix angle at base circle ° Eqs. (67), (71)
�
b
� helix angle at reference cone at mid-facewidth of hypoid °—
m
gear
� helix angle of virtual crossed axes helical gear ° Eq. (63)
s
auxiliary angle ° Eq. (86)
�
reference cone angle °—
�
4 © ISO 2000 – All rights reserved

Table 1 (continued)
Symbol Description Unit Reference
recess contact ratio — Eqs. (28), (29)
�
a
� approach contact ratio — Eqs. (28), (29)
f
contact ratio in normal section of virtual crossed axes — Eqs. (92), (93)
�
n
helical gear
addendum contact ratio of the pinion — Eq. (30)
�
� addendum contact ratio of the wheel — Eq. (31)
contact ratio — Eq. (45)
�
�
transverse contact ratio of virtual cylindrical gear — ISO 10300-1
�
v�
tip contact ratio of virtual cylindrical pinion — ISO 10300-1
�
v1
tip contact ratio of virtual cylindrical wheel — ISO 10300-1
�
v2
Hertzian auxiliary coefficient — Figure 7, Eqs. (57), (59)
�
mean coefficient of friction — Eqs. (1), (1a)
�
mC
� dynamic viscosity at oil temperature —
mPa�s
oil
heat conductivity —
� N/(s�K)
M
Poisson's ratio ——
�
� kinematic viscosity of the oil at 40 �C mm /s; cSt —
radius of curvature at tip of the pinion, wheel mm Eqs. (23), (24)
�
E1,2
� relative radius of curvature at pitch point in normal section mm Eq. (76)
Cn
radius of curvature at pitch point in normal section mm Eq. (75)
�
n1,2
relative radius of curvature at pitch point mm Eq. (3)
�
redC
Hertzian auxiliary coefficient — Figure 7, Eqs. (58), (60)
�
Hertzian auxiliary angle ° Eqs. (56) to (60)
�
flash temperature at pinion tooth tip when load sharing is K Eq. (19)
�
flaE
neglected
mean flash temperature K Eq. (18)
�
flaint
� mean flash temperature of hypoid gear K Eq. (50)
flainth
integral temperature K Eq. (17)
�
int
permissible integral temperature K Eq. (16)
�
intP
scuffing integral temperature (allowable integral K Eq. (94)
�
intS
temperature)
� mean flash temperature of the test gear K Eqs. (96), (99), (101)
flaintT
oil sump or spray temperature °C —
�
oil
� bulk temperature °C Eq. (20)
M-C
Table 1 (concluded)
Symbol Description Unit Reference
test bulk temperature Eqs. (95), (98), (100)
� �C
MT
axle angle of virtual crossed axes helical gear ° Eq. (72)
�
� axle angle of virtual crossed axes helical gear ° Eq. (65)
run-in grade — 5.2
�
E
parameter on the line of action — Eq. (10)
�
Subscripts:
1pinion
2 wheel
a tip diameter of the virtual gear
b base circle of the virtual gear
m mid-facewidth of bevel or hypoid gears
n normal section
s virtual crossed axes helical gear
t tangential direction
T test gear
4 Field of application
The calculation methods are based on results of the rig testing of gears run at pitch line velocities less than 80 m/s.
The equations can be used for gears which run at higher speeds, but with increasing uncertainty as speed
increases. The uncertainty concerns the estimation of bulk temperature, coefficient of friction, allowable
temperatures, etc. as speeds exceed the range with experimental background.
4.1 Scuffing damage
When once initiated, scuffing damage can lead to gross degradation of tooth flank surfaces, with increase of: power
loss, dynamic loading, noise and wear. It can also lead to tooth breakage if the severity of the operating conditions
is not reduced. In the event of scuffing due to an instantaneous overload, followed immediately by a reduction of
load, e.g. by load redistribution, the tooth flanks may self-heal by smoothing themselves to some extent. Even so,
the residual damage will continue to be a cause of increased power loss, dynamic loading and noise.
In most cases, the resistance of gears to scuffing can be improved by using a lubricant with enhanced E.P.
(extreme pressure) properties. It is important however, to be aware that some disadvantages attend the use of E.P.
oils — corrosion of copper, embrittlement of elastomers, lack of world-wide availability, etc. These disadvantages
are to be taken into consideration if optimum lubricant choice is to be made, which means: as few additives as
possible, but as many as necessary.
Due to continuous variation of different parameters, the complexity of the chemical properties and the thermo-
hydro-elastic processes in the instantaneous contact area, some scatter in the calculated assessments of
probability of scuffing risk is to be expected.
In contrast to the relatively long time of development of fatigue damage, one single momentary overload can initiate
scuffing damage of such severity that affected gears may no longer be used. This should be carefully considered
when choosing an adequate safety factor for gears, especially for gears required to operate at high circumferential
velocities.
6 © ISO 2000 – All rights reserved

4.2 Integral temperature criterion
This approach to the evaluation of the probability of scuffing is based on the assumption that scuffing is likely to
occur when the mean value of the contact temperatures along the path of contact is equal to or exceeds a
corresponding "critical value" . In the method presented herein, the sum of the bulk temperature and the weighted
mean of the integrated values of flash temperatures along the path of contact is the "integral temperature". The
bulk temperature is estimated as described under 6.1.5 and the mean value of the flash temperature is
approximated by substituting mean values of the coefficient of friction, the dynamic loading, etc., along the path of
contact. A weighting factor is introduced accounting for possible different influences of a real bulk temperature
value and a mathematically integrated mean flash temperature value on the scuffing phenomenon.
The probability of scuffing is assessed by comparing the integral temperature with a corresponding critical value
derived from the gear testing of lubricants for scuffing resistance (e.g. different FZG test procedures, the IAE and
the Ryder gear tests), or from gears which have scuffed in service.
5 Influence factors
5.1 Mean coefficient of friction�
mC
The actual coefficient of friction between the tooth flanks is an instantaneous and local value which depends on
several properties of the oil, surface roughness, lay of the surface irregularities such as those left by machining,
properties of the tooth flank materials, tangential velocities, forces at the surfaces and the dimensions. Assessment
of the instantaneous coefficient of friction is difficult since there is no method currently available for its
measurement.
The mean value for the coefficient of friction� along the path of contact was derived from measurements [1] and
mC
approximated by Equation (1). Although the local coefficient of friction is near to zero in the pitch point C, the mean
value can be approximated with the parameters at the pitch point and the oil viscosity � at oil temperature �
oil oil
when introduced into Equation (1).
02,
wK�
F I
Bt B�
�00, 5
2)
� = 0,045 � ��� XX� (1)
mC G J oil R L
���
H K
�C redC
The coefficient of friction of the integral temperature method takes account of the size of the gear in a different way
as the coefficient of friction of the flash temperature method. Equation (1) for calculating the coefficient of friction
should not be applied outside the field of the part where it is presented, e.g. coefficient of friction for thermal rating.
The equation for the calculation of � was derived from experiments in the following range of operating
mC
conditions. Extrapolation may lead to deviations between the calculated and the real coefficient of friction.
1m/su vu 50 m/s
At reference line velocities v lower than 1 m/s, higher coefficients of friction are expected. At reference line
velocities v higher than 50 m/s, the limiting value of v at v = 50 m/s has to be used in equation (1).
ΣC
w W 150 N/mm
Bt
For lower values of the specific normal tooth load w , the limiting value w = 150 N/mm has to be used in
Bt Bt
Equation (1).
v =2 · v · tan� ´ · cos� (2)
�C t t
�
u sin�
t
� � ��a (3)
redC
cos�
(1� u)
b
F
t
w = K · K · K · K · (4)
Bt A v B� B�
b
2� This formula for the coefficient of friction is derived from testing of gears with centre distance a � 100 mm.
0,2
��Fb
bt �0,05 0,25
��= 0,048��� Ra� X (1a)
mC oil L
��
�����
�C redC
where
0,2
��6
X��0,75 for polyglycols;
L
��
��v
�C
X = 1,0 for mineral oils;
L
X = 0,8 for polyalfaolefins;
L
X = 1,5 for traction fluids;
L
X = 1,3 for phosphate esters.
L
Equation (1a) represents results of tests within a range of a = 91,5 mm to 200 mm. The application of this equation makes it
necessary to adjust Figures 9, 10 and 11 for the scuffing temperature� accordingly.
intS
8 © ISO 2000 – All rights reserved

K is the helical load factor, scuffing takes account of increasing friction for increasing total contact ratio (see
B�
Figure 1).
Figure 1 — Helical load factor K
B�
K =1 for � u 2
B� �
K ��10,2����25� � for 2 �� � 3,5 (5)
didi
B�� � �
K =1,3 for� W 3,5
B� �
Ra =0,5 · (Ra + Ra ) (6)
1 2
Ra , Ra are the tooth flank roughness values of pinion and wheel measured on the new flanks as manufactured
1 2
(e.g. reference test gear Ra values are � 0,35 �m).
0,25
X =2,2 · (Ra/� ) (7)
R redC
where
X = 1,0 for mineral oils;
L
X = 0,8 for polyalfaolefins;
L
X = 0,7 for non water-soluble polyglycols;
L
X = 0,6 for water-soluble polyglycols;
L
X = 1,5 for traction fluids;
L
X = 1,3 for phosphate esters.
L
5.2 Run-in factor X
E
The present calculation methods presume that the gears are well run-in. In practice scuffing failure occurs very
often during the first few hours in service, e.g. in a full load test run, the acceptance run of vessels or when a new
set of gears is built into a production machinery when the gears are run under full load conditions before a proper
run-in. Investigations [1] show a 1/4 to 1/3 load carrying capacity of a newly manufactured gear flank as compared
to a properly run-in flank. This should be taken into account by a run-in factor X :
E
30� Ra
X ��11�� � (8)
bg
EE
�
redC
where
� = 1, full run-in (for carburized and ground gears full run-in can be assumed if Ra � 0,6 Ra );
E run-in new
� = 0, newly manufactured.
E
5.3 Thermal flash factor X
M
The thermal flash factor X accounts for the influence of the properties of pinion and gear materials on the flash
M
temperature.
Calculation of the thermal flash factor for an arbitrary point (index y) on the line of action (see Figure 2):
0,25
��
��
�
��11��� �
�� ��
��u
��
X�� (9)
M
��
��
11��vv �
��
� BB��11�� � �
�� ��
M1 M2
��
EE u
��
��12
a
Tipcircle1
b
Tipcircle2
Figure 2 — Parameter� on the line of action
10 © ISO 2000 – All rights reserved

tan�
y
� � � 1 (10)
y
tan��
t
If the materials of pinion and wheel are the same Equation (9) can be simplified to:
0,25
E
X � (11)
M
20,25
(1��vB)
M
In the above equations the thermal contact coefficient B is:
M
Bc()� � (12)
M� Mv
For case hardened steels with the following typical characteristic values:
2 2
� =50N/(s�K), c =3,8N/(mm �K), E = 206 000 N/mm and v=0,3
M v
follows
–0,75 0,5 –0,5
X = 50,0 K�N �s �m �mm
M
For the characteristic values of other materials, see [7].
5.4 Pressure angle factor X
��
The pressure angle factor X is used to account for the conversion of load and tangential speed from reference
��
circle to pitch circle.
Method A: Factor X
��-A
02,,5 025 02, 5
(sin ��cos��cos )
��
n
t
X ��12, 2 (13)
�
�� A
05,,05
(cos���� cos )
t t
Table 2 shows the values for the pressure angle factor X for a standard rack with pressure angle � =20�,the
�� n
typical range of standard working pressure angles a� and helix angles�.
t
Table 2 — Method B: Factor X
��-B
a�
� =0� � =10� � =20� � =30�
t
0,963 0,960 0,951 0,938
19�
20� 0,978 0,975 0,966 0,952
0,992 0,989 0,981 0,966
21�
1,007 1,004 0,995 0,981
22�
1,021 1,018 1,009 0,995
23�
24� 1,035 1,032 1,023 1,008
1,049 1,046 1,037 1,012
25�
As an approximation, for gears with normal pressure angle � =20�, the pressure angle factor can be
n
approximated as follows:
X =1
��-B
6 Calculation
6.1 Cylindrical gears
This part of ISO/TR 13989 contains equations which enable the assessment of the "probability of scuffing" (warm
scuffing) of oil lubricated, involute spur and helical gears.
It is assumed that the total tangential load is equally distributed between the two helices of double helical gears.
When, due to application of forces such as external axial forces, this is not the case, the influences of these are to
be taken into account separately. The two helices are to be treated as parallel single helical gears. Influences
affecting scuffing probability, for which quantitative assessments can be made, are included.
The equations are valid for gears with external or internal teeth which are conjugate to a basic rack as defined in
ISO 53. For internal gears negative values have to be introduced for the determination of the geometry factor X
BE
as presented in 6.1.10. They may also be considered as valid for similar gears of other basic rack form, of which
the transverse contact ratio is � u 2,5.
�
6.1.1 Scuffing safety factor S
intS
As uncertainties and inaccuracies in the assumptions cannot be excluded, it is necessary to introduce a safety
factor S . It must be pointed out that the scuffing safety factor is temperature related and is not a factor by which
intS
gear torque may be multiplied to arrive at same values for the integral temperature number � and the scuffing
int
integral temperature number� .
intS
�
intS
SS� W (14)
intS Smin
�
int
Recommendation for choosing S :
Smin
S � 1 High scuffing risk
Smin
1u S u 2 Critical range with moderate scuffing risk, influenced by the operating conditions of the actual
Smin
gear. Influencing factors are e.g. the tooth flank roughness, run-in effects, the accurate
knowledge of the load factors, the load capacity of lubricating oil, etc.
S � 2 Low scuffing risk
Smin
Given the relationship between the actual load and the integral temperature number, the corresponding load safety
factor S can be approximated by:
Sl
w � ��
Btmax intS oil
S�� (15)
Sl
w ���
Bteff int oil
6.1.2 Permissible integral temperature�
intP
�
intS
� = (16)
intP
S
Smin
The minimum required scuffing safety factor S is to be separately determined for each application.
Smin
6.1.3 Integral temperature�
int
� =� + C �� u� (17)
int M 2 flaint intP
where C is the weighting factor derived from experiments. For spur and helical gears C =1,5.
2 2
12 © ISO 2000 – All rights reserved

� =� � X (18)
flaint flaE �
6.1.4 Flash temperature at pinion tooth tip�
flaE
07,,5 05
()Kw��v
X
BB� t
E
������� ��� � (19)
flaE MBE
mC ��
02, 5
XX�
a QCa
6.1.5 Bulk temperature�
M
The bulk temperature is the temperature of the tooth surfaces immediately before they come into contact.
The bulk temperature is established by the thermal balance of the gear unit. There are several sources of heat in a
gear unit of which the most important are tooth and bearing friction. Other sources of heat such as seals and oil
flow contribute to some extent. At pitch line velocities in excess of 80 m/s, heat from the churning of oil in the mesh
and windage losses may become significant and should be taken into consideration (see Method A). The heat is
transferred to the environment via the housing walls by conduction, convection and radiation and for spray
lubrication conditions through the oil into an external heat exchanger.
Values obtained using the different calculation methods described below are to be distinguished by the subscripts
A, B, C.
6.1.5.1 Method A�
M-A
The bulk temperature as a mean value or as temperature distribution over the facewidth can be measured
experimentally or be determined by a theoretical analysis based on known power loss and heat transfer data, i.e.
by using thermal network methods.
6.1.5.2 Method B�
M-B
This method is not used for the integral temperature method (see the flash temperature method given in
ISO/TR 13989-1).
6.1.5.3 Method C�
M-C
An approximate value for the bulk temperature consists of the sum of the oil temperature and a part of a mean
value derived from the flash temperature over the path of contact according to method C.
� ��� CX� �� �X (20)
�
MC oil 1 mp flaint S
where
X = 1,2 for spray lubrication;
S
X = 1,0 for dip lubrication;
S
X = 0,2 for gears submerged in oil;
S
C is the constant accounting for heat transfer conditions, from test results C =0,7;
1 1
1+ n
p
X = (21)
mp
where n is the number of meshing gears.
p
6.1.6 Mean coefficient of friction�
mC
See 5.1.
6.1.7 Run-in factor X
E
See 5.2.
6.1.8 Thermal flash factor X
M
See 5.3.
6.1.9 Pressure angle factor X
��
See 5.4.
6.1.10 Geometry factor at tip of pinion X
BE
The geometry factor X takes into account Hertzian stress and sliding velocity at the pinion tooth tip. X is a
BE BE
function of the gear ratio u and the radius of curvature� at the pinion tooth tip E.
E
For internal gears the following parameters have to be introduced as negative values:
number of teeth z ,gear ratio u, centre distance a and all diameters
�
E2
� �
E1
z
2 u
X ��05, 1 �afu�1� (22)
BE
02, 5
z
���
ch
E1 E2
2 2
� ��05, dd� (23)
E1 a1 b1
� ��a sin���� (24)
E2 t E1
6.1.11 Approach factor X
Q
The approach factor X takes into account impact loads at the ingoing mesh (at tooth tip of driven gear) in areas of
Q
high sliding. It is represented by a function of the quotient of the approach contact ratio � over the recess contact
f
ratio � , see Figure 3.
a
�
f
X =1,00 for u15, (25)
Q
�
a
4 � �
f f
X��14, 0 � for 15,�� 3 (26)
Q
15 � �
a a
�
f
X =0,60 for 3u (27)
Q
�
a
���
U
f 2
when the pinion drives the wheel (28)
V
���
a 1
W
���
U
f 1
when the pinion is driven by the wheel (29)
V
���
a 2
W
14 © ISO 2000 – All rights reserved

L 2 O
F I
z d
a1
1 M P
��=.t��1an � (30)
1 t
G J
M P
2� H d K
b1
M P
N Q
L 2 O
z F I
d
2 a2
M P
��=.t��1an � (31)
2 t
G J
M P
2� H d K
b2
M P
N Q
When tooth tips are chamfered or rounded, the tip diameter d has to be substituted by the effective tip diameter
a
d at which the recess is starting.
Na
Figure 3 — Approach factor X
Q
6.1.12 Tip relief factor X
Ca
Elastic deformations of loaded teeth may cause high impact loads at tooth tips in areas of relatively high sliding.
The tip relief factor X takes account of the influences of profile modifications on such loads. X is a relative tip
Ca Ca
relief factor which depends on the actual amount of tip relief C related to the effective tip relief due to elastic
a
deformation C , see Figure 4.
eff
The curves in Figure 4 can be approximated by the equation
L O L O
F C I F C I
...

SLOVENSKI STANDARD
01-julij-2002
,]UDþXQQRVLOQRVWLJOHGHQDWRSORWQRUD]MHGDQMH]REQLKERNRYYDOMDVWLKVWRåþDVWLK
LQKLSRLGQLK]REQLNRYGHO0HWRGDSRYSUHþQHWHPSHUDWXUH
Calculation of scuffing load capacity of cylindrical, bevel and hypoid gears -- Part 2:
Integral temperature method
Calcul de la capacité de charge au grippage des engrenages cylindriques, coniques et
hypoïdes -- Partie 2: Méthode de la température intégrale
Ta slovenski standard je istoveten z: ISO/TR 13989-2:2000
ICS:
21.200 Gonila Gears
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

TECHNICAL ISO/TR
REPORT 13989-2
First edition
2000-03-15
Calculation of scuffing load capacity of
cylindrical, bevel and hypoid gears —
Part 2:
Integral temperature method
Calcul de la capacité de charge au grippage des engrenages cylindriques,
coniques et hypoïdes —
Partie 2: Méthode de la température intégrale
Reference number
©
ISO 2000
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ii © ISO 2000 – All rights reserved

Contents Page
Foreword.v
Introduction.vi
1 Scope .1
2 Normative references .1
3 Terms, definitions, symbols and units .1
3.1 Terms and definitions .1
3.2 Symbols and units.1
4 Field of application .6
4.1 Scuffing damage.6
4.2 Integral temperature criterion.7
5 Influence factors .7
5.1 Mean coefficient of friction� .7
mC
5.2 Run-in factor X .10
E
5.3 Thermal flash factor X .10
M
5.4 Pressure angle factor X .11
��
6 Calculation.12
6.1 Cylindrical gears.12
6.1.1 Scuffing safety factor S .12
intS
6.1.2 Permissible integral temperature� .12
intP
6.1.3 Integral temperature� .12
int
6.1.4 Flash temperature at pinion tooth tip� .13
flaE
6.1.5 Bulk temperature� .13
M
6.1.6 Mean coefficient of friction� .14
mC
6.1.7 Run-in factor X .14
E
6.1.8 Thermal flash factor X .14
M
6.1.9 Pressure angle factor X .14
��
6.1.10 Geometry factor at tip of pinion X .14
BE
6.1.11 Approach factor X .14
Q
6.1.12 Tip relief factor X .15
Ca
6.1.13 Contact ratio factor: X .16
ε
6.2 Bevel gears.19
6.2.1 Scuffing safety factor S .20
intS
6.2.2 Permissible integral temperature� .20
intP
6.2.3 Integral temperature� .20
int
6.2.4 Flash temperature at pinion tooth tip� .20
flaE
6.2.5 Bulk temperature� .20
M
6.2.6 Mean coefficient of friction� .20
mC
6.2.7 Run-in factor X .21
E
6.2.8 Thermal flash factor X .21
M
6.2.9 Pressure angle factor X .21
��
6.2.10 Geometry factor at tip of pinion X .21
BE
6.2.11 Approach factor X .21
Q
6.2.12 Tip relief factor X .21
Ca
6.2.13 Contact ratio factor X .22
ε
6.3 Hypoid gears .22
6.3.1 Scuffing safety factor S .22
intS
6.3.2 Permissible integral temperature� .22
intP
6.3.3 Integral temperature� .22
int
6.3.4 Bulk temperature� .22
M
6.3.5 Mean coefficient of friction� .23
mC
6.3.6 Run-in factor X .23
E
6.3.7 Geometry factor X .23
G
6.3.8 Approach factor X .24
Q
6.3.9 Tip relief factor X .25
Ca
6.3.10 Contact ratio factor X .25
ε
6.3.11 Calculation of virtual crossed axes helical gears .25
6.4 Scuffing integral temperature.29
6.4.1 Scuffing integral temperature� .29
intS
6.4.2 Relative welding factor X .33
WrelT
Annex A (informative) Examples.34
Annex B (informative) Contact-time-dependent scuffing temperature.44
Bibliography .48
iv © ISO 2000 – All rights reserved

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.
The main task of technical committees is to prepare International Standards, but in exceptional circumstances a
technical committee may propose the publication of a Technical Report of one of the following types:
� type 1, when the required support cannot be obtained for the publication of an International Standard, despite
repeated efforts;
� type 2, when the subject is still under technical development or where for any other reason there is the future
but not immediate possibility of an agreement on an International Standard;
� type 3, 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).
Technical Reports of types 1 and 2 are subject to review within three years of publication, to decide whether they
can be transformed into International Standards. Technical Reports of type 3 do not necessarily have to be
reviewed until the data they provide are considered to be no longer valid or useful.
Technical Reports are drafted in accordance with the rules given in the ISO/IEC Directives, Part 3.
Attention is drawn to the possibility that some of the elements of this part of ISO/TR 13989 may be the subject of
patent rights. ISO shall not be held responsible for identifying any or all such patent rights.
ISO/TR 13989-2, which is a Technical Report of type 2, was prepared by Technical Committee ISO/TC 60, Gears,
Subcommittee SC 2, Gear capacity calculation.
This document is being issued in the Technical Report (type 2) series of publications (according to
subclause G.3.2.2 of Part 1 of the ISO/IEC Directives, 1995) as a “prospective standard for provisional application”
in the field of scuffing load capacity of gears because there is an urgent need for guidance on how standards in this
field should be used to meet an identified need. In 1975, two methods to evaluate the risk of scuffing were
documented to be studied by ISO/TC 60. It was agreed that after a period of experience one method shall be selected.
Since the subject is still under technical development and there is a future possibility of an agreement on an
International Standard, the publication of a type 2 Technical Report was proposed.
This document is not to be regarded as an “International Standard”. It is proposed for provisional application so that
information and experience of its use in practice may be gathered. Comments on the content of this document
should be sent to the ISO Central Secretariat.
A review of this Technical Report (type 2) will be carried out not later than three years after its publication with the
options of: extension for another three years; conversion into an International Standard; or withdrawal.
ISO/TR 13989 consists of the following parts, under the general title Calculation of scuffing load capacity of
cylindrical, bevel and hypoid gears:
� Part 1: Flash temperature method
� Part 2: Integral temperature method
Annexes A and B of this part of ISO/TR 13989 are for information only.
Introduction
This part of ISO/TR 13989 describes the surface damage "warm scuffing" for cylindrical (spur and helical), bevel
and hypoid gears for generally used gear materials and different heat treatments. "Warm scuffing" is characterized
by typical scuffing and scoring marks, which can lead to increasing power loss, dynamic load, noise and wear. For
"cold scuffing", in general associated with low temperature and low speed, under approximately 4 m/s, and
through-hardened, heavily loaded gears, the equations are not suitable.
There is a particularly severe form of gear tooth surface damage in which seizure or welding together of areas of
tooth surfaces occurs, due to absence or breakdown of a lubricant film between the contacting tooth flanks of
mating gears, caused by high temperature and high pressure. This form of damage is termed "scuffing" and most
relevant when surface velocities are high. Scuffing may also occur for relatively low sliding velocities when tooth
surface pressures are high enough, either generally or, because of uneven surface geometry and loading, in
discrete areas.
Risk of scuffing damage varies with the properties of gear materials, the lubricant used, the surface roughness of
tooth flanks, the sliding velocities and the load. Excessive aeration or the presence of contaminants in the lubricant
such as metal particles in suspension, also increase the risk of scuffing damage. Consequences of the scuffing of
high speed gears include a tendency to high levels of dynamic loading due to increase of vibration, which usually
leads to further damage by scuffing, pitting or tooth breakage.
High surface temperatures due to high surface pressures and sliding velocities can initiate the breakdown of
lubricant films. On the basis of this hypothesis two approaches to relate temperature to lubricant film breakdown
are presented:
� the flash temperature method (presented in ISO/TR 13989-1), based on contact temperatures which vary
along the path of contact;
� the integral temperature method (presented in this part of ISO/TR 13989), based on the weighted average of
the contact temperatures along the path of contact.
The integral temperature method is based on the assumption that scuffing is likely to occur when the mean value of
the contact temperature (integral temperature) is equal to or exceeds a corresponding critical value. The risk of
scuffing of an actual gear unit can be predicted by comparing the integral temperature with the critical value,
derived from a gear test for scuffing resistance of lubricants. The calculation method takes account of all significant
influence parameters, i.e. the lubricant (mineral oil with and without EP-additives, synthetic oils), the surface
roughness, the sliding velocities, the load, etc.
In order to ensure that all types of scuffing and comparable forms of surface damage due to the complex
relationships between hydrodynamical, thermodynamical and chemical phenomena are dealt with, further methods
of assessment may be necessary. The development of such methods is the objective of ongoing research.
vi © ISO 2000 – All rights reserved

TECHNICAL REPORT ISO/TR 13989-2:2000(E)
Calculation of scuffing load capacity of cylindrical, bevel and
hypoid gears —
Part 2:
Integral temperature method
1 Scope
This part of ISO/TR 13989 specifies the integral temperature method for calculating the scuffing load capacity of
cylindrical, bevel and hypoid gears.
2 Normative references
The following normative documents contain provisions which, through reference in this text, constitute provisions of
this part of ISO/TR 13989. For dated references, subsequent amendments to, or revisions of, any of these
publications do not apply. However, parties to agreements based on this part of ISO/TR 13989 are encouraged to
investigate the possibility of applying the most recent editions of the normative documents indicated below. For
undated references, the latest edition of the normative document referred to applies. Members of ISO and IEC
maintain registers of currently valid International Standards.
ISO 53:1998, Cylindrical gears for general and heavy engineering — Standard basic rack tooth profile.
ISO 1122-1:1998, Vocabulary of gear terms — Part 1: Definitions related to geometry.
ISO 1328-1:1995, Cylindrical gears — ISO system of accuracy — Part 1: Definitions and allowable values of
deviations relevant to corresponding flanks of gear teeth.
ISO 6336-1:1996, Calculation of load capacity of spur and helical gears — Part 1: Basic principles, introduction and
general influence factors.
1)
ISO 10300-1:— , Calculation of load capacity of bevel gears — Part 1: Introduction and general influence factors.
3 Terms, definitions, symbols and units
3.1 Terms and definitions
For the purposes of this part of ISO/TR 13989, the terms and definitions given in ISO 1122-1 apply.
3.2 Symbols and units
The symbols used in this part of ISO/TR 13989 are given in Table 1.
1) To be published.
Table 1 — Symbols and units
Symbol Description Unit Reference
a centre distance mm —
a
virtual centre distance of virtual cylindrical gear mm ISO 10300-1
v
b face width, smaller value of pinion or wheel mm —
b
effective facewidth for scuffing mm Eq. (46)
eB
c specific heat capacity per unit volume N/(mm ·K) —
v
single stiffness N/(mm·µm) ISO 6336-1
c�
c mesh stiffness N/(mm·µm) ISO 6336-1
�
d referencecirclediameter mm —
d effective tip diameter mm —
Na
d tip diameter mm Eq. (69)
a
d base diameter mm Eq. (70)
b
d
diameter at mid-facewidth mm —
m
d reference circle of virtual crossed axes helical gear mm Eq. (68)
s
d reference diameter of virtual cylindrical gear mm ISO 10300-1
v
d tip diameter of virtual cylindrical gear mm ISO 10300-1
va
d base diameter of virtual cylindrical gear mm ISO 10300-1
vb
g recess path of contact of pinion, wheel mm Eqs. (90), (91)
an1,2
g approach path of contact of pinion, wheel mm Eqs. (90), (91)
fn1,2
g* sliding factor — Eq. (62)
h addendum at mid-facewidth of hypoid gear mm —
am
m module mm —
m normal module of hypoid gear at mid-facewidth mm —
mn
m normal module of virtual crossed axes helical gear mm Eq. (73)
sn
n number of meshing gears ——
p
p normal base pitch mm Eq. (74)
en
u gear ratio ——
u gear ratio of virtual cylindrical gear — ISO 10300-1
v
v reference line velocity m/s —
v tangential velocity of pinion, wheel of hypoid gear m/s Eqs. (77), (78)
t1,2
v
maximum sliding velocity at tip of pinion m/s Eq. (83)
g�1
v sliding velocity at pitch point m/s Eq. (82)
gs
v sliding velocity m/s Eqs. (84), (85)
g1,2
v sliding velocity m/s Eq. (87)
g�1
2 © ISO 2000 – All rights reserved

Table 1 (continued)
Symbol Description Unit Reference
v sliding velocity m/s Eq. (88)
g�1
v tangential speed at reference cone at mid-facewidth of m/s —
mt
bevel gear
v sums of tangential speeds at pitch point m/s Eqs. (2), (47), (81)
ΣC
tangential speed m/s Eq. (79)
v
Σs
v tangential speed m/s Eq. (80)
Σh
w specific tooth load, scuffing N/mm Eq. (4)
Bt
z number of teeth ——
z number of teeth of virtual cylindrical gear — ISO 10300-1
v
1/2
B thermal contact coefficient N/(mm·s ·K) Eq. (12)
M
C ,C ,C weighting factors ——
1 2 2H
C nominal tip reliefµm —
a
C
effective tip reliefµm Eqs. (37), (38), (49)
eff
E module of elasticity (Young's modulus) N/mm —
F
nominal tangential load at reference cone at mid-facewidth N —
mt
F normal tooth load N Eq. (51)
n
F nominal tangential load at reference circle N —
t
K application factor — ISO 6336-1,
A
ISO 10300-1
K dynamic factor — ISO 6336-1,
v
ISO 10300-1
K = K transverse load factor (scuffing) — 6.2.4, ISO 6336-1,
B� H�
ISO 10300-1
K = K face load factor (scuffing) — ISO 6336-1
B� H�
ISO 10300-1, 6.2.4,
Eqs. (52), (53)
K helical load factor (scuffing) — Eq. (5), 6.2.4, 6.3.5
B�
K bearing factor — 6.3.3
B�be
K transverse load factor — ISO 6336-1,
H�
ISO 10300-1
K face load factor — ISO 6336-1,
H�
ISO 10300-1
K bearing factor — ISO 10300-1
H�be
L contact parameter — Eq. (55)
Ra
arithmetic mean roughnessµm Eq. (6)
S scuffing safety factor — Eq. (14)
intS
S minimum required scuffing safety factor ——
Smin
Table 1 (continued)
Symbol Description Unit Reference
T torque of the pinion Nm —
T scuffing torque of test pinion Nm Eq. (96)
1T
X geometry factor at pinion tooth tip — Eq. (22)
BE
X
run-in factor — Eq. (8)
E
X tip relief factor — Eq. (32)
Ca
X geometry factor of hypoid gears — Eq. (54)
G
X lubricant factor — 5.1
L
X thermal flash factor — Eq. (9)
M
X approach factor — Eqs. (25), (26), (27)
Q
X roughness factor — Eq. (7)
R
X lubrication factor — 6.1.5.3
S
X welding factor of executed gear — Table 3
W
X welding factor of test gear — 6.4.2
WT
X relative welding factor — Eq. (102)
WrelT
X contact factor — Eq. (21)
mp
X pressure angle factor — Eqs. (13), (48)
��
X contact ratio factor — Eqs. (39) to (44)
ε
pressure angle °—
�
normal pressure angle at mid-facewidth of hypoid gear °—
�
mn
� normal pressure angle °—
n
normal pressure angle of crossed axes helical gear ° Eq. (64)
�
sn
� transverse pressure angle of crossed axes helical gear ° Eq. (66)
st
transverse pressure angle °—
�
t
� ´ transverse working pressure angle °—
t
transverse pressure angle of virtual cylindrical gear ° ISO 10300-1
�
vt
arbitrary angle ° Figure 2
�
y
helix angle °—
�
helix angle at base circle ° Eqs. (67), (71)
�
b
� helix angle at reference cone at mid-facewidth of hypoid °—
m
gear
� helix angle of virtual crossed axes helical gear ° Eq. (63)
s
auxiliary angle ° Eq. (86)
�
reference cone angle °—
�
4 © ISO 2000 – All rights reserved

Table 1 (continued)
Symbol Description Unit Reference
recess contact ratio — Eqs. (28), (29)
�
a
� approach contact ratio — Eqs. (28), (29)
f
contact ratio in normal section of virtual crossed axes — Eqs. (92), (93)
�
n
helical gear
addendum contact ratio of the pinion — Eq. (30)
�
� addendum contact ratio of the wheel — Eq. (31)
contact ratio — Eq. (45)
�
�
transverse contact ratio of virtual cylindrical gear — ISO 10300-1
�
v�
tip contact ratio of virtual cylindrical pinion — ISO 10300-1
�
v1
tip contact ratio of virtual cylindrical wheel — ISO 10300-1
�
v2
Hertzian auxiliary coefficient — Figure 7, Eqs. (57), (59)
�
mean coefficient of friction — Eqs. (1), (1a)
�
mC
� dynamic viscosity at oil temperature —
mPa�s
oil
heat conductivity —
� N/(s�K)
M
Poisson's ratio ——
�
� kinematic viscosity of the oil at 40 �C mm /s; cSt —
radius of curvature at tip of the pinion, wheel mm Eqs. (23), (24)
�
E1,2
� relative radius of curvature at pitch point in normal section mm Eq. (76)
Cn
radius of curvature at pitch point in normal section mm Eq. (75)
�
n1,2
relative radius of curvature at pitch point mm Eq. (3)
�
redC
Hertzian auxiliary coefficient — Figure 7, Eqs. (58), (60)
�
Hertzian auxiliary angle ° Eqs. (56) to (60)
�
flash temperature at pinion tooth tip when load sharing is K Eq. (19)
�
flaE
neglected
mean flash temperature K Eq. (18)
�
flaint
� mean flash temperature of hypoid gear K Eq. (50)
flainth
integral temperature K Eq. (17)
�
int
permissible integral temperature K Eq. (16)
�
intP
scuffing integral temperature (allowable integral K Eq. (94)
�
intS
temperature)
� mean flash temperature of the test gear K Eqs. (96), (99), (101)
flaintT
oil sump or spray temperature °C —
�
oil
� bulk temperature °C Eq. (20)
M-C
Table 1 (concluded)
Symbol Description Unit Reference
test bulk temperature Eqs. (95), (98), (100)
� �C
MT
axle angle of virtual crossed axes helical gear ° Eq. (72)
�
� axle angle of virtual crossed axes helical gear ° Eq. (65)
run-in grade — 5.2
�
E
parameter on the line of action — Eq. (10)
�
Subscripts:
1pinion
2 wheel
a tip diameter of the virtual gear
b base circle of the virtual gear
m mid-facewidth of bevel or hypoid gears
n normal section
s virtual crossed axes helical gear
t tangential direction
T test gear
4 Field of application
The calculation methods are based on results of the rig testing of gears run at pitch line velocities less than 80 m/s.
The equations can be used for gears which run at higher speeds, but with increasing uncertainty as speed
increases. The uncertainty concerns the estimation of bulk temperature, coefficient of friction, allowable
temperatures, etc. as speeds exceed the range with experimental background.
4.1 Scuffing damage
When once initiated, scuffing damage can lead to gross degradation of tooth flank surfaces, with increase of: power
loss, dynamic loading, noise and wear. It can also lead to tooth breakage if the severity of the operating conditions
is not reduced. In the event of scuffing due to an instantaneous overload, followed immediately by a reduction of
load, e.g. by load redistribution, the tooth flanks may self-heal by smoothing themselves to some extent. Even so,
the residual damage will continue to be a cause of increased power loss, dynamic loading and noise.
In most cases, the resistance of gears to scuffing can be improved by using a lubricant with enhanced E.P.
(extreme pressure) properties. It is important however, to be aware that some disadvantages attend the use of E.P.
oils — corrosion of copper, embrittlement of elastomers, lack of world-wide availability, etc. These disadvantages
are to be taken into consideration if optimum lubricant choice is to be made, which means: as few additives as
possible, but as many as necessary.
Due to continuous variation of different parameters, the complexity of the chemical properties and the thermo-
hydro-elastic processes in the instantaneous contact area, some scatter in the calculated assessments of
probability of scuffing risk is to be expected.
In contrast to the relatively long time of development of fatigue damage, one single momentary overload can initiate
scuffing damage of such severity that affected gears may no longer be used. This should be carefully considered
when choosing an adequate safety factor for gears, especially for gears required to operate at high circumferential
velocities.
6 © ISO 2000 – All rights reserved

4.2 Integral temperature criterion
This approach to the evaluation of the probability of scuffing is based on the assumption that scuffing is likely to
occur when the mean value of the contact temperatures along the path of contact is equal to or exceeds a
corresponding "critical value" . In the method presented herein, the sum of the bulk temperature and the weighted
mean of the integrated values of flash temperatures along the path of contact is the "integral temperature". The
bulk temperature is estimated as described under 6.1.5 and the mean value of the flash temperature is
approximated by substituting mean values of the coefficient of friction, the dynamic loading, etc., along the path of
contact. A weighting factor is introduced accounting for possible different influences of a real bulk temperature
value and a mathematically integrated mean flash temperature value on the scuffing phenomenon.
The probability of scuffing is assessed by comparing the integral temperature with a corresponding critical value
derived from the gear testing of lubricants for scuffing resistance (e.g. different FZG test procedures, the IAE and
the Ryder gear tests), or from gears which have scuffed in service.
5 Influence factors
5.1 Mean coefficient of friction�
mC
The actual coefficient of friction between the tooth flanks is an instantaneous and local value which depends on
several properties of the oil, surface roughness, lay of the surface irregularities such as those left by machining,
properties of the tooth flank materials, tangential velocities, forces at the surfaces and the dimensions. Assessment
of the instantaneous coefficient of friction is difficult since there is no method currently available for its
measurement.
The mean value for the coefficient of friction� along the path of contact was derived from measurements [1] and
mC
approximated by Equation (1). Although the local coefficient of friction is near to zero in the pitch point C, the mean
value can be approximated with the parameters at the pitch point and the oil viscosity � at oil temperature �
oil oil
when introduced into Equation (1).
02,
wK�
F I
Bt B�
�00, 5
2)
� = 0,045 � ��� XX� (1)
mC G J oil R L
���
H K
�C redC
The coefficient of friction of the integral temperature method takes account of the size of the gear in a different way
as the coefficient of friction of the flash temperature method. Equation (1) for calculating the coefficient of friction
should not be applied outside the field of the part where it is presented, e.g. coefficient of friction for thermal rating.
The equation for the calculation of � was derived from experiments in the following range of operating
mC
conditions. Extrapolation may lead to deviations between the calculated and the real coefficient of friction.
1m/su vu 50 m/s
At reference line velocities v lower than 1 m/s, higher coefficients of friction are expected. At reference line
velocities v higher than 50 m/s, the limiting value of v at v = 50 m/s has to be used in equation (1).
ΣC
w W 150 N/mm
Bt
For lower values of the specific normal tooth load w , the limiting value w = 150 N/mm has to be used in
Bt Bt
Equation (1).
v =2 · v · tan� ´ · cos� (2)
�C t t
�
u sin�
t
� � ��a (3)
redC
cos�
(1� u)
b
F
t
w = K · K · K · K · (4)
Bt A v B� B�
b
2� This formula for the coefficient of friction is derived from testing of gears with centre distance a � 100 mm.
0,2
��Fb
bt �0,05 0,25
��= 0,048��� Ra� X (1a)
mC oil L
��
�����
�C redC
where
0,2
��6
X��0,75 for polyglycols;
L
��
��v
�C
X = 1,0 for mineral oils;
L
X = 0,8 for polyalfaolefins;
L
X = 1,5 for traction fluids;
L
X = 1,3 for phosphate esters.
L
Equation (1a) represents results of tests within a range of a = 91,5 mm to 200 mm. The application of this equation makes it
necessary to adjust Figures 9, 10 and 11 for the scuffing temperature� accordingly.
intS
8 © ISO 2000 – All rights reserved

K is the helical load factor, scuffing takes account of increasing friction for increasing total contact ratio (see
B�
Figure 1).
Figure 1 — Helical load factor K
B�
K =1 for � u 2
B� �
K ��10,2����25� � for 2 �� � 3,5 (5)
didi
B�� � �
K =1,3 for� W 3,5
B� �
Ra =0,5 · (Ra + Ra ) (6)
1 2
Ra , Ra are the tooth flank roughness values of pinion and wheel measured on the new flanks as manufactured
1 2
(e.g. reference test gear Ra values are � 0,35 �m).
0,25
X =2,2 · (Ra/� ) (7)
R redC
where
X = 1,0 for mineral oils;
L
X = 0,8 for polyalfaolefins;
L
X = 0,7 for non water-soluble polyglycols;
L
X = 0,6 for water-soluble polyglycols;
L
X = 1,5 for traction fluids;
L
X = 1,3 for phosphate esters.
L
5.2 Run-in factor X
E
The present calculation methods presume that the gears are well run-in. In practice scuffing failure occurs very
often during the first few hours in service, e.g. in a full load test run, the acceptance run of vessels or when a new
set of gears is built into a production machinery when the gears are run under full load conditions before a proper
run-in. Investigations [1] show a 1/4 to 1/3 load carrying capacity of a newly manufactured gear flank as compared
to a properly run-in flank. This should be taken into account by a run-in factor X :
E
30� Ra
X ��11�� � (8)
bg
EE
�
redC
where
� = 1, full run-in (for carburized and ground gears full run-in can be assumed if Ra � 0,6 Ra );
E run-in new
� = 0, newly manufactured.
E
5.3 Thermal flash factor X
M
The thermal flash factor X accounts for the influence of the properties of pinion and gear materials on the flash
M
temperature.
Calculation of the thermal flash factor for an arbitrary point (index y) on the line of action (see Figure 2):
0,25
��
��
�
��11��� �
�� ��
��u
��
X�� (9)
M
��
��
11��vv �
��
� BB��11�� � �
�� ��
M1 M2
��
EE u
��
��12
a
Tipcircle1
b
Tipcircle2
Figure 2 — Parameter� on the line of action
10 © ISO 2000 – All rights reserved

tan�
y
� � � 1 (10)
y
tan��
t
If the materials of pinion and wheel are the same Equation (9) can be simplified to:
0,25
E
X � (11)
M
20,25
(1��vB)
M
In the above equations the thermal contact coefficient B is:
M
Bc()� � (12)
M� Mv
For case hardened steels with the following typical characteristic values:
2 2
� =50N/(s�K), c =3,8N/(mm �K), E = 206 000 N/mm and v=0,3
M v
follows
–0,75 0,5 –0,5
X = 50,0 K�N �s �m �mm
M
For the characteristic values of other materials, see [7].
5.4 Pressure angle factor X
��
The pressure angle factor X is used to account for the conversion of load and tangential speed from reference
��
circle to pitch circle.
Method A: Factor X
��-A
02,,5 025 02, 5
(sin ��cos��cos )
��
n
t
X ��12, 2 (13)
�
�� A
05,,05
(cos���� cos )
t t
Table 2 shows the values for the pressure angle factor X for a standard rack with pressure angle � =20�,the
�� n
typical range of standard working pressure angles a� and helix angles�.
t
Table 2 — Method B: Factor X
��-B
a�
� =0� � =10� � =20� � =30�
t
0,963 0,960 0,951 0,938
19�
20� 0,978 0,975 0,966 0,952
0,992 0,989 0,981 0,966
21�
1,007 1,004 0,995 0,981
22�
1,021 1,018 1,009 0,995
23�
24� 1,035 1,032 1,023 1,008
1,049 1,046 1,037 1,012
25�
As an approximation, for gears with normal pressure angle � =20�, the pressure angle factor can be
n
approximated as follows:
X =1
��-B
6 Calculation
6.1 Cylindrical gears
This part of ISO/TR 13989 contains equations which enable the assessment of the "probability of scuffing" (warm
scuffing) of oil lubricated, involute spur and helical gears.
It is assumed that the total tangential load is equally distributed between the two helices of double helical gears.
When, due to application of forces such as external axial forces, this is not the case, the influences of these are to
be taken into account separately. The two helices are to be treated as parallel single helical gears. Influences
affecting scuffing probability, for which quantitative assessments can be made, are included.
The equations are valid for gears with external or internal teeth which are conjugate to a basic rack as defined in
ISO 53. For internal gears negative values have to be introduced for the determination of the geometry factor X
BE
as presented in 6.1.10. They may also be considered as valid for similar gears of other basic rack form, of which
the transverse contact ratio is � u 2,5.
�
6.1.1 Scuffing safety factor S
intS
As uncertainties and inaccuracies in the assumptions cannot be excluded, it is necessary to introduce a safety
factor S . It must be pointed out that the scuffing safety factor is temperature related and is not a factor by which
intS
gear torque may be multiplied to arrive at same values for the integral temperature number � and the scuffing
int
integral temperature number� .
intS
�
intS
SS� W (14)
intS Smin
�
int
Recommendation for choosing S :
Smin
S � 1 High scuffing risk
Smin
1u S u 2 Critical range with moderate scuffing risk, influenced by the operating conditions of the actual
Smin
gear. Influencing factors are e.g. the tooth flank roughness, run-in effects, the accurate
knowledge of the load factors, the load capacity of lubricating oil, etc.
S � 2 Low scuffing risk
Smin
Given the relationship between the actual load and the integral temperature number, the corresponding load safety
factor S can be approximated by:
Sl
w � ��
Btmax intS oil
S�� (15)
Sl
w ���
Bteff int oil
6.1.2 Permissible integral temperature�
intP
�
intS
� = (16)
intP
S
Smin
The minimum required scuffing safety factor S is to be separately determined for each application.
Smin
6.1.3 Integral temperature�
int
� =� + C �� u� (17)
int M 2 flaint intP
where C is the weighting factor derived from experiments. For spur and helical gears C =1,5.
2 2
12 © ISO 2000 – All rights reserved

� =� � X (18)
flaint flaE �
6.1.4 Flash temperature at pinion tooth tip�
flaE
07,,5 05
()Kw��v
X
BB� t
E
������� ��� � (19)
flaE MBE
mC ��
02, 5
XX�
a QCa
6.1.5 Bulk temperature�
M
The bulk temperature is the temperature of the tooth surfaces immediately before they come into contact.
The bulk temperature is established by the thermal balance of the gear unit. There are several sources of heat in a
gear unit of which the most important are tooth and bearing friction. Other sources of heat such as seals and oil
flow contribute to some extent. At pitch line velocities in excess of 80 m/s, heat from the churning of oil in the mesh
and windage losses may become significant and should be taken into consideration (see Method A). The heat is
transferred to the environment via the housing walls by conduction, convection and radiation and for spray
lubrication conditions through the oil into an external heat exchanger.
Values obtained using the different calculation methods described below are to be distinguished by the subscripts
A, B, C.
6.1.5.1 Method A�
M-A
The bulk temperature as a mean value or as temperature distribution over the facewidth can be measured
experimentally or be determined by a theoretical analysis based on known power loss and heat transfer data, i.e.
by using thermal network methods.
6.1.5.2 Method B�
M-B
This method is not used for the integral temperature method (see the flash temperature method given in
ISO/TR 13989-1).
6.1.5.3 Method C�
M-C
An approximate value for the bulk temperature consists of the sum of the oil temperature and a part of a mean
value derived from the flash temperature over the path of contact according to method C.
� ��� CX� �� �X (20)
�
MC oil 1 mp flaint S
where
X = 1,2 for spray lubrication;
S
X = 1,0 for dip lubrication;
S
X = 0,2 for gears submerged in oil;
S
C is the constant accounting for heat transfer conditions, from test results C =0,7;
1 1
1+ n
p
X = (21)
mp
where n is the number of meshing gears.
p
6.1.6 Mean coefficient of friction�
mC
See 5.1.
6.1.7 Run-in factor X
E
See 5.2.
6.1.8 Thermal flash factor X
M
See 5.3.
6.1.9 Pressure angle factor X
��
See 5.4.
6.1.10 Geometry factor at tip of pinion X
BE
The geometry factor X takes into account Hertzian stress and sliding velocity at the pinion tooth tip. X is a
BE BE
function of the gear ratio u and the radius of curvature� at the pinion tooth tip E.
E
For internal gears the following parameters have to be introduced as negative values:
number of teeth z ,gear ratio u, centre distance a and all diameters
�
E2
� �
E1
z
2 u
X ��05, 1 �afu�1� (22)
BE
02, 5
z
���
ch
E1 E2
2 2
� ��05, dd� (23)
E1 a1 b1
� ��a sin���� (24)
E2 t E1
6.1.11 Approach factor X
Q
The approach factor X takes into account impact loads at the ingoing mesh (at tooth
...

RAPPORT ISO/TR
TECHNIQUE 13989-2
Première édition
2000-03-15
Calcul de la capacité de charge au grippage
des engrenages cylindriques, coniques et
hypoïdes —
Partie 2:
Méthode de la température intégrale
Calculation of scuffing load capacity of cylindrical, bevel and hypoid
gears —
Part 2: Integral temperature method
Numéro de référence
©
ISO 2000
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ii © ISO 2000 – Tous droits réservés

Sommaire Page
Avant-propos.v
Introduction.vii
1 Domaine d’application.1
2Références normatives .1
3 Termes, définitions, symboles et unités .1
3.1 Termes et définitions.1
3.2 Symboles et unités .2
4 Domaine d'application.6
4.1 Détérioration par grippage.6
4.2 Critère de la température intégrale .7
5 Facteurs d'influence .7
5.1 Coefficient de frottement moyen� .7
mC
5.2 Facteur de rodage X .10
E
5.3 Facteur thermique éclair X .10
M
5.4 Facteur d'angle de pression X .11
��
6 Calcul .12
6.1 Engrenages cylindriques .12
6.1.1 Coefficient de sécurité au grippage S .12
intS
6.1.2 Température intégrale admissible� .12
intP
6.1.3 Température intégrale� .13
int
6.1.4 Température-éclair en tête de dent du pignon� .13
flaE
6.1.5 Température de masse� .13
M
6.1.6 Coefficient moyen de frottement� .14
mC
6.1.7 Facteur de rodage X .14
E
6.1.8 Facteur thermique éclair X .14
M
6.1.9 Facteur d'angle de pression X .14
��
6.1.10 Facteur géométrique en tête du pignon X .14
BE
6.1.11 Facteur d'approche X .14
Q
6.1.12 Facteur de dépouille de tête X .15
Ca
6.1.13 Facteur de rapport de conduite X .17
�
6.2 Engrenages coniques.20
6.2.1 Coefficient de sécurité au grippage S .20
intS
6.2.2 Température intégrale admissible� .20
intP
6.2.3 Température intégrale� .20
int
6.2.4 Température-éclair en tête de dent du pignon� .20
flaE
6.2.5 Température de masse� .20
M
6.2.6 Coefficient de frottement moyen� .21
mC
6.2.7 Facteur de rodage X .21
E
6.2.8 Facteur thermique éclair X .21
M
6.2.9 Facteur d'angle de pression X .21
��
6.2.10 Facteur géométrique en tête du pignon X .21
BE
6.2.11 Facteur d'approche X .21
Q
6.2.12 Facteur de dépouille de tête X .22
Ca
6.2.13 Facteur de rapport de conduite X .22
�
6.3 Engrenages hypoïdes.22
6.3.1 Coefficient de sécurité au grippage S .22
intS
6.3.2 Température intégrale admissible� .22
intP
6.3.3 Température intégrale� .23
int
6.3.4 Température de masse� .23
M
6.3.5 Coefficient de frottement moyen� .23
mC
6.3.6 Facteur de rodage X .23
E
6.3.7 Facteur géométrique X .23
G
6.3.8 Facteur d'approche X .25
Q
6.3.9 Facteur de dépouille de tête X .25
Ca
6.3.10 Facteur de rapport de conduite X .25
�
6.3.11 Calcul des engrenages gauches hélicoïdaux équivalents .25
6.4 Température intégrale de grippage.29
6.4.1 Température intégrale de grippage� .29
intS
6.4.2 Facteur relatif de soudure X .33
WrelT
Annexe A (informative) Exemples.34
Annexe B (informative) Température de grippage en fonction de la durée de contact.44
Bibliographie .48
iv © ISO 2000 – Tous droits réservés

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 étudealedroit de fairepartieducomité
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.
La tâche principale des comités techniques est d'élaborer les Normes internationales. Exceptionnellement, un
comité technique peut proposer la publication d'un rapport technique de l'un des types suivants:
— type 1, lorsque, en dépit de maints efforts, l'accord requis ne peut être réalisé en faveur de la publication d'une
Norme internationale;
— type 2, lorsque le sujet en question est encore en cours de développement technique ou lorsque, pour toute
autre raison, la possibilité d'un accord pour la publication d'une Norme internationale peut être envisagée pour
l'avenir mais pas dans l'immédiat;
— type 3, 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).
Les rapports techniques des types 1 et 2 font l'objet d'un nouvel examen trois ans au plus tard après leur
publicationafindedécider éventuellement de leur transformation en Normes internationales. Les rapports
techniques de type 3 ne doivent pas nécessairement être révisés avant que les données fournies ne soient plus
jugées valables ou utiles.
Les rapports techniques sont rédigés conformément aux règles données dans les Directives ISO/CEI, Partie 3.
L’attention est appelée sur le fait que certains des éléments delaprésente partie de l’ISO TR 13989 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’Amendement au Rapport technique ISO/TR 13989-2: a étéélaboré par le comité technique ISO/TC 60,
Engrenages, sous-comité SC 2, Calcul de la capacité des engrenages.
Le présent document est publié dans la série des Rapports techniques de type 2 (conformément au paragraphe
G.3.2.2. de la partie 1 des Directives ISO/CEI, 1995) comme «norme prospective d’application provisoire» dans le
domaine de [à décrire] en raison de l’urgence d’avoir une indication quant à la manière dont il convient d’utiliser les
normes dans ce domaine pour répondre à un besoin déterminé.
Ce document ne doit pas être considéré comme une «Norme internationale». Il est proposé pour une mise en
œuvre provisoire, dans le but de recueillir des informations et d’acquérir de l’expérience quant à son application
dans la pratique. Il est de règle d’envoyer les observations éventuelles relatives au contenu de ce document au
Secrétariat central de l’ISO.
Il sera procédéà un nouvel examen de ce Rapport technique de type 2 trois ans au plus tard après sa publication,
avec la faculté d’en prolonger la validité pendant trois autres années, de le transformer en Norme internationale ou
de l’annuler.
L'ISO TR 13989 comprend les parties suivantes, présentées sous le titre général Calcul de la capacité de charge
au grippage des engrenages cylindriques, coniques et hypoïdes:
� Partie 1: Méthode de la température-éclair
� Partie 2: Méthode de la température intégrale
Les annexes A et B de la présente partie de l’ISO/TR 13989 sont données uniquement à titre d’information.
vi © ISO 2000 – Tous droits réservés

Introduction
La présente partie de l’ISO/TR 13989 décrit la détérioration de surface d'engrenages cylindriques «grippage à
chaud» (à denture droite et hélicoïdale), coniques et hypoïdes, pour les matériaux d'engrenages généralement
utilisés combinésavecdifférents traitements thermiques. Le «grippage à chaud» est caractérisé par des marques
de grippage et de griffures typiques qui peuvent donner lieu à une augmentation de la perte de puissance, de la
charge dynamique, du bruit et de l'usure. Pour le «grippage à froid»,engénéral associéà des engrenages à basse
température et faible vitesse, tournant à des vitesses inférieures à 4 m/s environ, trempés à cœur et soumis à des
charges élevées, les équations ne conviennent pas
Il s'agit là d'une forme particulièrement sévère de détérioration de la surface de la denture d'un engrenage, au cours
de laquelle un arrachement ou une soudure par fusion des surfaces en contact apparaît, due à l'absence ou à la
rupture du film de lubrifiant entre les flancs de dents en contact d'engrenages conjugués, due à des températures et
des pressions élevées.Cette formededétérioration est appelée «grippage»; elle est d'autant plus importante que
les vitesses de surface sont élevées. Le grippage peut également apparaître à de faibles vitesses de glissement
lorsque les pressions à la surface des dentures sont suffisamment élevées, soit de manière uniforme, soit dans des
zones discrètes du fait d'une géométrie et d'une distribution de charge sur les flancs inégales.
Le risque de détérioration par grippage varie selon les propriétés des matériaux des dentures, le lubrifiant utilisé,la
rugosité de surface des flancs de denture, les vitesses de glissement et la charge. Une aération excessive ou la
présence de contaminants dans le lubrifiant, tels que des particules métalliques en suspension, augmente
également le risque de détérioration par grippage. En conséquence du grippage, les engrenages à grande vitesse
peuvent subir des niveaux de charge dynamique élevés du fait de l'augmentation des vibrations qui conduisent
généralement à une détérioration accrue par grippage, formation de piqûres ou rupture de dent.
Les températures superficielles élevées, induites par des pressions de contact et des vitesses de glissement
élevées peuvent conduire à la rupture des films de lubrifiant. Sur la base de cette hypothèse, deux approches
permettant de corréler la température et la rupture du film de lubrifiant sont présentées:
� la méthode de la température-éclair (présentée dans l’ISO/TR 13989-1), basée sur les températures de contact
qui varient sur la longueur de conduite;
� la méthode de la température intégrale (présentée dans la présente partie de l’ISO/TR 13989), baséesur la
moyenne pondérée des températures de contact sur la longueur de conduite.
La méthode de la température intégrale est basée sur l'hypothèse que le grippage apparaît probablement lorsque la
valeur moyenne de la température de contact (température intégrale) est supérieure ou égale à une valeur critique
correspondante. Le risque de grippage d'une transmission par engrenages réelle peut être prédit en comparant la
température intégrale à la valeur critique, issue d'essais sur engrenages de la résistance des lubrifiants au
grippage. La méthode de calcul tient compte de tous les paramètres d'influence significatifs, c'est-à-dire le lubrifiant
(huile minérale sans ou avec additifs EP, huile synthétique), la rugosité de surface, les vitesses de glissement, la
charge, etc.
Il est admis que d'autres méthodes peuvent être nécessaires afin de s'assurer que tous les types de grippage et
formes comparables de détérioration de surface dus aux interactions complexes entre phénomènes
hydrodynamiques, thermodynamiques et chimiques, sont traités. Le développement de ces méthodes fait
actuellement l'objet de recherches poussées.
RAPPORT TECHNIQUE ISO/TR 13989-2:2000(F)
Calcul de la capacité de charge au grippage des engrenages
cylindriques, coniques et hypoïdes —
Partie 2:
Méthode de la température intégrale
1 Domaine d’application
La présente partie de l’ISO/TR 13989 spécifielaméthode de la température intégrale pour calculer la capacité de
charge au grippage des engrenages cylindriques, coniques et hypoïdes.
2Références normatives
Les documents normatifs suivants contiennent des dispositions qui, par suite de la référence qui y est faite,
constituent des dispositions valables pour la présente partie de l’ISO/TR 13989. Pour les références datées, les
amendements ultérieurs ou les révisions de ces publications ne s’appliquent pas. Toutefois, les parties prenantes
aux accords fondés sur la présente partie de l'ISO/TR 13989 sont invitées à rechercher la possibilité d'appliquer les
éditions les plus récentes des documents normatifs indiqués ci-après. Pour les références non datées, la dernière
édition du document normatif en référence s’applique. Les membres de l'ISO et de la CEI possèdent le registre des
Normes internationales en vigueur.
ISO 53:1998, Engrenages cylindriques de mécanique générale et de grosse mécanique — Tracé de référence.
ISO 1122-1:1998, Vocabulaire des engrenages — Partie 1: Définitions géométriques.
ISO 1328-1:1995, Engrenages cylindriques — Système ISO de précision — Partie 1: Définitions et valeurs
admissibles des écarts pour les flancs homologues de la denture.
ISO 6336-1:1996, Calcul de la capacité de charge des engrenages cylindriques à dentures droite et hélicoïdale —
Partie 1: Principes de base, introduction et facteurs généraux d'influence.
1)
ISO 10300-1:— , Calcul de la capacité de charge des engrenages coniques — Partie 1: Introduction et facteurs
généraux d’influence.
3 Termes, définitions, symboles et unités
3.1 Termes et définitions
Pour les besoins de la présente partie de l'ISO/TR 13989, les termes et définitions donnés dans l’ISO 1122-1
s'appliquent.
1) À publier.
3.2 Symboles et unités
Les symboles utilisés dans la présente partie de l’ISO/TR 13989 sont donnés dans le Tableau 1.
Tableau 1 — Symboles et unités
Symbole Description Unité Référence
a entraxe mm —
entraxe équivalent de l'engrenage cylindrique à denture
a
mm ISO 10300-1
v
droite équivalent
b largeur de denture, plus petite valeur du pignon ou de la roue mm —
b largeur de denture effective pour le grippage mm Éq. (46)
eB
c
capacité thermique spécifique par unité de volume N/(mm ·K) —
v
c� raideur simple N/(mm·µm) ISO 6336-1
c
raideur d'engrènement N/(mm·µm) ISO 6336-1
�
d diamètrederéférence mm —
d
diamètre actif de tête mm —
Na
d
diamètredetête mm Éq. (69)
a
d
diamètredebase mm Éq. (70)
b
d diamètre à mi-largeur de la denture mm —
m
cercle de référence d'une roue équivalente d'un
d
mm Éq. (68)
s
engrenage gauche hélicoïdal
diamètrederéférence de la roue cylindrique à denture
d
mm ISO 10300-1
v
droite équivalente
diamètredetête de la roue cylindrique à denture droite
d
mm ISO 10300-1
va
équivalente
diamètre de base de la roue cylindrique à denture droite
d
mm ISO 10300-1
vb
équivalente
g longueur de retraite du pignon, de la roue mm Éqs. (90), (91)
an1,2
g
longueur d'approche du pignon, de la roue mm Éqs. (90), (91)
fn1,2
g*
facteur de glissement —Éq. (62)
h
saillie à mi-largeur de la denture d'engrenage hypoïde mm —
am
m module mm —
module réel à mi-largeur de la denture d'engrenage
m
mm —
mn
hypoïde
m
module réel d'engrenage gauche hélicoïdal équivalent mm Éq. (73)
sn
n
nombre de roues dentées en prise ——
p
p
pas de base réel mm Éq. (74)
en
u rapport d'engrenage ——
u
rapport d'engrenage de l'engrenage cylindrique équivalent — ISO 10300-1
v
2 © ISO 2000 – Tous droits réservés

Tableau 1 — Symboles et unités (suite)
Symbole Description Unité Référence
v vitesse de la ligne de référence m/s —
vitesse tangentielle du pignon, de la roue d'un engrenage
v
m/s Éqs. (77), (78)
t1,2
hypoïde
v
vitesse de glissement maximale à la tête de pignon m/s Éq. (83)
g�1
v
vitesse de glissement au point primitif m/s Éq. (82)
gs
v
vitesse de glissement m/s Éqs. (84), (85)
g1,2
v
vitesse de glissement m/s Éq. (87)
g�1
v
vitesse de glissement m/s Éq. (88)
g�1
vitesse tangentielle au cône de référence à mi-largeur de
v m/s —
mt
la denture d'engrenage conique
v
somme des vitesses tangentielles au point primitif m/s Éqs. (2), (47), (81)
�C
v
vitesse tangentielle m/s Éq. (79)
�s
v
vitesse tangentielle m/s Éq. (80)
�h
w
charge spécifique sur les dents, grippage N/mm Éq. (4)
Bt
z nombre de dents ——
z nombre de dents de l'engrenage cylindrique équivalent — ISO 10300-1
v
1/2
B
coefficient de contact thermique N/(mm·s ·K) Éq. (12)
M
C ,C ,C
facteurs de pondération ——
1 2 2H
C
dépouille de tête nominaleµm —
a
C
dépouille de tête effectiveµm Éqs. (37), (38), (49)
eff
E module d'élasticité (module de Young) N/mm —
charge tangentielle nominale au cône de référence à mi-
F N —
mt
largeur de la denture
F
charge réelle sur les dents N Éq. (51)
n
F
charge tangentielle nominale au cercle de référence N —
t
K
facteur d'application — ISO 6336-1, ISO 10300-1
A
K
facteur dynamique — ISO 6336-1, ISO 10300-1
v
= K facteur de distribution transversale de la charge
6.2.4, ISO 6336-1,
H�
K
—
B�
ISO 10300-1
(grippage)
= K facteur de distribution longitudinale de la charge ISO 6336-1 ISO 10300-1,
H�
K
—
B�
6.2.4, Éqs. (52), (53)
(grippage)
K
facteur de charge hélicoïdale (grippage) —Éq. (5), 6.2.4, 6.3.5
B�
K
facteur de portée — 6.3.3
B�be
K
facteur de distribution transversale de la charge — ISO 6336-1, ISO 10300-1
H�
K
facteur de distribution longitudinale de la charge — ISO 6336-1, ISO 10300-1
H�
Tableau 1 — Symboles et unités (suite)
Symbole Description Unité Référence
K
facteur de portée — ISO 10300-1
H�be
L paramètre de contact —Éq. (55)
Ra rugosité moyenne arithmétiqueµm Éq. (6)
S
coefficient de sécurité au grippage —Éq. (14)
intS
S
coefficient de sécurité au grippage minimal exigé— —
Smin
T
couple sur le pignon Nm —
T
couple de grippage sur le pignon d'essai Nm Éq. (96)
1T
X
facteur géométrique en tête de dent du pignon —Éq. (22)
BE
X
facteur de rodage —Éq. (8)
E
X
facteur de dépouille de tête —Éq. (32)
Ca
X
facteur géométrique des engrenages hypoïdes —Éq. (54)
G
X
facteur lubrifiant — 5.1
L
X
facteur thermique éclair —Éq. (9)
M
X
facteur d'approche —Éqs. (25), (26), (27)
Q
X
facteur de rugosité—Éq. (7)
R
X facteur de lubrification — 6.1.5.3
S
X facteur de soudure de l'engrenage fabriqué— Tableau 3
W
X
facteur de soudure de l'engrenage d'essai — 6.4.2
WT
X
facteur relatif de soudure —Éq. (102)
WrelT
X
facteur de contact —Éq. (21)
mp
X
facteur d'angle de pression —Éqs. (13), (48)
��
X
facteur de rapport de conduite —Éqs. (39) à (44)
�
angle de pression °—
�
angle de pression réel à mi-largeur de denture pour
�
°—
mn
l'engrenage hypoïde
�
angle de pression réel °—
n
� angle de pression réel de l'engrenage gauche hélicoïdal °Éq. (64)
sn
angle de pression apparent de l'engrenage gauche
� °Éq. (66)
st
hélicoïdal
�
angle de pression apparent °—
t
� � angle de pression apparent de fonctionnement °—
t
angle de pression apparent pour l'engrenage cylindrique
� ° ISO 10300-1
vt
équivalent
� angle d'incidence arbitraire ° Figure 2
y
4 © ISO 2000 – Tous droits réservés

Tableau 1 — Symboles et unités (suite)
Symbole Description Unité Référence
� angle d'hélice °—
�
angle d'hélice de base °Éqs. (67), (71)
b
angle d'héliceaucône de référence à mi-largeur de la
�
°—
m
denture pour l'engrenage hypoïde
angle d'hélice pour l'engrenage gauche hélicoïdal
�
°Éq. (63)
S
équivalent
� angle auxiliaire °Éq. (86)
� angle du cône de référence °—
� rapport de retrait —Éqs. (28), (29)
a
�
rapport d'approche —Éqs. (28), (29)
f
rapport de conduite du profil réel pour l'engrenage gauche
� —Éqs. (92), (93)
n
hélicoïdal équivalent
� rapport de conduite de saillie du pignon —Éq. (30)
�
rapport de conduite de saillie de la roue —Éq. (31)
� rapport de conduite —Éq. (45)
�
rapport de conduite apparent pour l'engrenage cylindrique
� — ISO 10300-1
v�
équivalent
rapport de conduite de tête pour le pignon cylindrique
� — ISO 10300-1
v1
équivalent
rapport de conduite de tête pour la roue cylindrique
�
— ISO 10300-1
v2
équivalente
� coefficient auxiliaire hertzien — Figure 7, Éqs. (57), (59)
�
coefficient de frottement moyen —Éqs. (1), (1a)
mC
�
viscosité dynamique à la température de l'huile mPa�s —
oil
� conductivité thermique N/(s�K) —
M
� coefficient de Poisson ——
� viscosité cinématique de l'huile à 40 °Cmm /s; cSt —
� rayondecourbureentête du pignon, de la roue mm Éqs. (23), (24)
E1,2
rayon de courbure équivalent au point primitif dans le plan
� mm Éq. (76)
Cn
normal
�
rayon de courbure au point primitif dans le plan normal mm Éq. (75)
n1,2
� rayon de courbure équivalent au point primitif mm Éq. (3)
redC
� coefficient auxiliaire hertzien — Figure 7, Éqs. (58), (60)
� angle auxiliaire hertzien °Éqs. (56) à (60)
température-éclair à la tête de dent de pignon lorsque la
� répartition de charge entre dents n'est pas prise en K Éq. (19)
flaE
compte
Tableau 1 — Symboles et unités (suite)
Symbole Description Unité Référence
�
température-éclair moyenne K Éq. (18)
flaint
� température-éclair moyenne pour l'engrenage hypoïde K Éq. (50)
flainth
�
température intégrale K Éq. (17)
int
� température intégrale admissible K Éq. (16)
intP
température intégrale de grippage (température intégrale
�
K Éq. (94)
intS
acceptable)
�
température-éclair moyenne de l'engrenage d'essai K Éqs. (96), (99), (101)
flaintT
� température de l'huile du bain ou de l'injection °C —
oil
�
température de masse °C Éq. (20)
M-C
� température de masse de l'essai Éqs. (95), (98), (100)
�C
MT
angle des axes de l'engrenage gauche hélicoïdal
� °Éq. (72)
équivalent
� angle des axes d'engrenage gauche hélicoïdal équivalent °Éq. (65)
� degré de rodage — 5.2
E
� paramètre sur la ligne de conduite —Éq. (10)
Indices:
1 au pignon
2 à la roue
aaudiamètredetête de l'engrenage équivalent
b au cercle de base de l'engrenage équivalent
m à mi-largeur de denture de l'engrenage conique ou hypoïde
n auplannormal
s à l'engrenage gauche hélicoïdal équivalent
t à la direction tangentielle
T à l'engrenage d'essai
4 Domaine d'application
Les méthodes de calcul sont fondées sur des résultats d'essai obtenus sur banc d'engrenages avec des vitesses
tangentielles inférieures à 80 m/s. Les équations peuvent être utilisées pour des engrenages tournant à des
vitesses plus élevées, en sachant que l'incertitude augmente en fonction de la vitesse. Cette incertitude concerne
l'estimation de la température de masse, du coefficient de frottement, des températures admissibles, etc., lorsque
les vitesses dépassent le domaine couvert par les retours expérimentaux.
4.1 Détérioration par grippage
Une fois initiée, la détérioration par grippage peut entraîner une dégradation globale de la surface des flancs des
dents avec une augmentation: de la perte de puissance, de la charge dynamique, du bruit et de l'usure. Elle peut
également donner lieu à une rupture des dents, si la sévérité des conditions de fonctionnement n'est pas réduite.
En cas de grippage dûà une surcharge instantanée, immédiatement suivie d'une réduction de charge, par exemple
par une redistribution de la charge, les flancs des dents peuvent «s'auto-réparer» en se rodant eux-mêmes dans
6 © ISO 2000 – Tous droits réservés

une certaine mesure. Même ainsi, la détérioration résiduelle restera une cause d'augmentation de la perte de
puissance, de la charge dynamique et du bruit.
Dans la plupart des cas, la résistance des engrenages au grippage peut être améliorée au moyen d'un lubrifiant
ayant des propriétésEP (extrême pression) augmentées. Il est cependant important de noter que l'utilisation des
huiles EP comporte certains inconvénients — corrosion du cuivre, fragilisation des élastomères, difficulté
d'approvisionnement, etc. Ces inconvénients doivent être pris en compte pour un choix optimal de l'huile, ce qui
signifie: aussi peu d'additifs que possible, autant d'additifs que nécessaire.
Du fait de la variation constante des divers paramètres, la complexité des propriétés chimiques et des processus
thermo-hydroélastiques dans la zone de contact instantané, on peut prévoir une certaine dispersion dans
l'évaluation de la probabilité calculée du risque de grippage.
Contrairement au temps de développement de la détérioration par fatigue, qui est relativement lent, une surcharge
temporaire unique peut amorcer une détérioration par grippage d'une sévérité telle que les engrenages affectésne
peuvent plus être utilisés. Il convient de tenir compte de ces considérations lors du choix du coefficient de sécurité
approprié sur l'engrenage considéré,spécialement pour les engrenages qui doivent fonctionner à des vitesses
tangentielles élevées.
4.2 Critère de la température intégrale
Cette approche de l'évaluation de la probabilité de grippage est basée sur l'hypothèse selon laquelle le grippage
risque d'apparaître lorsque la valeur moyenne des températures de contact sur la longueur de conduite est
supérieure ou égale à une «valeur critique» correspondante. Dans la méthode présentéeici,lasomme de la
température de masse et la moyenne pondérée des valeurs intégrées des températures-éclair sur la longueur de
conduite constitue la «température intégrale».Latempérature de masse est estimée comme décrit en 6.1.5 et une
approximation de la valeur moyenne de la température-éclair est obtenue en utilisant des valeurs moyennes du
coefficient de frottement, de la charge dynamique, etc., sur la longueur de conduite. Un facteur de pondération est
introduit afin de tenir compte des éventuelles influences différentes d'une valeur réelle de température de masse et
d'une valeur moyenne, mathématiquement intégrée, de la température-éclair sur le phénomène de grippage.
La probabilité de grippage est évaluée en comparant la température intégrale à une valeur critique correspondante,
résultant d'essais de lubrifiants sur des engrenages, afin de vérifier leur résistance au grippage (par exemple
différentes procédures d'essai FZG, les essais d'engrenages IAE et Ryder), ou à partir d'engrenages qui ont grippé
en fonctionnement.
5 Facteurs d'influence
5.1 Coefficient de frottement moyen�
mC
Le coefficient réel de frottement entre les flancs de dent est une valeur instantanée et locale qui dépend de
plusieurs propriétés de l'huile, de la rugosité de surface, de la disposition des irrégularités de surface, telles que
celles laissées par l'usinage, les propriétés des matériaux de flanc de dent, des vitesses tangentielles, des forces
au niveau des surfaces ainsi que des dimensions. Il est difficile d'évaluer le coefficient de frottement instantané car
il n'y a pas actuellement de méthode disponible pour le quantifier par mesure.
La valeur moyenne du coefficient de frottement � sur la longueur de conduite a été déduite des mesures [1] et
mC
calculée par approximation au moyen de l'équation (1). Bien que le coefficient de frottement local soit proche de
zéro au niveau du point primitif C, on peut en calculer sa valeur moyenne par approximation au moyen des
paramètres déterminés au niveau du point primitif et de la viscosité de l'huile � à la température de l'huile �
oil oil
lorsqu'ils sont introduits dans l'équation (1).
02,
FwK�
I
Bt B�
�00, 5
2�
� = 0,045 � ��� XX� (1)
mC oil R L
G J
���
H K
�C redC
Le coefficient de frottement de la méthode de la température intégrale tient compte de la dimension de l'engrenage
d'une manière différente du coefficient de frottement de la méthode de la température-éclair. Il est recommandé de
ne pas appliquer l'équation (1) pour le calcul du coefficient de frottement, hors du domaine d'application de la partie
où il est présenté, par exemple le coefficient de frottement pour la capacité thermique.
L'équation utilisée pour le calcul de � a été déduite d'expérimentations effectuées dans le domaine ci-aprèsde
mC
conditions de fonctionnement. Il est admis qu'une extrapolation donne lieu à des écarts entre le coefficient de
frottement calculé et le coefficient de frottement réel.
1m/su vu 50 m/s
Aux vitesses tangentielles v inférieures à 1 m/s, des coefficients de frottement plus élevés sont attendus. Aux
vitesses tangentielles v supérieures à 50 m/s, la valeur limite de v à v = 50 m/s doit être utilisée dans
�C
l'équation (1).
w W 150 N/mm
Bt
Pour des valeurs inférieures de la charge spécifique réelle sur les dents, w , la valeur limite w = 150 N/mm doit
Bt Bt
être utilisée dans l'équation (1).
v =2 · v · tan� ´ · cos� (2)
�C t t
�
u sin�
t
� � ��a (3)
redC
cos�
(1� u)
b
F
t
w = K · K · K · K · (4)
Bt A v B� B�
b
2) Cette formule du coefficient de frottement est déduite d'essais d'engrenages d'entraxe a � 100 mm.
0,2
��Fb
bt �0,05 0,25
��= 0,048��� Ra� X (1a)
mC oil L
��
�����
�C redC
où
0,2
��
X��0,75 pour les polyglycols;
L
��
��v
�
C
X = 1,0 pour les huiles minérales;
L
X = 0,8 pour les polyalfaoléfines;
L
X = 1,5 pour les fluides de traction;
L
X = 1,3 pour les esters de phosphate.
L
L'équation (1a) représente les résultats des essais effectués sur une gamme de valeurs comprises entre a = 91,5 mm et
a = 200 mm. L'application de cette équation est rendue nécessaire pour ajuster les données indiquées aux Figures 9, 10 et 11
en fonction de la température de grippage� .
intS
8 © ISO 2000 – Tous droits réservés

K facteur de charge hélicoïdale, le grippage tient compte de l'augmentation du frottement en fonction de
B�
l'augmentation du rapport total de conduite (voir la Figure 1).
Figure 1 — Facteur de charge hélicoïdale K
B�
K = 1 pour� u 2
B� �
K ��10,2����25� � pour 2�� � 3,5 (5)
didi �
B�� �
K = 1,3 pour� W 3,5
B� �
Ra =0,5 · (Ra + Ra ) (6)
1 2
Ra , Ra sont les valeurs de rugosité des flancs de dent de pignon et de roue, mesurées sur le flanc à l'état neuf
1 2
(par exemple, les valeurs Ra de l'engrenage d'essai de référence sont� 0,35�m).
0,25
X =2,2 · (Ra/� ) (7)
R redC
où
X = 1,0 pour les huiles minérales
L
X = 0,8 pour les polyalphaoléfines
L
X = 0,7 pour les polyglycols non solubles dans l'eau
L
X = 0,6 pour les polyglycols solubles dans l'eau
L
X = 1,5 pour les fluides de traction
L
X = 1,3 pour les esters de phosphate
L
5.2 Facteur de rodage X
E
Les méthodes de calcul actuelles supposent que les engrenages sont bien rodés. En pratique, la défaillance par
grippage a souvent lieu au cours des toutes premières heures de fonctionnement, par exemple lors d'un essai à
pleine charge, lors de l'essai de réception des navires ou lorsqu'un nouveau jeu de roues dentées est intégré dans
une machine de production et qu'elles sont soumises à des conditions de pleine charge avant un bon rodage. Les
recherches [1] montrent que la capacité de charge d'un flanc de roue dentée nouvellement fabriquée est de 1/4 à
1/3 comparéà un flanc correctement rodé. Il convient de prendre ces considérations en compte par un facteur de
rodage X :
E
30� Ra
X ��11bg�� � (8)
EE
�
redC
où
� = 1, rodage complet (pour des engrenages cémentéset rectifiés, on peut considérer qu'il y a rodage
E
complet si Ra � 0,6 Ra );
rodage état neuf
� = 0, en sortie de fabrication.
E
5.3 Facteur thermique éclair X
M
Le facteur thermique éclair X tient compte de l'influence des matériaux du pignon et de la roue sur la température-
M
éclair.
Calcul du facteur thermique éclair pour un point quelconque (indice y) sur la ligne de conduite (voir la Figure 2):
0,25
��
��
�
11��� �
��
�� ��
2 ��u
��
X�� (9)
M
��
��
11��vv �
��
� BB��11�� � �
�� ��
M1 M2
��u
��EE
��12
a
Cercle de tête 1
b
Cercle de tête 2
Figure 2 — Paramètre������sur la ligne d'action
10 © ISO 2000 – Tous droits réservés

tan�
y
� � � 1 (10)
y
tan��
t
Si les matériaux du pignon et de la roue sont les mêmes, l'équation (9) peut être simplifiée en:
0,25
E
X � (11)
M
20,25
(1��vB)
M
Dans les équations ci-dessus, le coefficient de contact thermique B est:
M
Bc()� � (12)
M� Mv
Pour des aciers cémentés, avec les valeurs caractéristiques typiques:
2 2
� =50N/(s·K), c =3,8N/(mm ·K), E = 206 000 N/mm et v = 0,3, il s'en suit:
M v
–0,75 0,5 –0,5
X = 50,0 K·N ·s ·m ·mm
M
Pour les valeurs caractéristiques d'autres matériaux, voir [7].
5.4 Facteur d'angle de pression X
��
Le facteur d'angle de pression X est utilisé pour tenir compte de la conversion de la charge et de la vitesse
��
tangentielle du cercle de référence au cercle primitif de fonctionnement.
Méthode A: Facteur X
��-A
02,,5 025 02, 5
(sin ��cos��cos )
��
n
t
X ��12, 2 (13)
�
�� A
05,,05
(cos���� cos )
t t
Le Tableau 2 indique les valeurs du facteur d'angle de pression X pour un tracé de référence avec un angle de
��
pression � =20°, le domaine type des angles de pression de fonctionnement normalisés a� et les angles
n
t
d'hélice�.
Tableau 2 — Méthode B: Facteur X
��-B
X
��-B
� ´
t
� =0° � =10° � =20° � =30°
19° 0,963 0,960 0,951 0,938
20° 0,978 0,975 0,966 0,952
21° 0,992 0,989 0,981 0,966
22° 1,007 1,004 0,995 0,981
23° 1,021 1,018 1,009 0,995
24° 1,035 1,032 1,023 1,008
25° 1,049 1,046 1,037 1,012
Pour des roues dentées ayant un angle de pression réel � =20°, l'approximation du facteur d'angle de pression
n
peut être obtenue comme suit:
X =1
��-B
6 Calcul
6.1 Engrenages cylindriques
La présente partie de l'ISO/TR 13989 comporte des équations qui permettent d'évaluer la «probabilité de grippage»
(à chaud) d'engrenages à denture droite et hélicoïdaleendéveloppante de cercle lubrifiés à l'huile.
Pour les engrenages à denture hélicoïdale double, on suppose que la charge tangentielle totale est également
répartie entre les deux hélices. Lorsque ce n'est pas le cas du fait de l'application de forces telles que des forces
axiales externes, son influence doit être prise en compte séparément. Les deux hélices doivent être considérées
comme de simples engrenages hélicoïdaux parallèles. Les influences qui affectent la probabilité de grippage, et
pour lesquelles des évaluations quantitatives peuvent être réalisées, sont également présentées.
Les équations sont valables pour des engrenages à denture extérieure ou intérieure, conjuguée à un tracé de
référence tel que défini dans l'ISO 53. Pour les engrenages à denture intérieure, des valeurs négatives doivent être
introduites afin de déterminer le facteur géométrique X , tel que présenté en 6.1.10. Il est également admis de
BE
considérer ces équations comme valables pour des engrenages similaires issus d'une autre forme de tracé de
référence dont le rapport de conduite apparent est� u 2,5.
�
6.1.1 Coefficient de sécurité au grippage S
intS
Étant donné qu'il n'est pas possible d'exclure des incertitudes et des inexactitudes dans les hypothèses, il est
nécessaire d'introduire un coefficient de sécurité S .Ilest à souligner que le coefficient de sécurité au grippage
intS
porte sur la température et que ce n'est pas un coefficient par lequel il est admis de multiplier le couple appliqué sur
l'engrenage pour obtenir des valeurs de température intégrale � et de température intégrale de grippage �
int intS
identiques.
�
intS
SS� W (14)
intS Smin
�
int
Recommandations pour le choix de S :
Smin
S � 1 Risque de grippage élevé
Smin
1u S u 2 Domaine critique avec un risque modéré de grippage, influencé par les conditions de
Smin
fonctionnement de l'engrenage proprement dit. Les facteurs d'influence sont, par exemple, la
rugosité du flanc de dent, les effets du rodage, la connaissance précise des facteurs de
charge, la capacité de charge de l'huile de lubrification, etc.
S � 2 Risque de grippage faible
Smin
Étant donné le rapport entre la charge réelle et la température intégrale, il est possible d'approcher le coefficient de
sécurité S sur la charge transmise correspondante par:
Sl
w � ��
Btmax intS oil
S�� (15)
Sl
w ���
Bteff int oil
6.1.2 Température intégrale admissible �
intP
�
intS
� = (16)
intP
S
Smin
Le coefficient de sécurité de grippage minimum exigé S doit être déterminé séparément pour
...