ISO/TR 13989-1:2000

Calculation of scuffing load capacity of cylindrical, bevel and hypoid gears - Part 1: Flash temperature method

Calcul de la capacité de charge au grippage des engrenages cylindriques, coniques et hypoïdes — Partie 1: Méthode de la température-éclair

Le concept fondamental selon Blok est applicable à tous les éléments de machine ayant des zones de contact mobiles. Les formules de température-éclair sont valables pour une zone de contact hertzien en forme de bande ou quasiment en forme de bande et pour des conditions de fonctionnement caractérisées par des nombres de Péclet suffisamment élevés.

Izračun nosilnosti glede na toplotno razjedanje zobnih bokov valjastih, stožčastih in hipoidnih zobnikov - 1. del: Metoda trenutne temperature

General Information

• Status: Withdrawn
• Publication Date: 29-Mar-2000
• Withdrawal Date: 29-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 13989-1:2002 - Calculation of scuffing load capacity of cylindrical, bevel and hypoid gears -- Part 1: Flash temperature method

Standards Content (Sample)

TECHNICAL ISO/TR
REPORT 13989-1
First edition
2000-03-15
Calculation of scuffing load capacity of
cylindrical, bevel and hypoid gears —
Part 1:
Flash temperature method
Calcul de la capacité de charge au grippage des engrenages cylindriques,
coniques et hypoïdes —
Partie 1: Méthode de la température-éclair
Reference number
©
ISO 2000
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ii © ISO 2000 – All rights reserved

Contents Page
Foreword.iv
Introduction.v
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 Scuffing and wear.6
4.1 Occurrence of scuffing and wear.6
4.2 Transition diagram.6
4.3 Friction at incipient scuffing.8
5 Basic formulae .8
5.1 Contact temperature.8
5.2 Flash temperature formula.9
5.3 Transverse unit load.10
5.4 Distribution of overall bulk temperatures .11
5.5 Rough approximation of a bulk temperature.12
6 Coefficient of friction.12
6.1 Mean coefficient of friction, method A.13
6.2 Mean coefficient of friction, method B .13
6.3 Mean coefficient of friction, method C .13
7 Parameter on the line of action .14
8 Approach factor .16
9 Load sharing factor .17
9.1 Buttressing factor.17
9.2 Spur gears with unmodified profiles .18
9.3 Spur gears with profile modification .19
9.4 Narrow helical gears with unmodified profiles.20
9.5 Narrow helical gears with profile modification.20
9.6 Wide helical gears with unmodified profiles.21
9.7 Wide helical gears with profile modification.21
9.8 Narrow bevel gears.22
9.9 Wide bevel gears.23
10 Scuffing temperature and safety.24
10.1 Scuffing temperature.24
10.2 Structural factor .24
10.3 Contact exposure time .25
10.4 Scuffing temperature in gear tests.26
10.5 Safety range .26
Annexe A (informative) Flash temperature formula presentation.28
Annexe B (informative) Optimal profile modification .35
Bibliography .37
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 13989-1, 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 13989 are for information only.
iv © ISO 2000 – All rights reserved

Introduction
Since 1990 the flash temperature method, presented in this part of ISO/TR 13989, was enriched with research for
short exposure times, consideration of transition diagrams, new approximations for the coefficient of friction, and
completely renewed load sharing factors. In 1991 Prof. Blok contributed an extension of the flash temperature
formula which made it directly applicable to hypoid gears.
The integral temperature, presented in ISO/TR 13989-2, averages the flash temperature and supplements empirical
influence factors to the hidden load sharing factor. The resulting value approximates the maximum contact
temperature, thus yielding about the same assessment of scuffing risk as the flash temperature method of this part
of ISO/TR 13989. The integral temperature method is less sensitive for those cases where there are local
temperature peaks, usually in gearsets that have low contact ratio or contact near the base circle or other sensitive
geometries.
The 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. 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. According to Blok [12][13][14][15][16][17], high contact temperatures of lubricant and tooth surfaces
at the instantaneous contact position may effect a break-down of the lubricant film at the contact interface.
The interfacial contact temperature is conceived as the sum of two components:
� the interfacial bulk temperature of the moving interface, which, if varying, does so only comparatively slowly.
For evaluating this component, it may be suitably averaged from the two overall bulk temperatures of the two
rubbing teeth. The latter two bulk temperatures follow from the thermal network theory [18].
� the rapidly fluctuating flash temperature of the moving faces in contact. Special attention has to be paid to the
coefficient of friction. A common practice is the use of a coefficient of friction valid for regular working
conditions, although it may be stated that at incipient scuffing the coefficient of friction has significantly higher
values.
The complex relationship between mechanical, hydrodynamical, thermodynamical and chemical phenomena was
the objective of extensive research and experiments, which may induce various empirical influence factors. A direct
suppletion of empirical influence factors may enforce the related functional factors in the main formula to be fixated
to average values. However, correct treatment of functional factors (e.g. coefficient of friction, load sharing factor,
thermal contact coefficient) keeps the main formula intact, in confirmation with the experiments and practice.
Next to the maximum contact temperature, the progress of the contact temperature along the path of contact
provides necessary information to the gear design.
TECHNICAL REPORT ISO/TR 13989-1:2000(E)
Calculation of scuffing load capacity of cylindrical, bevel and
hypoid gears —
Part 1:
Flash temperature method
1 Scope
This part of ISO/TR 13989 specifies methods and formulae for evaluating the risk of scuffing, based on Blok's
contact temperature concept.
The fundamental concept according to Blok is applicable to all machine elements with moving contact zones. The
flash temperature formulae are valid for a band-shaped or approximately band-shaped Hertzian contact zone and
working conditions characterized by sufficiently high Péclet numbers.
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 1122-1:1998, Vocabulary of gear terms — Part 1: Definitions related to geometry.
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.
ISO 10825:1995, Gears — Wear and damage to gear teeth — Terminology.
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 and ISO 10825 apply.
3.2 Symbols and units
The symbols used in this part of ISO/TR 13989 are given in Table 1. The units of length metre, millimetre and
micrometre are chosen in accordance with common practice. To achieve a "coherent" system, the units for B , c ,
M �
X are adapted to the mixed application of metre and millimetre or millimetre and micrometre.
M
1) To be published.
Table 1 — Symbols and units
Symbol Description Unit Reference
a centre distance mm Eq. (A.5)
a
b facewidth, smaller value for pinion or wheel mm Eq. (11)
b effective facewidth mm Eq. (12)
eff
b semi-width of Hertzian contact band mm Eq. (3)
H
½ ½ ½
B thermal contact coefficient N/(mm �m �s �K) Eq (A.13)
M
½ ½ ½
B thermal contact coefficient of pinion N/(mm �m �s �K) Eq. (3)
M1
½ ½ ½
B thermal contact coefficient of wheel N/(mm �m �s �K) Eq. (3)
M2
C tip relief of pinion �m Eq. (48)
a1
C tip relief of wheel �m Eq. (46)
a2
C optimal tip relief �m Eq. (46)
eff
C equivalent tip relief of pinion �mEq.(B.2)
eq1
C equivalent tip relief of wheel �mEq.(B.3)
eq2
C root relief of pinion �mEq.(B.3)
f1
C root relief of wheel �mEq.(B.2)
f2
c specific heat per unit mass of pinion J/(kg�K) Eq. (9)
M1
c specific heat per unit mass of wheel J/(kg�K) Eq. (10)
M2
c mesh stiffness N/(mm��m) Eq. (B.1)
�
d reference diameter of pinion mm Eq. (34)
d reference diameter of wheel mm Eq. (35)
d tip diameter of pinion mm Eq. (34)
a1
d tip diameter of wheel mm Eq. (35)
a2
E modulus of elasticity of pinion N/mm Eq. (A.10)
E modulus of elasticity of wheel N/mm Eq. (A.10)
E reduced modulus of elasticity N/mm Eq. (A.9)
r
F external axial force N Eq. (18)
ex
F normal load in wear test N Fig. 1
n
F nominal tangential force N Eq. (11)
t
H auxiliary dimension mm Eq. (B.3)
H auxiliary dimension mm Eq. (B.2)
h tip height in mean cone of pinion mm Eq. (43)
am1
h tip height in mean cone of wheel mm Eq. (44)
am2
2 © ISO 2000 – All rights reserved

Table 1 — Symbols and units (continued)
Symbol Description Unit Reference
K application factor — Eq. (11)
A
K transverse load factor (scuffing) — Eq. (11)
B�
K face load factor (scuffing) — Eq. (11)
B�
K transverse load factor (contact stress) — Eq. (15)
H�
K face load factor (contact stress) — Eq. (14)
H�
K multiple path factor — Eq. (11)
mp
K dynamic factor — Eq. (11)
v
m normal module mm Eq. (B.2)
n
n revolutions per minute of pinion r/min Eq. (5)
n number of mesh contacts — Eq. (16)
p
Pe Péclet number of pinion material — Eq. (9)
Pe Péclet number of wheel material — Eq. (10)
Q quality grade — Eq. (57)
Ra tooth flank surface roughness of pinion �m Eq. (28)
Ra tooth flank surface roughness of wheel �m Eq. (28)
R cone distance of mean cone mm Eq. (A.16)
m
r reference radius in mean cone of pinion mm Eq. (43)
m1
r reference radius in mean cone of wheel mm Eq. (44)
m2
S safety factor for scuffing — Eq. (100)
B
S load stage (in FZG test) — Eq. (99)
FZG
t contact exposure time of pinion �s Eq. (95)
t contact exposure time of wheel �s Eq. (96)
t contact exposure time at bend of curve �s Eq. (97)
c
t longest contact exposure time �s Eq. (95)
max
u gear ratio — Eq. (A.6)
u virtual ratio — Eq. (B.6)
v
v sliding velocity m/s Fig. 1
g
v tangential velocity of pinion m/s Eq. (3)
g1
v tangential velocity of wheel m/s Eq. (3)
g2
v sum of tangential velocities in pitch point m/s Eq. (25)
g�C
v pitch line velocity m/s Eq. (26)
t
w normal unit load N/mm Eq. (3)
Bn
w transverse unit load N/mm Eq. (5)
Bt
Table 1 — Symbols and units (continued)
Symbol Description Unit Reference
X buttressing factor — Eq. (54)
but
X buttressing value — Eq. (51)
butA
X buttressing value — Eq. (51)
butE
X geometry factor — Eq. (A.5)
G
X approach factor — Eq. (3)
J
X lubricant factor — Eq. (25)
L
–¾ –½ –½
X thermo-elastic factor K�N �s �m �mm Eq. (5)
M
X multiple mating pinion factor — Eq. (22)
mp
X roughness factor — Eq. (25)
R
X lubrication system factor — Eq. (22)
S
X structural factor — Eq. (94)
W
X angle factor — Eq. (A.6)
��
X load sharing factor — Eq. (3)
�
X gradient of the scuffing temperature — Eq. (97)
�
z number of teeth of pinion — Eq. (30)
z number of teeth of wheel — Eq. (30)
� transverse tip pressure angle of pinion ° Eq. (31)
a1
� transverse tip pressure angle of wheel ° Eq. (30)
a2
� transverse pressure angle ° Eq. (34)
t
� normal working pressure angle ° Eq. (A.2)
wn
� transverse working pressure angle ° Eq. (7)
wt
� pinion pressure angle at arbitrary point ° Eq. (29)
y1
� helix angle ° Eq. (18)
� base helix angle ° Eq. (49)
b
� base helix angle in midcone ° Eq. (50)
bm
� working helix angle ° Eq. (A.2)
w
� parameter on the line of action at point A — Eq. (24)
A
� parameter on the line of action at point AA — Eq. (68)
AA
� parameter on the line of action at point AB — Eq. (66)
AB
� parameter on the line of action at point AU — Eq. (49)
AU
� parameter on the line of action at point B — Eq. (31)
B
� parameter on the line of action at point BB — Eq. (70)
BB
� parameter on the line of action at point D — Eq. (32)
D
� parameter on the line of action at point DD — Eq. (72)
DD
� parameter on the line of action at point DE — Eq. (67)
DE
4 © ISO 2000 – All rights reserved

Table 1 — Symbols and units (continued)
Symbol Description Unit Reference
� parameter on the line of action at point E — Eq. (24)
E
� parameter on the line of action at point EE — Eq. (74)
EE
� parameter on the line of action at point EU — Eq. (49)
EU
� parameter on the line of action at point M — Eq. (86)
M
� parameter on the line of action at arbitrary point — Eq. (7)
y
� angle of direction of tangential velocity of pinion — Eq. (3)
� angle of direction of tangential velocity of wheel — Eq. (3)
� pitch cone angle of pinion ° Eq. (37)
� pitch cone angle of wheel ° Eq. (39)
� transverse contact ratio — Eq. (76)
�
� overlap ratio — Eq. (52)
�
� absolute (dynamic) viscosity at oil temperature mPa�s Eq. (27)
oil
� contact temperature °CEq.(1)
B
� maximum contact temperature °CEq.(2)
Bmax
� flash temperature K Eq. (1)
fl
� average flash temperature K Eq. (22)
flm
� maximum flash temperature K Eq. (2)
flmax
� maximum flash temperature at test K Eq. (94)
flmaxT
� bulk temperature °C Eq. (22)
M
� interfacial bulk temperature °CEq.(1)
Mi
� bulk temperature of pinion teeth °C Eq. (20)
M1
� bulk temperature of wheel teeth °C Eq. (20)
M2
� bulk temperature at test °C Eq. (94)
MT
� oil temperature before reaching the mesh °C Eq. (22)
oil
� scuffing temperature °C Eq. (94)
S
� scuffing temperature at long contact time °C Eq. (97)
Sc
� heat conductivity of pinion N/(s�K) Eq. (9)
M1
� heat conductivity of wheel N/(s�K) Eq. (10)
M2
� coefficient of friction in pin-and-ring test — Fig. 1
� mean coefficient of friction — Eq. (3)
m
� Poisson's ratio of pinion material — Eq. (A.10)
� Poisson's ratio of wheel material — Eq. (A.10)
Table 1 — Symbols and units (concluded)
Symbol Description Unit Reference
� density of pinion material kg/m³ Eq. (9)
M1
� density of wheel material kg/m³ Eq. (10)
M2
� relative radius of curvature at pitch point mm Eq. (25)
relC
� radius of curvature at arbitrary point of pinion mm Eq. (5)
y1
� radius of curvature at arbitrary point of wheel mm Eq. (5)
y2
� relative radius of curvature at arbitrary point mm Eq. (5)
yrel
� shaft angle ° Eq. (A.15)
� quill shaft twist ° Eq. (17)
a
The term wheel is used for the mating gear of a pinion.
4 Scuffing and wear
4.1 Occurrence of scuffing and wear
When gear teeth are completely separated by a full fluid film of lubricant, there is no contact between the asperities
of the tooth surfaces, and usually there is no scuffing or wear. Here, the coefficient of friction is rather low. In
exceptional cases a damage similar to scuffing may be caused by a sudden thermal instability [19] in a thick oil film,
which phenomenon is not treated here.
For thinner elastohydrodynamic films, incidental asperity contact takes place. As the mean film thickness
decreases, the number of contacts increases acccordingly. Abrasive wear, adhesive wear or scuffing becomes
possible. Abrasive wear may occur due to the rolling action of the gear teeth or the presence of abrasive particles in
the lubricant. Adhesive wear occurs by localized welding and subsequent detachment and transfer of particles from
one or both of the meshing teeth. Abrasive or adhesive wear may not be harmful if it is mild and if it subsides with
time, as in a normal run-in process.
In contrast to mild wear, scuffing is a severe form of adhesive wear that can result in progressive damage to the
gear teeth. In contrast to pitting and fatigue breakage which show a distinct incubation period, a short transient
overloading can result in scuffing failure.
Excessive aeration or the presence in the lubricant of contaminants such as metal particles in suspension, or water,
also increases the risk of scuffing damage. After scuffing, high speed gears tend to suffer high levels of dynamic
loading due to vibration which usually cause further damage by scuffing, pitting or tooth breakage.
)
In most cases, the resistance of gears to scuffing can be improved by using a lubricant with enhanced anti-scuff
additives. It is important, however, to be aware that some disadvantages attend the use of anti-scuff additives:
corrosion of copper, embrittlement of elastomers, lack of world-wide availability, etc.
The methods described are not suitable for "cold scuffing" which is in general associated with low speed, under
approx. 4 m/s, through-hardened heavily loaded gears of rather poor quality.
4.2 Transition diagram
The lubrication condition of sliding concentrated steel contacts, which operate in a liquid lubricant, can be described
[20][21][22][23] in terms of transition diagrams. A transition diagram according to Figure 1 is considered to be
applicable to contacts functioning at constant oil bath temperature.
2) The less correct designation Extreme Pressure, EP, is replaced by anti-scuff.
6 © ISO 2000 – All rights reserved

At combinations of normal force F and relative sliding velocity v which fall below the line A1-S, in region I, see
n g
Figure 1, the lubrication condition is characterized by a coefficient of friction of about 0,1 and a specific wear rate of
�2 3 �6 3
10 mm /(N�m) to 10 mm /(N�m) (i.e. volume wear per unit of normal force, per unit of sliding distance).
If, with v not above a value according to point S, the load is increased into region II, a transition into a second
g
condition of lubrication occurs. This mild wear lubrication condition is characterized by a coefficient of friction of
3 3
about 0,3 to 0,4 and a specific wear rate of 1 mm /(N�m) to5mm /(N�m).
Figure 1 — Transition diagram for contraform contacts with example of calculated contact temperatures
If load is increased still further, a transition into a third condition of lubrication, region III, occurs at intersection of the
line A2-S. This region is characterized by a coefficient of friction equal to 0,4 to 0,5. The wear rate, however, is
3 3
considerably higher, i.e. 100 mm /(N�m) to 1 000 mm /(N�m), than in regions I and II and the worn surfaces show
evidence of severe wear in the form of scuffing. If load increases at relative sliding velocities beyond point S, a
direct transition from region I to region III takes place.
There is strong evidence that the position of the line A1-S-A3 depends upon lubricant viscosity [24] as well as upon
Hertzian contact pressure [20][21]. At combinations of F and v that fall below this line, it is believed that the
n g
surfaces are kept apart by a thin lubricant film which is, however, penetrated by roughness asperities. In this
context, the term "partial elastohydrodynamic lubrication" has been used [21].
In region III liquid film effects are completely absent. This region is identical to the region of "incipient scuffing" [25].
There is evidence that the transition which occurs at intersecting the line A2-S is associated with reaching a critical
value of the contact temperature. This is the fundamental concept according to Blok.
The transition diagram shown is applicable to newly assembled, i.e. unoxidized steel contacts, as occur in gears,
cams and followers, etc. It has been found that the diagram is applicable to four-ball as well as to pin-and-ring test
results.
Along curve A1-S-A3 temperature ranges from an oil bath, respectively overall bulk, respectively interfacial bulk
temperature, of 28 °Cat v = 0,001 m/s to a contact temperature of 498 °Cat v = 10 m/s. This temperature
g g
behaviour strongly suggests that the collapse of (partial) elasto-hydrodynamical lubrication does not occur at a
constant contact or interfacial bulk temperature, for instance being associated with melting of chemisorbed material.
Instead, the pronounced decrease of load carrying capacity with increasing sliding velocity is supposed to be due to
decreasing viscosity [24][26][27][28][29].
Contrary to the above, calculated contact temperatures along curve A2-S-A3 tend to attain a constant value, e.g. in
the case of AISI 52100 steel specimens approximately 500 °C; see Figure 1. This suggests that the II-III transition is
associated with a transformation in the steel, causing the wear mechanism of surfaces to change from mildly
adhesive to severely adhesive, perhaps involving a mechanism of thermo-elastic instability [30][31].
Therefore, the results indicate scuffing is associated with a critical magnitude of the contact temperature. For steel,
lubricated with mineral oils, the critical magnitude does not depend on load, velocity and geometry, and equals near
500 °C.
4.3 Friction at incipient scuffing
As shown in the transition diagram, Figure 1, in the case of scuffing the coefficient of friction leaps from about 0,25
to about 0,5. The corresponding contact temperature proves to be about 500 °C. This contact temperature is the
sum of a measured interfacial bulk temperature of 28 °C and a calculated flash temperature of 470 °C. During the
flash temperature calculation use is made of the coefficient of friction just before transition, � = 0,35. If this method
has to be applied not only for pin-and-ring tests but also (during the design stage) for gear transmissions, one shall
agree upon the choice of the value of the critical magnitude of the contact temperature on one hand and the value
of the coefficient of friction to be used in the calculations on the other.
A gear load capacity can be predicted
� on the safe side, with the coefficient of friction � = 0,50;
� accurately, with the coefficient of friction between � = 0,25 and� = 0,35, dependent on the lubricant;
� according to previous practice, with a low coefficient of friction of regular working conditions, provided that the
limiting contact temperature is correspondingly low.
In terms of previous practice, for non-additive and low-additive mineral oils, each combination of oil and rolling
materials has a critical scuffing temperature which, in general, is constant regardless of the operating conditions,
load, velocity and geometry.
For high-additive and certain kinds of synthetic lubricants the critical scuffing temperature may well vary from one
set of operating conditions to another. So, this critical temperature must then be determined for each such set
separately from tests which closely simulate the operating condition of the gearset.
5 Basic formulae
5.1 Contact temperature
As already mentioned in the introduction, the contact temperature is the sum of the interfacial bulk temperature,
� , see 5.3, and the flash temperature,��� , see 5.2,
Mi fl
����� (1)
BMi fl
a
Position in the path of contact.
Figure 2 — Contact temperature along the path of contact
8 © ISO 2000 – All rights reserved

Only the flash temperature varies along the path of contact; see Figure 2.
The maximum contact temperature is
����� (2)
Bmax Mi flmax
where� is the maximum value of � , being located either at the approach path or at the recess path.
flmax fl
Prediction of the probability of scuffing is possible by comparing the calculated maximum contact temperature with
a critical magnitude. This critical magnitude of the contact temperature can be evaluated from any gear scuffing
test, or can be provided by field investigations.
For a reliable evaluation of the scuffing risk, it is important that an accurate value of the gear bulk temperature be
used for the analysis.
5.2 Flash temperature formula
The flash temperature formula of Blok [12][14][16][32] in a most general representation, for (approximately)
band-shaped contact and tangential velocities differently directed (as for hypoid gears), see annex A, reads
abs()vv�
���XX�w
g1 g2
mJ� Bn
���11, 1 � (3)
fl
()2� b Bv��(sin��)�B�(v�sin )
H M1 g1 12Mg22
For cylindrical or bevel gears, with band-shaped contact and parallel tangential velocities, the general
representation, see annex A, reads
abs()vv�
���XX � w
g1 g2
mJ� Bn
���11, 1 � (4)
fl
()2� b Bv��()B�v
H M1 g1 M2 g2
or, in an equivalent representation,
abs��� / u
ej
y1 y2
X n
F I
M 3 1
����25,(2 �� � ��� w)� � (5)
G J
fl m JB�t
H K
50 60 4
�
yrel
where
� is the mean coefficient of friction (see clause 6);
m
X is the thermo-elastic factor (see annex A);
M
–¾ ½ –½
X =50K�N �s �m �mm for commonly applied steel;
M
X is the approach factor (see clause 8);
J
X is the load sharing factor (see clause 9);
�
w is the transverse unit load (see 5.3), in N/mm;
Bt
n is the rotational speed of pinion, in r/min;
� is the local relative radius of curvature, in mm:
yrel
� ��
y1 y2
� � (6)
yrel
���
y1 y2
� is the local radius of curvature of pinion flank, in mm:
y1
1��
y
��� ��a sin (cylindrical gears) (7)
y1 wt
1� u
� is the local radius of curvature of wheel flank, in mm:
y2
u��
y
��� ��a sin (cylindrical gears) (8)
y2 wt
1� u
For bevel gears, see equations (37) and (38).�
For an adapted representation, see annex A.
Two Péclet numbers have to be sufficiently high, which is satisfied in almost all cases where scuffing may occur.
For lower Péclet numbers the heat flow from the contact band into the gear teeth causes a different temperature
distribution for which formulae (3) to (6) are not valid.
vb��� �c
g1 H M1 M1
Pe � � 5 (9)
��� sin
M1 1
vb��� �c
g2 H M2 M2
Pe � � 5 (10)
��� sin
M2 2
where
� is the density of pinion material, in kg/m ;
M1
� is the density of wheel material, in kg/m ;
M2
c is the specific heat per unit mass of pinion, in J/(kg�K);
M1
c is the specific heat per unit mass of wheel, in J/(kg�K);
M2
� is the heat conductivity of pinion, in N/(s�K);
M1
� is the heat conductivity of wheel, in N/(s�K).
M2
For cylindrical and bevel gears, sin� =sin� =1.
1 2
5.3 Transverse unit load
The transverse unit load for cylindrical gears and bevel gears is
F
t
wK��K�K �K �K � (cylindrical gears) (11)
Bt A v B� Bm� p
b
F
t
wK��K�K �K �K � (bevel gears) (12)
Bt A v B� Bm� p
b
eff
where
F is the nominal tangential force on pitch circle, in N;
t
b is the facewidth, in mm;
b =0,85b (13)
eff
10 © ISO 2000 – All rights reserved

K is the application factor (see ISO 6336-1 for cylindrical gears, ISO 10300-1 for bevel gears);
A
K is the dynamic factor (see ISO 6336-1 for cylindrical gears, ISO 10300-1 for bevel gears);
v
K is the face load factor;
B�
K = K (see ISO 6336-1 for cylindrical gears, ISO 10300-1 for bevel gears); (14)
B� H�
K is the transverse load factor:
B�
K = K (see ISO 6336-1 for cylindrical gears, ISO 10300-1 for bevel gears); (15)
B� H�
K is the multiple-path factor:
mp
The multiple-path factor K accounts for the maldistribution in multiple-path transmissions depending on accuracy
mp
and flexibility of the branches. If no relevant analysis is available, the following may apply.
� for epicyclical gear trains with n planets (n W 3)
p p
Kn��10,25 �3 (16)
mp p
� for dual tandem gears with quill shaft twist �� degrees under full load
K ��10(,2/�) (17)
mp
� for double helical gears with an external axial force F
ex
F
ex
K ��1 (18)
mp
F � tan�
t
� for other cases
K =1 (19)
mp
5.4 Distribution of overall bulk temperatures
The friction loss most typical of gear transmissions is the one caused by the meshing zone. In this source the heat
is generated mainly by tooth friction. The mechanical "pumping" energy expended for sideways expulsion of
superfluous oil may sometimes be far from negligible. The other unavoidable friction loss is from the bearings,
either of the rolling or the sliding type. In high speed gear transmissions, sliding bearings may well generate much
more frictional heat than gears. Other heat sources are oil churning and friction from seals. All the above heat
sources have the following features in common:
� in each of these sources the fluid friction depends on some oil viscosity representative of the operating
condition;
� all of the heat sources are thermally interconnected through transmission elements to the sinks, such as the
ambient air or the cooling system.
The thermal interconnection allows calculation concepts such as:
� finite element methods for discrete components;
� bondgraph methods,
� thermal network analogue methods [18].
The interfacial bulk temperature� may be suitably averaged from the two overall bulk temperatures of the teeth in
Mi
contact, � and � . The following formula is valid to a good approximation (at high values of the Péclet
M1 M2
numbers):
Bv�����B�v�
M1 g1 M1 M2 g2 M2
� � (20)
Mi
Bv��B�v
M1 g1 M2 g2
�
B v
M1 g1
In a fairly wide range of the ratio a simple arithmetic average is valid to a reasonable approximation
�
Bv
M2 g2
���� �� (21)
bg
Mi M1 M2
Bulk temperatures in excess of 150 °C for long periods may have an adverse effect on the surface durability.
5.5 Rough approximation of a bulk temperature
For very rough inquiry the bulk temperature may be estimated by the sum of the oil temperature, taking into account
some impediment in heat transfer for spray lubrication, and a part which depends mainly on the flash temperature,
of which the maximum value is taken.
��= + 0,47��XX �� (22)
Moil mp flm
S
where�
X = 1,2 for spray lubrication;
S
X = 1,0 for dip lubrication;
S
X = 1,0 for meshes with additional spray for cooling purpose;
S
X = 0,2 for gears submerged in oil, provided sufficient cooling;
S
1� n
p
X � for a pinion with n mating gears; (23)
p
mp
� is the average of flash temperature along path of contact, in °C:
flm
E
��� d
fl y
z
A
� � (24)
flm
���
EA
However, for a reliable evaluation of the scuffing risk, it is important that instead of a rough approximation, an
accurate value of the gear bulk temperature be used for the analysis.
6 Coefficient of friction
Several factors influencing the friction between gear teeth vary throughout a meshing cycle. On one of the two
mating tooth faces the relative motion is uniformly accelerating, on the other it is uniformly decelerating. Only at
pitch point position pure rolling occurs. In any other meshing position combined rolling and sliding will occur. Also
the load acting on two mating tooth faces will vary from one meshing position to another. These conditions cause a
continuous variation of the film thickness, the lubrication regime and the coefficient of friction. Even in a similar
meshing position the coefficient of friction may vary for different teeth and different time.
12 © ISO 2000 – All rights reserved

The local coefficient of friction is considered to be a representative quantity valid for the local point concerned,
smoothing various influences. The geometrically determined variation of the local coefficient of friction is difficult to
calculate or to measure, hence instead of a local value, a representative mean value of the coefficient of friction will
be applied.�
A mean value (along the path of contact) of the coefficient of friction has commonly been applied, and even that
value is uncertain. Too often, in test reports on friction, important influential quantities were neglected, for instance
the bulk temperature which determines the inlet viscosity and therefore the lubrication regime.
3)
The mean coefficient of friction � depends on the geometry of the path of contact, the tangential velocities, the
m
normal load, the inlet viscosity (which is identical with viscosity at teeth bulk temperature), the pressure-viscosity
coefficient, the reduced modulus of elasticity, the surface roughness, the normal relative radius of curvature.
Depending on further investigations, other quantities and influences may have to be accounted for, either in the
formula or in the description of the field of application. The number of quantities may be reduced by dimension
analysis [33], and a possible neglect of some minor influential quantities.
The coefficient of friction may be measured or estimated according to various methods. The limiting contact
temperature shall be chosen correspondingly to the coefficient of friction.
6.1 Mean coefficient of friction, method A
The coefficient of friction at the onset of scuffing may be measured in gear tests or pin-and-ring tests. The limiting
contact temperature is correspondingly high.
6.2 Mean coefficient of friction, method B
According to previous practice, whereby low coefficients of friction of regular working conditions are used, the final
calculation of the coefficient of friction may be made with some appropriate formula, i.e. one containing a value of
absolute (dynamic) viscosity � that corresponds to the gear bulk temperature. The limiting contact temperature is
L
correspondingly low, see clause 10.
6.3 Mean coefficient of friction, method C
If at the start of a calculation the bulk temperature is not yet known, the mean coefficient of friction of common
working conditions may be estimated by
02,
F I
w
Bt
���0,060 ��XX (25)
m
G J LR
v ��
H gC� relC K
where
w is the transverse unit load, see equation (11) or (12), in N/mm;
Bt
v is the sum of tangential velocities in pitch point, in m/s:
g�C
vv=2��sin� (26)
gC� twt
v is the pitch line velocity, in m/s (if v � 50 m/s, substitute the value 50 in equation (26), instead of v );
t t t
� is the transverse relative radius of curvature, in mm (see equation (6) for � =0);
relC y
3) The mean coefficient of friction is defined as the mean value of the local coefficients of friction along the path of contact.
Although the actual local coefficient of friction at the pitch point will differ from the mean coefficient of friction defined for the
whole path of contact, that mean coefficient of friction may be expressed in terms related to the pitch point.
ISO/TR 1398
...

SLOVENSKI STANDARD
01-julij-2002
,]UDþXQQRVLOQRVWLJOHGHQDWRSORWQRUD]MHGDQMH]REQLKERNRYYDOMDVWLKVWRåþDVWLK
LQKLSRLGQLK]REQLNRYGHO0HWRGDWUHQXWQHWHPSHUDWXUH
Calculation of scuffing load capacity of cylindrical, bevel and hypoid gears -- Part 1:
Flash temperature method
Calcul de la capacité de charge au grippage des engrenages cylindriques, coniques et
hypoïdes -- Partie 1: Méthode de la température-éclair
Ta slovenski standard je istoveten z: ISO/TR 13989-1:2000
ICS:
21.200 Gonila Gears
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

TECHNICAL ISO/TR
REPORT 13989-1
First edition
2000-03-15
Calculation of scuffing load capacity of
cylindrical, bevel and hypoid gears —
Part 1:
Flash temperature method
Calcul de la capacité de charge au grippage des engrenages cylindriques,
coniques et hypoïdes —
Partie 1: Méthode de la température-éclair
Reference number
©
ISO 2000
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ii © ISO 2000 – All rights reserved

Contents Page
Foreword.iv
Introduction.v
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 Scuffing and wear.6
4.1 Occurrence of scuffing and wear.6
4.2 Transition diagram.6
4.3 Friction at incipient scuffing.8
5 Basic formulae .8
5.1 Contact temperature.8
5.2 Flash temperature formula.9
5.3 Transverse unit load.10
5.4 Distribution of overall bulk temperatures .11
5.5 Rough approximation of a bulk temperature.12
6 Coefficient of friction.12
6.1 Mean coefficient of friction, method A.13
6.2 Mean coefficient of friction, method B .13
6.3 Mean coefficient of friction, method C .13
7 Parameter on the line of action .14
8 Approach factor .16
9 Load sharing factor .17
9.1 Buttressing factor.17
9.2 Spur gears with unmodified profiles .18
9.3 Spur gears with profile modification .19
9.4 Narrow helical gears with unmodified profiles.20
9.5 Narrow helical gears with profile modification.20
9.6 Wide helical gears with unmodified profiles.21
9.7 Wide helical gears with profile modification.21
9.8 Narrow bevel gears.22
9.9 Wide bevel gears.23
10 Scuffing temperature and safety.24
10.1 Scuffing temperature.24
10.2 Structural factor .24
10.3 Contact exposure time .25
10.4 Scuffing temperature in gear tests.26
10.5 Safety range .26
Annexe A (informative) Flash temperature formula presentation.28
Annexe B (informative) Optimal profile modification .35
Bibliography .37
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 13989-1, 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 13989 are for information only.
iv © ISO 2000 – All rights reserved

Introduction
Since 1990 the flash temperature method, presented in this part of ISO/TR 13989, was enriched with research for
short exposure times, consideration of transition diagrams, new approximations for the coefficient of friction, and
completely renewed load sharing factors. In 1991 Prof. Blok contributed an extension of the flash temperature
formula which made it directly applicable to hypoid gears.
The integral temperature, presented in ISO/TR 13989-2, averages the flash temperature and supplements empirical
influence factors to the hidden load sharing factor. The resulting value approximates the maximum contact
temperature, thus yielding about the same assessment of scuffing risk as the flash temperature method of this part
of ISO/TR 13989. The integral temperature method is less sensitive for those cases where there are local
temperature peaks, usually in gearsets that have low contact ratio or contact near the base circle or other sensitive
geometries.
The 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. 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. According to Blok [12][13][14][15][16][17], high contact temperatures of lubricant and tooth surfaces
at the instantaneous contact position may effect a break-down of the lubricant film at the contact interface.
The interfacial contact temperature is conceived as the sum of two components:
� the interfacial bulk temperature of the moving interface, which, if varying, does so only comparatively slowly.
For evaluating this component, it may be suitably averaged from the two overall bulk temperatures of the two
rubbing teeth. The latter two bulk temperatures follow from the thermal network theory [18].
� the rapidly fluctuating flash temperature of the moving faces in contact. Special attention has to be paid to the
coefficient of friction. A common practice is the use of a coefficient of friction valid for regular working
conditions, although it may be stated that at incipient scuffing the coefficient of friction has significantly higher
values.
The complex relationship between mechanical, hydrodynamical, thermodynamical and chemical phenomena was
the objective of extensive research and experiments, which may induce various empirical influence factors. A direct
suppletion of empirical influence factors may enforce the related functional factors in the main formula to be fixated
to average values. However, correct treatment of functional factors (e.g. coefficient of friction, load sharing factor,
thermal contact coefficient) keeps the main formula intact, in confirmation with the experiments and practice.
Next to the maximum contact temperature, the progress of the contact temperature along the path of contact
provides necessary information to the gear design.
TECHNICAL REPORT ISO/TR 13989-1:2000(E)
Calculation of scuffing load capacity of cylindrical, bevel and
hypoid gears —
Part 1:
Flash temperature method
1 Scope
This part of ISO/TR 13989 specifies methods and formulae for evaluating the risk of scuffing, based on Blok's
contact temperature concept.
The fundamental concept according to Blok is applicable to all machine elements with moving contact zones. The
flash temperature formulae are valid for a band-shaped or approximately band-shaped Hertzian contact zone and
working conditions characterized by sufficiently high Péclet numbers.
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 1122-1:1998, Vocabulary of gear terms — Part 1: Definitions related to geometry.
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.
ISO 10825:1995, Gears — Wear and damage to gear teeth — Terminology.
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 and ISO 10825 apply.
3.2 Symbols and units
The symbols used in this part of ISO/TR 13989 are given in Table 1. The units of length metre, millimetre and
micrometre are chosen in accordance with common practice. To achieve a "coherent" system, the units for B , c ,
M �
X are adapted to the mixed application of metre and millimetre or millimetre and micrometre.
M
1) To be published.
Table 1 — Symbols and units
Symbol Description Unit Reference
a centre distance mm Eq. (A.5)
a
b facewidth, smaller value for pinion or wheel mm Eq. (11)
b effective facewidth mm Eq. (12)
eff
b semi-width of Hertzian contact band mm Eq. (3)
H
½ ½ ½
B thermal contact coefficient N/(mm �m �s �K) Eq (A.13)
M
½ ½ ½
B thermal contact coefficient of pinion N/(mm �m �s �K) Eq. (3)
M1
½ ½ ½
B thermal contact coefficient of wheel N/(mm �m �s �K) Eq. (3)
M2
C tip relief of pinion �m Eq. (48)
a1
C tip relief of wheel �m Eq. (46)
a2
C optimal tip relief �m Eq. (46)
eff
C equivalent tip relief of pinion �mEq.(B.2)
eq1
C equivalent tip relief of wheel �mEq.(B.3)
eq2
C root relief of pinion �mEq.(B.3)
f1
C root relief of wheel �mEq.(B.2)
f2
c specific heat per unit mass of pinion J/(kg�K) Eq. (9)
M1
c specific heat per unit mass of wheel J/(kg�K) Eq. (10)
M2
c mesh stiffness N/(mm��m) Eq. (B.1)
�
d reference diameter of pinion mm Eq. (34)
d reference diameter of wheel mm Eq. (35)
d tip diameter of pinion mm Eq. (34)
a1
d tip diameter of wheel mm Eq. (35)
a2
E modulus of elasticity of pinion N/mm Eq. (A.10)
E modulus of elasticity of wheel N/mm Eq. (A.10)
E reduced modulus of elasticity N/mm Eq. (A.9)
r
F external axial force N Eq. (18)
ex
F normal load in wear test N Fig. 1
n
F nominal tangential force N Eq. (11)
t
H auxiliary dimension mm Eq. (B.3)
H auxiliary dimension mm Eq. (B.2)
h tip height in mean cone of pinion mm Eq. (43)
am1
h tip height in mean cone of wheel mm Eq. (44)
am2
2 © ISO 2000 – All rights reserved

Table 1 — Symbols and units (continued)
Symbol Description Unit Reference
K application factor — Eq. (11)
A
K transverse load factor (scuffing) — Eq. (11)
B�
K face load factor (scuffing) — Eq. (11)
B�
K transverse load factor (contact stress) — Eq. (15)
H�
K face load factor (contact stress) — Eq. (14)
H�
K multiple path factor — Eq. (11)
mp
K dynamic factor — Eq. (11)
v
m normal module mm Eq. (B.2)
n
n revolutions per minute of pinion r/min Eq. (5)
n number of mesh contacts — Eq. (16)
p
Pe Péclet number of pinion material — Eq. (9)
Pe Péclet number of wheel material — Eq. (10)
Q quality grade — Eq. (57)
Ra tooth flank surface roughness of pinion �m Eq. (28)
Ra tooth flank surface roughness of wheel �m Eq. (28)
R cone distance of mean cone mm Eq. (A.16)
m
r reference radius in mean cone of pinion mm Eq. (43)
m1
r reference radius in mean cone of wheel mm Eq. (44)
m2
S safety factor for scuffing — Eq. (100)
B
S load stage (in FZG test) — Eq. (99)
FZG
t contact exposure time of pinion �s Eq. (95)
t contact exposure time of wheel �s Eq. (96)
t contact exposure time at bend of curve �s Eq. (97)
c
t longest contact exposure time �s Eq. (95)
max
u gear ratio — Eq. (A.6)
u virtual ratio — Eq. (B.6)
v
v sliding velocity m/s Fig. 1
g
v tangential velocity of pinion m/s Eq. (3)
g1
v tangential velocity of wheel m/s Eq. (3)
g2
v sum of tangential velocities in pitch point m/s Eq. (25)
g�C
v pitch line velocity m/s Eq. (26)
t
w normal unit load N/mm Eq. (3)
Bn
w transverse unit load N/mm Eq. (5)
Bt
Table 1 — Symbols and units (continued)
Symbol Description Unit Reference
X buttressing factor — Eq. (54)
but
X buttressing value — Eq. (51)
butA
X buttressing value — Eq. (51)
butE
X geometry factor — Eq. (A.5)
G
X approach factor — Eq. (3)
J
X lubricant factor — Eq. (25)
L
–¾ –½ –½
X thermo-elastic factor K�N �s �m �mm Eq. (5)
M
X multiple mating pinion factor — Eq. (22)
mp
X roughness factor — Eq. (25)
R
X lubrication system factor — Eq. (22)
S
X structural factor — Eq. (94)
W
X angle factor — Eq. (A.6)
��
X load sharing factor — Eq. (3)
�
X gradient of the scuffing temperature — Eq. (97)
�
z number of teeth of pinion — Eq. (30)
z number of teeth of wheel — Eq. (30)
� transverse tip pressure angle of pinion ° Eq. (31)
a1
� transverse tip pressure angle of wheel ° Eq. (30)
a2
� transverse pressure angle ° Eq. (34)
t
� normal working pressure angle ° Eq. (A.2)
wn
� transverse working pressure angle ° Eq. (7)
wt
� pinion pressure angle at arbitrary point ° Eq. (29)
y1
� helix angle ° Eq. (18)
� base helix angle ° Eq. (49)
b
� base helix angle in midcone ° Eq. (50)
bm
� working helix angle ° Eq. (A.2)
w
� parameter on the line of action at point A — Eq. (24)
A
� parameter on the line of action at point AA — Eq. (68)
AA
� parameter on the line of action at point AB — Eq. (66)
AB
� parameter on the line of action at point AU — Eq. (49)
AU
� parameter on the line of action at point B — Eq. (31)
B
� parameter on the line of action at point BB — Eq. (70)
BB
� parameter on the line of action at point D — Eq. (32)
D
� parameter on the line of action at point DD — Eq. (72)
DD
� parameter on the line of action at point DE — Eq. (67)
DE
4 © ISO 2000 – All rights reserved

Table 1 — Symbols and units (continued)
Symbol Description Unit Reference
� parameter on the line of action at point E — Eq. (24)
E
� parameter on the line of action at point EE — Eq. (74)
EE
� parameter on the line of action at point EU — Eq. (49)
EU
� parameter on the line of action at point M — Eq. (86)
M
� parameter on the line of action at arbitrary point — Eq. (7)
y
� angle of direction of tangential velocity of pinion — Eq. (3)
� angle of direction of tangential velocity of wheel — Eq. (3)
� pitch cone angle of pinion ° Eq. (37)
� pitch cone angle of wheel ° Eq. (39)
� transverse contact ratio — Eq. (76)
�
� overlap ratio — Eq. (52)
�
� absolute (dynamic) viscosity at oil temperature mPa�s Eq. (27)
oil
� contact temperature °CEq.(1)
B
� maximum contact temperature °CEq.(2)
Bmax
� flash temperature K Eq. (1)
fl
� average flash temperature K Eq. (22)
flm
� maximum flash temperature K Eq. (2)
flmax
� maximum flash temperature at test K Eq. (94)
flmaxT
� bulk temperature °C Eq. (22)
M
� interfacial bulk temperature °CEq.(1)
Mi
� bulk temperature of pinion teeth °C Eq. (20)
M1
� bulk temperature of wheel teeth °C Eq. (20)
M2
� bulk temperature at test °C Eq. (94)
MT
� oil temperature before reaching the mesh °C Eq. (22)
oil
� scuffing temperature °C Eq. (94)
S
� scuffing temperature at long contact time °C Eq. (97)
Sc
� heat conductivity of pinion N/(s�K) Eq. (9)
M1
� heat conductivity of wheel N/(s�K) Eq. (10)
M2
� coefficient of friction in pin-and-ring test — Fig. 1
� mean coefficient of friction — Eq. (3)
m
� Poisson's ratio of pinion material — Eq. (A.10)
� Poisson's ratio of wheel material — Eq. (A.10)
Table 1 — Symbols and units (concluded)
Symbol Description Unit Reference
� density of pinion material kg/m³ Eq. (9)
M1
� density of wheel material kg/m³ Eq. (10)
M2
� relative radius of curvature at pitch point mm Eq. (25)
relC
� radius of curvature at arbitrary point of pinion mm Eq. (5)
y1
� radius of curvature at arbitrary point of wheel mm Eq. (5)
y2
� relative radius of curvature at arbitrary point mm Eq. (5)
yrel
� shaft angle ° Eq. (A.15)
� quill shaft twist ° Eq. (17)
a
The term wheel is used for the mating gear of a pinion.
4 Scuffing and wear
4.1 Occurrence of scuffing and wear
When gear teeth are completely separated by a full fluid film of lubricant, there is no contact between the asperities
of the tooth surfaces, and usually there is no scuffing or wear. Here, the coefficient of friction is rather low. In
exceptional cases a damage similar to scuffing may be caused by a sudden thermal instability [19] in a thick oil film,
which phenomenon is not treated here.
For thinner elastohydrodynamic films, incidental asperity contact takes place. As the mean film thickness
decreases, the number of contacts increases acccordingly. Abrasive wear, adhesive wear or scuffing becomes
possible. Abrasive wear may occur due to the rolling action of the gear teeth or the presence of abrasive particles in
the lubricant. Adhesive wear occurs by localized welding and subsequent detachment and transfer of particles from
one or both of the meshing teeth. Abrasive or adhesive wear may not be harmful if it is mild and if it subsides with
time, as in a normal run-in process.
In contrast to mild wear, scuffing is a severe form of adhesive wear that can result in progressive damage to the
gear teeth. In contrast to pitting and fatigue breakage which show a distinct incubation period, a short transient
overloading can result in scuffing failure.
Excessive aeration or the presence in the lubricant of contaminants such as metal particles in suspension, or water,
also increases the risk of scuffing damage. After scuffing, high speed gears tend to suffer high levels of dynamic
loading due to vibration which usually cause further damage by scuffing, pitting or tooth breakage.
)
In most cases, the resistance of gears to scuffing can be improved by using a lubricant with enhanced anti-scuff
additives. It is important, however, to be aware that some disadvantages attend the use of anti-scuff additives:
corrosion of copper, embrittlement of elastomers, lack of world-wide availability, etc.
The methods described are not suitable for "cold scuffing" which is in general associated with low speed, under
approx. 4 m/s, through-hardened heavily loaded gears of rather poor quality.
4.2 Transition diagram
The lubrication condition of sliding concentrated steel contacts, which operate in a liquid lubricant, can be described
[20][21][22][23] in terms of transition diagrams. A transition diagram according to Figure 1 is considered to be
applicable to contacts functioning at constant oil bath temperature.
2) The less correct designation Extreme Pressure, EP, is replaced by anti-scuff.
6 © ISO 2000 – All rights reserved

At combinations of normal force F and relative sliding velocity v which fall below the line A1-S, in region I, see
n g
Figure 1, the lubrication condition is characterized by a coefficient of friction of about 0,1 and a specific wear rate of
�2 3 �6 3
10 mm /(N�m) to 10 mm /(N�m) (i.e. volume wear per unit of normal force, per unit of sliding distance).
If, with v not above a value according to point S, the load is increased into region II, a transition into a second
g
condition of lubrication occurs. This mild wear lubrication condition is characterized by a coefficient of friction of
3 3
about 0,3 to 0,4 and a specific wear rate of 1 mm /(N�m) to5mm /(N�m).
Figure 1 — Transition diagram for contraform contacts with example of calculated contact temperatures
If load is increased still further, a transition into a third condition of lubrication, region III, occurs at intersection of the
line A2-S. This region is characterized by a coefficient of friction equal to 0,4 to 0,5. The wear rate, however, is
3 3
considerably higher, i.e. 100 mm /(N�m) to 1 000 mm /(N�m), than in regions I and II and the worn surfaces show
evidence of severe wear in the form of scuffing. If load increases at relative sliding velocities beyond point S, a
direct transition from region I to region III takes place.
There is strong evidence that the position of the line A1-S-A3 depends upon lubricant viscosity [24] as well as upon
Hertzian contact pressure [20][21]. At combinations of F and v that fall below this line, it is believed that the
n g
surfaces are kept apart by a thin lubricant film which is, however, penetrated by roughness asperities. In this
context, the term "partial elastohydrodynamic lubrication" has been used [21].
In region III liquid film effects are completely absent. This region is identical to the region of "incipient scuffing" [25].
There is evidence that the transition which occurs at intersecting the line A2-S is associated with reaching a critical
value of the contact temperature. This is the fundamental concept according to Blok.
The transition diagram shown is applicable to newly assembled, i.e. unoxidized steel contacts, as occur in gears,
cams and followers, etc. It has been found that the diagram is applicable to four-ball as well as to pin-and-ring test
results.
Along curve A1-S-A3 temperature ranges from an oil bath, respectively overall bulk, respectively interfacial bulk
temperature, of 28 °Cat v = 0,001 m/s to a contact temperature of 498 °Cat v = 10 m/s. This temperature
g g
behaviour strongly suggests that the collapse of (partial) elasto-hydrodynamical lubrication does not occur at a
constant contact or interfacial bulk temperature, for instance being associated with melting of chemisorbed material.
Instead, the pronounced decrease of load carrying capacity with increasing sliding velocity is supposed to be due to
decreasing viscosity [24][26][27][28][29].
Contrary to the above, calculated contact temperatures along curve A2-S-A3 tend to attain a constant value, e.g. in
the case of AISI 52100 steel specimens approximately 500 °C; see Figure 1. This suggests that the II-III transition is
associated with a transformation in the steel, causing the wear mechanism of surfaces to change from mildly
adhesive to severely adhesive, perhaps involving a mechanism of thermo-elastic instability [30][31].
Therefore, the results indicate scuffing is associated with a critical magnitude of the contact temperature. For steel,
lubricated with mineral oils, the critical magnitude does not depend on load, velocity and geometry, and equals near
500 °C.
4.3 Friction at incipient scuffing
As shown in the transition diagram, Figure 1, in the case of scuffing the coefficient of friction leaps from about 0,25
to about 0,5. The corresponding contact temperature proves to be about 500 °C. This contact temperature is the
sum of a measured interfacial bulk temperature of 28 °C and a calculated flash temperature of 470 °C. During the
flash temperature calculation use is made of the coefficient of friction just before transition, � = 0,35. If this method
has to be applied not only for pin-and-ring tests but also (during the design stage) for gear transmissions, one shall
agree upon the choice of the value of the critical magnitude of the contact temperature on one hand and the value
of the coefficient of friction to be used in the calculations on the other.
A gear load capacity can be predicted
� on the safe side, with the coefficient of friction � = 0,50;
� accurately, with the coefficient of friction between � = 0,25 and� = 0,35, dependent on the lubricant;
� according to previous practice, with a low coefficient of friction of regular working conditions, provided that the
limiting contact temperature is correspondingly low.
In terms of previous practice, for non-additive and low-additive mineral oils, each combination of oil and rolling
materials has a critical scuffing temperature which, in general, is constant regardless of the operating conditions,
load, velocity and geometry.
For high-additive and certain kinds of synthetic lubricants the critical scuffing temperature may well vary from one
set of operating conditions to another. So, this critical temperature must then be determined for each such set
separately from tests which closely simulate the operating condition of the gearset.
5 Basic formulae
5.1 Contact temperature
As already mentioned in the introduction, the contact temperature is the sum of the interfacial bulk temperature,
� , see 5.3, and the flash temperature,��� , see 5.2,
Mi fl
����� (1)
BMi fl
a
Position in the path of contact.
Figure 2 — Contact temperature along the path of contact
8 © ISO 2000 – All rights reserved

Only the flash temperature varies along the path of contact; see Figure 2.
The maximum contact temperature is
����� (2)
Bmax Mi flmax
where� is the maximum value of � , being located either at the approach path or at the recess path.
flmax fl
Prediction of the probability of scuffing is possible by comparing the calculated maximum contact temperature with
a critical magnitude. This critical magnitude of the contact temperature can be evaluated from any gear scuffing
test, or can be provided by field investigations.
For a reliable evaluation of the scuffing risk, it is important that an accurate value of the gear bulk temperature be
used for the analysis.
5.2 Flash temperature formula
The flash temperature formula of Blok [12][14][16][32] in a most general representation, for (approximately)
band-shaped contact and tangential velocities differently directed (as for hypoid gears), see annex A, reads
abs()vv�
���XX�w
g1 g2
mJ� Bn
���11, 1 � (3)
fl
()2� b Bv��(sin��)�B�(v�sin )
H M1 g1 12Mg22
For cylindrical or bevel gears, with band-shaped contact and parallel tangential velocities, the general
representation, see annex A, reads
abs()vv�
���XX � w
g1 g2
mJ� Bn
���11, 1 � (4)
fl
()2� b Bv��()B�v
H M1 g1 M2 g2
or, in an equivalent representation,
abs��� / u
ej
y1 y2
X n
F I
M 3 1
����25,(2 �� � ��� w)� � (5)
G J
fl m JB�t
H K
50 60 4
�
yrel
where
� is the mean coefficient of friction (see clause 6);
m
X is the thermo-elastic factor (see annex A);
M
–¾ ½ –½
X =50K�N �s �m �mm for commonly applied steel;
M
X is the approach factor (see clause 8);
J
X is the load sharing factor (see clause 9);
�
w is the transverse unit load (see 5.3), in N/mm;
Bt
n is the rotational speed of pinion, in r/min;
� is the local relative radius of curvature, in mm:
yrel
� ��
y1 y2
� � (6)
yrel
���
y1 y2
� is the local radius of curvature of pinion flank, in mm:
y1
1��
y
��� ��a sin (cylindrical gears) (7)
y1 wt
1� u
� is the local radius of curvature of wheel flank, in mm:
y2
u��
y
��� ��a sin (cylindrical gears) (8)
y2 wt
1� u
For bevel gears, see equations (37) and (38).�
For an adapted representation, see annex A.
Two Péclet numbers have to be sufficiently high, which is satisfied in almost all cases where scuffing may occur.
For lower Péclet numbers the heat flow from the contact band into the gear teeth causes a different temperature
distribution for which formulae (3) to (6) are not valid.
vb��� �c
g1 H M1 M1
Pe � � 5 (9)
��� sin
M1 1
vb��� �c
g2 H M2 M2
Pe � � 5 (10)
��� sin
M2 2
where
� is the density of pinion material, in kg/m ;
M1
� is the density of wheel material, in kg/m ;
M2
c is the specific heat per unit mass of pinion, in J/(kg�K);
M1
c is the specific heat per unit mass of wheel, in J/(kg�K);
M2
� is the heat conductivity of pinion, in N/(s�K);
M1
� is the heat conductivity of wheel, in N/(s�K).
M2
For cylindrical and bevel gears, sin� =sin� =1.
1 2
5.3 Transverse unit load
The transverse unit load for cylindrical gears and bevel gears is
F
t
wK��K�K �K �K � (cylindrical gears) (11)
Bt A v B� Bm� p
b
F
t
wK��K�K �K �K � (bevel gears) (12)
Bt A v B� Bm� p
b
eff
where
F is the nominal tangential force on pitch circle, in N;
t
b is the facewidth, in mm;
b =0,85b (13)
eff
10 © ISO 2000 – All rights reserved

K is the application factor (see ISO 6336-1 for cylindrical gears, ISO 10300-1 for bevel gears);
A
K is the dynamic factor (see ISO 6336-1 for cylindrical gears, ISO 10300-1 for bevel gears);
v
K is the face load factor;
B�
K = K (see ISO 6336-1 for cylindrical gears, ISO 10300-1 for bevel gears); (14)
B� H�
K is the transverse load factor:
B�
K = K (see ISO 6336-1 for cylindrical gears, ISO 10300-1 for bevel gears); (15)
B� H�
K is the multiple-path factor:
mp
The multiple-path factor K accounts for the maldistribution in multiple-path transmissions depending on accuracy
mp
and flexibility of the branches. If no relevant analysis is available, the following may apply.
� for epicyclical gear trains with n planets (n W 3)
p p
Kn��10,25 �3 (16)
mp p
� for dual tandem gears with quill shaft twist �� degrees under full load
K ��10(,2/�) (17)
mp
� for double helical gears with an external axial force F
ex
F
ex
K ��1 (18)
mp
F � tan�
t
� for other cases
K =1 (19)
mp
5.4 Distribution of overall bulk temperatures
The friction loss most typical of gear transmissions is the one caused by the meshing zone. In this source the heat
is generated mainly by tooth friction. The mechanical "pumping" energy expended for sideways expulsion of
superfluous oil may sometimes be far from negligible. The other unavoidable friction loss is from the bearings,
either of the rolling or the sliding type. In high speed gear transmissions, sliding bearings may well generate much
more frictional heat than gears. Other heat sources are oil churning and friction from seals. All the above heat
sources have the following features in common:
� in each of these sources the fluid friction depends on some oil viscosity representative of the operating
condition;
� all of the heat sources are thermally interconnected through transmission elements to the sinks, such as the
ambient air or the cooling system.
The thermal interconnection allows calculation concepts such as:
� finite element methods for discrete components;
� bondgraph methods,
� thermal network analogue methods [18].
The interfacial bulk temperature� may be suitably averaged from the two overall bulk temperatures of the teeth in
Mi
contact, � and � . The following formula is valid to a good approximation (at high values of the Péclet
M1 M2
numbers):
Bv�����B�v�
M1 g1 M1 M2 g2 M2
� � (20)
Mi
Bv��B�v
M1 g1 M2 g2
�
B v
M1 g1
In a fairly wide range of the ratio a simple arithmetic average is valid to a reasonable approximation
�
Bv
M2 g2
���� �� (21)
bg
Mi M1 M2
Bulk temperatures in excess of 150 °C for long periods may have an adverse effect on the surface durability.
5.5 Rough approximation of a bulk temperature
For very rough inquiry the bulk temperature may be estimated by the sum of the oil temperature, taking into account
some impediment in heat transfer for spray lubrication, and a part which depends mainly on the flash temperature,
of which the maximum value is taken.
��= + 0,47��XX �� (22)
Moil mp flm
S
where�
X = 1,2 for spray lubrication;
S
X = 1,0 for dip lubrication;
S
X = 1,0 for meshes with additional spray for cooling purpose;
S
X = 0,2 for gears submerged in oil, provided sufficient cooling;
S
1� n
p
X � for a pinion with n mating gears; (23)
p
mp
� is the average of flash temperature along path of contact, in °C:
flm
E
��� d
fl y
z
A
� � (24)
flm
���
EA
However, for a reliable evaluation of the scuffing risk, it is important that instead of a rough approximation, an
accurate value of the gear bulk temperature be used for the analysis.
6 Coefficient of friction
Several factors influencing the friction between gear teeth vary throughout a meshing cycle. On one of the two
mating tooth faces the relative motion is uniformly accelerating, on the other it is uniformly decelerating. Only at
pitch point position pure rolling occurs. In any other meshing position combined rolling and sliding will occur. Also
the load acting on two mating tooth faces will vary from one meshing position to another. These conditions cause a
continuous variation of the film thickness, the lubrication regime and the coefficient of friction. Even in a similar
meshing position the coefficient of friction may vary for different teeth and different time.
12 © ISO 2000 – All rights reserved

The local coefficient of friction is considered to be a representative quantity valid for the local point concerned,
smoothing various influences. The geometrically determined variation of the local coefficient of friction is difficult to
calculate or to measure, hence instead of a local value, a representative mean value of the coefficient of friction will
be applied.�
A mean value (along the path of contact) of the coefficient of friction has commonly been applied, and even that
value is uncertain. Too often, in test reports on friction, important influential quantities were neglected, for instance
the bulk temperature which determines the inlet viscosity and therefore the lubrication regime.
3)
The mean coefficient of friction � depends on the geometry of the path of contact, the tangential velocities, the
m
normal load, the inlet viscosity (which is identical with viscosity at teeth bulk temperature), the pressure-viscosity
coefficient, the reduced modulus of elasticity, the surface roughness, the normal relative radius of curvature.
Depending on further investigations, other quantities and influences may have to be accounted for, either in the
formula or in the description of the field of application. The number of quantities may be reduced by dimension
analysis [33], and a possible neglect of some minor influential quantities.
The coefficient of friction may be measured or estimated according to various methods. The limiting contact
temperature shall be chosen correspondingly to the coefficient of friction.
6.1 Mean coefficient of friction, method A
The coefficient of friction at the onset of scuffing may be measured in gear tests or pin-and-ring tests. The limiting
contact temperature is correspondingly high.
6.2 Mean coefficient of friction, method B
According to previous practice, whereby low coefficients of friction of regular working conditions are used,
...

RAPPORT ISO/TR
TECHNIQUE 13989-1
Première édition
2000-03-15
Calcul de la capacité de charge au grippage
des engrenages cylindriques, coniques et
hypoïdes —
Partie 1:
Méthode de la température-éclair
Calculation of scuffing load capacity of cylindrical, bevel and hypoid
gears —
Part 1: Flash temperature method
Numéro de référence
©
ISO 2000
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ii © ISO 2000 – Tous droits réservés

Sommaire Page
Avant-propos.iv
Introduction.vi
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 Grippage et usure .6
4.1 Apparition du grippage et de l'usure .6
4.2 Diagramme de transition.7
4.3 Frottement à l'amorçage du grippage.8
5 Formules de base .9
5.1 Température de contact .9
5.2 Formule de la température-éclair .9
5.3 Charge unitaire apparente .11
5.4 Répartition des températures de masse globales.12
5.5 Approximation grossière de la température de masse.13
6 Coefficient de frottement .13
6.1 Coefficient de frottement moyen, méthode A.14
6.2 Coefficient de frottement moyen, méthode B.14
6.3 Coefficient de frottement moyen, méthode C.14
7 Paramètre sur la ligne d'action.15
8 Facteur d'approche.17
9 Facteur de répartition de charge.18
9.1 Facteur de contrefort.18
9.2 Engrenages à denture droite à profils non corrigés .19
9.3 Engrenages à denture droite à profils corrigés.20
9.4 Engrenages à denture hélicoïdale étroits à profils non corrigés.21
9.5 Engrenages à denture hélicoïdale étroits à profils corrigés .21
9.6 Engrenages à denture hélicoïdale larges à profils non corrigés.22
9.7 Engrenages à denture hélicoïdale larges à profils corrigés.22
9.8 Engrenages coniques étroits.23
9.9 Engrenages coniques larges.24
10 Température de grippage et sécurité.25
10.1 Température de grippage.25
10.2 Facteur de structure .25
10.3 Durée de contact.26
10.4 Température de grippage dans les essais d'engrenage.27
10.5 Domaine de sécurité.27
Annexe A (informative) Présentation de la formule de la température-éclair.29
Annexe B (informative) Correction de profil optimale .36
Bibliographie .38
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'ISO/TR 13989-1, rapport technique du type 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 la capacité de charge au grippage des engrenages 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. En 1975, deux méthodes de calcul pour évaluer le risque de grippage ont été documentées pour être
étudiées par le comité technique ISO/TC 60. Il a été admis qu'après une période d'expérimentation, une seule
méthode doit être adoptée. Étant donné que le sujet est encore en développement technique et qu'il y a une
possibilité future d'un accord en tant que Norme internationale, la publication en tant que Rapport technique de
type2a été proposée.
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.
iv © ISO 2000 – Tous droits réservés

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.
Introduction
Depuis 1990, la méthode de la température-éclair, présentée dans la présente partie de l’ISO/TR 13989, a été
enrichie par des recherches sur les temps de contact de courte durée, sur la prise en compte des diagrammes de
transition, sur de nouvelles approximations sur le coefficient de frottement et sur un renouvellement complet des
facteurs de répartition de charge. En 1991, le Professeur Blok a apporté une extension de la formulation de la
température-éclair, la rendant directement applicable aux engrenages hypoïdes.
La méthode de la température intégrale, présentée dans l’ISO/TR 13989-2, moyenne la température-éclair et ajoute
des facteurs d'influence empiriques au facteur de répartition de charge. Les valeurs résultantes arrondissent la
température maximale de contact, donnant alors à peu de chose prèsla même évaluation du risque de grippage
que la méthode de la température-éclair delaprésente partie de l’ISO/TR 13989. La méthode de la température
intégrale est moins sensible dans les cas présentant des pics de température localisés, habituellement dans les
ensembles d'engrenages qui ont des faibles rapports de conduite ou qui présentent des contacts au voisinage du
cercle de base ou des géométries sensibles.
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. Par opposition au
développement relativement long de la détérioration par fatigue, une surcharge instantanée unique peut initier la
détérioration par grippage avec une telle sévérité que l'engrenage ne pourra être utilisé plus longtemps. D'après
Blok [12][13][14][15][16][17], des températures de contact élevées du lubrifiant et des surfaces de denture au point
de contact instantané peuvent entraîner une rupture du film de lubrifiant à l'interface du contact.
La température de contact à l'interface résulte de la somme de deux composantes:
� la température de masse de l'interface en mouvement, qui, si elle varie, le fait comparativement lentement.
Pour évaluer cette composante, elle peut être moyennée à partir des deux températures de masse des deux
dentures frottantes. Ces deux dernières températures de masse se déduisent de la théorie des réseaux
thermiques [18];
� la fluctuation rapide de la température-éclair des surfaces en contact en mouvement. Une attention toute
particulière doit être apportée au coefficient de frottement. La pratique habituelle est d'utiliser un coefficient de
frottement valide pour des conditions de fonctionnement normales, bien qu'il soit établi qu'au commencement
du grippage le coefficient de frottement atteint des valeurs plus élevées.
Les relations complexes entre les phénomènes mécaniques, hydrodynamiques, thermodynamiques et chimiques
furent l'objet d'importantes recherches et expérimentations, qui peuvent induire différents facteurs d'influence
empiriques. Une suppléance directe des facteurs d'influence empiriques peut renforcer les paramètres fonctionnels
associés dans la formule de base et les fixer à des valeurs moyennes. Cependant, un traitement correct des
paramètres fonctionnels (c'est-à-dire coefficient de frottement, facteur de répartition de charge, coefficient
thermique de contact) garde la formule principale intacte, ce qui est confirmé avec l'expérimentation et la pratique.
À côté de la température maximale de contact, l'évolution de la température de contact le long de la ligne d'action
fournit l'information nécessaire pour la conception de l'engrenage.
vi © ISO 2000 – Tous droits réservés

RAPPORT TECHNIQUE ISO/TR 13989-1:2000(F)
Calcul de la capacité de charge au grippage des engrenages
cylindriques, coniques et hypoïdes —
Partie 1:
Méthode de la température-éclair
1 Domaine d’application
La présente partie de l’ISO/TR 13989 spécifie les méthodes et les formules pour l'évaluation des risques de
grippage, en se basant sur le concept de la température de contact de Blok.
Le concept fondamental selon Blok est applicable à tous les éléments de machine ayant des zones de contact
mobiles. Les formules de température-éclair sont valables pour une zone de contact hertzien en forme de bande ou
quasiment en forme de bande et pour des conditions de fonctionnement caractérisées par des nombres de Péclet
suffisamment élevés.
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 1122-1:1998, Vocabulaire des engrenages — Partie 1: Définitions géométriques.
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.
ISO 10825:1995, Engrenages — Usure et défauts des dentures — Terminologie.
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 et
l’ISO 10825 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. Les unitésde
longueur mètre, millimètre et micromètre sont choisies conformément à l'usage en la matière. Pour obtenir un
système cohérent, les unités pour B , c�, X sont adaptées à l'application combinéedemètre et millimètreoude
M M
millimètre et micromètre.
Tableau 1 — Symboles et unités
Symbole Description Unité Référence
a entraxe mm Éq. (A.5)
largeur de denture, plus petite valeur du pignon ou de la
b mm Éq. (11)
a
roue
b
largeur de denture effective mm Éq. (12)
eff
b
demi-largeur de la bande de contact hertzien mm Éq. (3)
H
½ ½ ½
B
coefficient de contact thermique Éq. (A.13)
M
N/(mm �m �s �K)
½ ½ ½
B
coefficient de contact thermique du pignon Éq. (3)
M1
N/(mm �m �s �K)
½ ½ ½
B
coefficient de contact thermique de la roue Éq. (3)
M2
N/(mm �m �s �K)
C
dépouille de tête du pignon �m Éq. (48)
a1
C dépouille de tête de la roue Éq. (46)
�m
a2
C dépouille de tête optimale Éq. (46)
�m
eff
C
dépouille de tête équivalente du pignon Éq. (B.2)
�m
eq1
C
dépouille de tête équivalente de la roue �m Éq. (B.3)
eq2
C
dépouille de pied du pignon �m Éq. (B.3)
f1
C
dépouille de pied de la roue �m Éq. (B.2)
f2
c
chaleur spécifique par unité de masse du pignon J/(kg�K) Éq. (9)
M1
c
chaleur spécifique par unité de masse de la roue J/(kg�K) Éq. (10)
M2
raideur d'engrènement (Éq. B.1)
c� N/(mm��m)
d
diamètrederéférence du pignon mm Éq. (34)
d
diamètrederéférencedelaroue mm Éq. (35)
d
diamètredetête du pignon mm Éq. (34)
a1
d
diamètredetête de la roue mm Éq. (35)
a2
E
module d'élasticité du pignon N/mm Éq. (A.10)
E
module d'élasticité de la roue N/mm Éq. (A.10)
E module d'élasticité réduit N/mm Éq. (A.9)
r
F force axiale externe N Éq. (18)
ex
F
charge réelle de l'essai d'usure N Figure. 1
n
F
force tangentielle nominale N Éq. (11)
t
2 © ISO 2000 – Tous droits réservés

Tableau 1 — Symboles et unités (suite)
Symbole Description Unité Référence
H
dimension auxiliaire mm Éq. (B.3)
H
dimension auxiliaire mm Éq. (B.2)
h saillie de denture au cône moyen du pignon mm Éq. (43)
am1
h saillie de denture au cônemoyendelaroue mm Éq. (44)
am2
K
facteur d'application —Éq. (11)
A
K
facteur de distribution transversale de la charge (grippage) —Éq. (11)
B�
K
facteur de distribution longitudinale de la charge (grippage) —Éq. (11)
B�
facteur de distribution transversale de la charge
K
—Éq. (15)
H�
(pression de contact)
facteur de distribution longitudinale de la charge
K
—Éq. (14)
H�
(pression de contact)
K
facteur d'engrènement multiple —Éq. (11)
mp
K
facteur dynamique —Éq. (11)
v
m
module normal mm Éq. (B.2)
n
n
vitesse de rotation par minute du pignon r/min Éq. (5)
n nombre de contacts d'engrènement —Éq. (16)
p
Pe nombre de Péclet du matériau du pignon —Éq. (9)
Pe
nombre de Péclet du matériau de la roue —Éq. (10)
Q
classe de précision —Éq. (57)
Ra
rugosité de surface du flanc de dent du pignon �m Éq. (28)
Ra
rugosité de surface du flanc de dent de la roue �m Éq. (28)
R
génératriceducône complémentaire moyen mm Éq. (A.16)
m
r rayonderéférenceducône moyen du pignon mm Éq. (43)
m1
r rayonderéférenceducônemoyendelaroue mm Éq. (44)
m2
S
coefficient de sécurité relatif au grippage —Éq. (100)
B
S
niveau de charge (en essai FZG) —Éq. (99)
FZG
t
durée de contact sur le pignon �s Éq. (95)
t
durée de contact sur la roue �s Éq. (96)
t
durée de contact au coude de la courbe �s Éq. (97)
c
t
durée de contact la plus longue �s Éq. (95)
max
u rapport d'engrenage —Éq. (A.6)
u
rapport équivalent —Éq. (B.6)
v
v
vitesse de glissement m/s Figure 1
g
v
vitesse tangentielle du pignon m/s Éq. (3)
g1
Tableau 1 — Symboles et unités (suite)
Symbole Description Unité Référence
v
vitesse tangentielle de la roue m/s Éq. (3)
g2
v
somme des vitesses tangentielles au point primitif m/s Éq. (25)
g�C
v
vitesse tangentielle au primitif de fonctionnement m/s Éq. (26)
t
w
charge unitaire normale N/mm Éq. (3)
Bn
w
charge unitaire apparente N/mm Éq. (5)
Bt
X
facteur de contrefort —Éq. (54)
but
X
valeur de contrefort —Éq. (51)
butA
X valeur de contrefort —Éq. (51)
butE
X facteur géométrique —Éq. (A.5)
G
X facteur d'approche —Éq. (3)
J
X facteur lubrifiant —Éq. (25)
L
–¾ –½ –½
X
facteur thermoélastique Éq. (5)
M
K�N �s �m �mm
X
facteur de pignons conjugués multiples —Éq. (22)
mp
X facteur de rugosité—Éq. (25)
R
X facteur système de lubrification —Éq. (22)
S
X facteur de structure —Éq. (94)
W
X facteur d'angle —Éq. (A.6)
��
X
facteur de répartition de charge —Éq. (3)
�
X
gradient de la température de grippage —Éq. (97)
�
z nombre de dents du pignon —Éq. (30)
z nombre de dents de la roue —Éq. (30)
�
angle de pression de tête apparent du pignon °Éq. (31)
a1
� angle de pression de tête apparent de la roue °Éq. (30)
a2
angle de pression apparent °Éq. (34)
�
t
� angle de pression de fonctionnement normal °Éq. (A.2)
wn
�
angle de pression de fonctionnement apparent °Éq. (7)
wt
�
angle d'incidence du pignon en un point quelconque °Éq. (29)
y1
� angle d'hélice °Éq. (18)
�
angle d'hélice de base °Éq. (49)
b
�
angle d'hélicedebaseaudemi-cône °Éq. (50)
bm
� angle d'hélice de fonctionnement °Éq. (A.2)
w
�
paramètre sur la ligne d'action au point A —Éq. (24)
A
� paramètre sur la ligne d'action au point AA —Éq. (68)
AA
4 © ISO 2000 – Tous droits réservés

Tableau 1 — Symboles et unités (suite)
Symbole Description Unité Référence
�
paramètre sur la ligne d'action au point AB —Éq. (66)
AB
� paramètre sur la ligne d'action au point AU —Éq. (49)
AU
�
paramètre sur la ligne d'action au point B —Éq. (31)
B
� paramètre sur la ligne d'action au point BB —Éq. (70)
BB
� paramètre sur la ligne d'action au point D —Éq. (32)
D
�
paramètre sur la ligne d'action au point DD —Éq. (72)
DD
� paramètre sur la ligne d'action au point DE —Éq. (67)
DE
� paramètre sur la ligne d'action au point E —Éq. (24)
E
�
paramètre sur la ligne d'action au point EE —Éq. (74)
EE
�
paramètre sur la ligne d'action au point EU —Éq. (49)
EU
� paramètre sur la ligne d'action au point M —Éq. (86)
M
�
paramètre sur la ligne d'action en un point arbitraire —Éq. (7)
y
�
angle de direction de la vitesse tangentielle du pignon —Éq. (3)
� angle de direction de la vitesse tangentielle de la roue —Éq. (3)
�
angle du cône primitif de fonctionnement du pignon °Éq. (37)
�
angle du cône primitif de fonctionnement de la roue °Éq. (39)
�
rapport de conduite apparent —Éq. (76)
�
�
rapport de recouvrement —Éq. (52)
�
�
viscosité absolue (dynamique) à la température de l'huile mPa�s Éq. (27)
oil
�
température de contact °C Éq. (1)
B
� température de contact maximale °C Éq. (2)
Bmax
�
température-éclair K Éq. (1)
fl
�
température-éclair moyenne K Éq. (22)
flm
� température-éclair maximale K Éq. (2)
flmax
�
température-éclair maximale en cours d'essai K Éq. (94)
flmaxT
�
température de masse °C Éq. (22)
M
� température de masse interfaciale °C Éq. (1)
Mi
�
température de masse des dents du pignon °C Éq. (20)
M1
�
température de masse des dents de la roue °C Éq. (20)
M2
�
température de masse en cours d'essai °C Éq. (94)
MT
� température d'huile avant d'atteindre l'engrènement °C Éq. (22)
oil
Tableau 1 — Symboles et unités (suite)
Symbole Description Unité Référence
�
température de grippage °C Éq. (94)
S
� température de grippage pour une durée de contact longue °C Éq. (97)
Sc
�
conductivité thermique du pignon N/(s�K) Éq. (9)
M1
�
conductivité thermique de la roue N/(s�K) Éq. (10)
M2
� coefficient de frottement dans l'essai pion-disque — Figure 1
�
coefficient de frottement moyen —Éq. (3)
m
�
coefficient de Poisson du matériau du pignon —Éq. (A.10)
� coefficient de Poisson du matériau de la roue —Éq. (A.10)
�
densité du matériau du pignon kg/m Éq. (9)
M1
� densité du matériau de la roue kg/m Éq. (10)
M2
� rayon de courbure relatif au point primitif mm Éq. (25)
relC
�
rayon de courbure en un point quelconque du pignon mm Éq. (5)
y1
� rayon de courbure en un point quelconque de la roue mm Éq. (5)
y2
� rayon de courbure relatif en un point quelconque mm Éq. (5)
yrel
� angle des axes °Éq. (A.15)
� torsion d'arbre torsible °Éq. (17)
a
Le terme roue est utilisé pour le mobile conjugué d'un pignon.
4 Grippage et usure
4.1 Apparition du grippage et de l'usure
Lorsque les dents d'engrenage sont entièrement séparées par un film fluide complet de lubrifiant, il n'y a pas de
contact entre les aspérités de surface des dents et, habituellement, il n'y a pas de grippage ou d'usure. Dans ce
cas, le coefficient de frottement est plutôt faible. Dans des cas exceptionnels, une détérioration semblable au
grippage peut être provoquée par une instabilité thermique soudaine [19] dans un film d'huile épais, mais ce
phénomène n'est pas traité dans la présente partie de l’ISO/TR 13989.
Pour des films élastohydrodynamiques plus minces, il y a contact fortuit des aspérités. Au fur et à mesure que
l'épaisseur moyenne du film décroît, le nombre de contacts augmente. L'usure par abrasion, l'usure par micro-
soudage ou le grippage deviennent alors possibles. L'usure par abrasion peut apparaîtredufaitdel'actionde
roulement des dents d'engrenage ou du fait de la présence de particules abrasives dans le lubrifiant. L'usure par
adhésion est due à une soudure par fusion locale suivie d'un arrachement et d'un transfert des particules de l'une
ou des deux dents en prise. L'usure abrasive ou par adhésion peut ne pas être nuisible si elle est modérée et si elle
s'atténue avec le temps, comme lors d'un processus normal de rodage.
Contrairement à l'usure modérée, le grippage est une forme grave d'usure par adhésion qui peut entraîner une
détérioration progressive des dents des roues. Contrairement à la formation de piqûres et à la rupture de fatigue qui
présentent une période d'incubation, une surcharge provisoire de courte durée peut entraîner une défaillance par
grippage.
Une aération excessive ou la présence de contaminants dans le lubrifiant, tels que des particules métalliques en
suspension ou de l'eau, augmente également le risque de détérioration par grippage. Après grippage, les
engrenages à grande vitesse sont soumis à des charges dynamiques élevées produites par des vibrations qui
conduisent généralement à une détérioration ultérieure par grippage, pitting ou rupture de dent.
6 © ISO 2000 – Tous droits réservés

Dans la plupart des cas, la résistance des engrenages au grippage peut être améliorée en utilisant un lubrifiant
2�
enrichi d'additifs antigrippage . Il est cependant important de noter que l'utilisation de ces additifs antigrippage
comporte certains inconvénients: corrosion du cuivre, fragilisation des élastomères, difficulté d'approvisionnement,
etc.
Les méthodes décrites dans la présente partie de l’ISO/TR 13989 ne s'appliquent pas au «grippage à froid», qui est
en général associéà des engrenages traités dans la masse, lourdement chargés, de précision plutôtmédiocre et
travaillant à faible vitesse, inférieure à environ 4 m/s.
4.2 Diagramme de transition
Les conditions de lubrification de contacts concentrésglissantsde pièces en acier, qui fonctionnent dans un
lubrifiant liquide, peuvent être décrites [20] [21] [22] [23] en termes de diagrammes de transition. Un diagramme de
transition conforme à la Figure 1 est considéré applicable aux contacts fonctionnant à des températures de bain
d'huile constante.
Pour des combinaisons de force réelle F et de vitesse de glissement relative v situées au-dessous de la
n g
ligne A1-S, dans la région I, voir la Figure 1, les conditions de lubrification sont caractérisées par un coefficient de
�2 3 �6 3
frottement de l'ordre de 0,1 et d'un taux d'usure spécifique de 10 �mm /(N�m) à 10 mm /(N�m) (c'est-à-dire une
usure volumique par unité de force normale, par unité de distance de glissement).
Si, avec v n'étant pas supérieure à une valeur correspondant au point S, on augmente la charge dans la région II, il
g
y a transition dans un second état de lubrification. Cette condition de lubrification à usure modérée est caractérisée
3 3
par un coefficient de frottement d'environ 0,3 à 0,4 et d'un taux d'usure spécifique de 1 mm /(N�m) à5mm /(N�m).
Figure 1 — Diagramme de transition pour des contacts de contre-forme avec exemple de températures de
contact calculées
Si la charge augmente encore, il y a transition en un troisième état de lubrification, région III, à l'intersection de la
ligne A2-S. Cette région est caractérisée par un coefficient de frottement de 0,4 à 0,5. Le taux d'usure est
3 3
cependant beaucoup plus élevé que dans les régions I et II, c'est-à-dire de 100 mm /(N�m) à 1 000 mm /(N�m) et
lessurfacesusées révèlent des traces d'usure sévères sous la forme de grippage. Si la charge augmente à des
vitesses de glissement relatives au-delà du point S, il y a transition directe de la région I à la région III.
2� L'appellation moins correcte Extrême Pression, EP, est remplacée par anti-grippage.
Il est prouvé que la position de la ligne A1-S-A3 dépend de la viscosité du lubrifiant [24] ainsi que de la pression de
contact hertzienne [20][21]. Pour des combinaisons de F et v s'inscrivant sous cette ligne, on considère que les
n g
surfaces sont séparées par un mince film de lubrifiant, qui est cependant traversé par les aspérités de rugosité.
Dans ce contexte, le terme «lubrification élastohydrodynamique partielle» est utilisé [21].
La région III présente une absence totale d'effets de film liquide. Cette région est identique à la région d'«amorce de
grippage» [25]. Il est prouvé que la transition qui apparaît à l'intersection de ligne A2-S est associée à l'atteinte
d'une valeur critique de la température de contact. Cela est le concept fondamental selon Blok.
Le diagramme de transition présenté ci-avant est applicable à des contacts acier fraîchement employés, c'est-à-
dire non oxydés, tels qu'ils se présentent dans les engrenages, cames et galets, etc. Le diagramme s'est révélé
applicable aux résultats d'essai quatre-billes et pion-disque.
Sur la courbe A1-S-A3, la température s'étend d'une température de bain d'huile, respectivement une température
de masse globale, une température de masse interfaciale, de 28 °C à v = 0,001 m/s à une température de contact
g
de 498 °C à v = 10 m/s. Ce comportement thermique suggère fortement qu'une dégradation de la lubrification
g
élastohydrodynamique (partielle) n'apparaît pas à une température de masse interfaciale ou à une température de
contact constante lorsqu'elle est associée par exemple à la fusion d'un matériau à absorption chimique. Par contre,
la réduction prononcée de la capacité de charge, au fur et à mesure de l'augmentation de la vitesse de glissement,
est supposée être due à une diminution de la viscosité [24][26][27][28][29].
Au contraire, les températures de contact calculées sur la courbe A2-S-A3 ont tendance à atteindre une valeur
constante, par exemple dans le cas d'éprouvettes d'acier AISI 52100, environ 500 °C; voir la Figure 1. Cela suggère
que la transition II-III est associée à une transformation de l'acier qui entraîne une modification du mécanisme
d'usure des surfaces, qui, de adhésive modérée devient adhésive sévère, et implique probablement un mécanisme
d'instabilité thermoélastique [30][31].
Par conséquent, les résultats indiquent que le grippage est associéà une valeur critique de la température de
contact. Pour l'acier lubrifiéà l'huile minérale, la valeur critique ne dépend ni de la charge ni de la vitesse ni de la
géométrie et est voisine de 500 °C.
4.3 Frottement à l'amorçage du grippage
Comme illustré dans le diagramme de transition, Figure 1, en cas de grippage, le coefficient de frottement passe de
0,25 à environ 0,5. La température de contact correspondante est d'environ 500 °C. Cette température de contact
est la somme d'une température de masse interfaciale mesuréede28 °C et d'une température-éclair calculéede
470 °C. Lors du calcul de la température-éclair, on utilise le coefficient de frottement juste avant transition, � = 0,35.
Si cette méthode doit être appliquée non seulement aux essais pion-disque, mais également (lors de l'étape de
conception) aux transmissions par engrenages, il faut convenir du choix de la valeur critique de la température de
contact d'une part, et de la valeur du coefficient de frottement à utiliser dans les calculs d'autre part.
Il est possible de prédire une capacité de charge de l'engrenage
� conservative, avec le coefficient de frottement� =0,50;
� précise, avec le coefficient de frottement compris entre� =0,25et� = 0,35 selon le lubrifiant ;
� selonlaméthode précédente, avec un faible coefficient de frottement dans des conditions de fonctionnement
normales, à condition que la température de contact limite soit par conséquent faible.
Selonlaméthode précédente, pour les huiles minérales sans additifs et à faible teneur en additifs, toute
combinaison d'huiles et de matériaux de roulement à une température de grippage critique qui, en général, est
constante quelles que soient les conditions de fonctionnement, de charge, de vitesse et de géométrie.
Pour les lubrifiants à haute teneur en additifs et certains types de lubrifiants synthétiques, il est admis que la
température de grippage critique varie d'un ensemble de conditions de fonctionnement à l'autre. Ainsi, cette
température critique doit alors être déterminéeséparément pour chaque ensemble de conditions de
fonctionnement, sur la base d'essais qui simulent étroitement les conditions de fonctionnement de l'engrenage.
8 © ISO 2000 – Tous droits réservés

5 Formulesdebase
5.1 Température de contact
Comme indiqué dans l’introduction, la température de contact est la somme de la température de masse
interfaciale,� ,voir 5.4,etdelatempérature-éclair� ,voir 5.2.
Mi fl
��= +� (1)
BMi fl
Seule la température-éclair varie sur la ligne de conduite, voir la Figure 2.
La température de contact maximale est
����� (2)
Bmax Mi flmax
où� est la valeur maximale de� , localisée sur le segment d'approche ou de retrait.
flmax fl
Il est possible de prédire la probabilité de grippage en comparant la température de contact maximale calculée à
une valeur critique. Cette valeur critique de la température de contact peut être évaluée par un éventuel essai de
grippage de l'engrenage, ou peut être obtenue par des analyses sur des applications.
Pour une évaluation fiable du risque de grippage, il est important d'utiliser dans l'analyse une valeur précise de la
température de masse de l'engrenage.
a
Position sur la ligne de conduite
Figure 2 — Température de contact sur la ligne de conduite
5.2 Formule de la température-éclair
La formuledelatempérature-éclair de Blok [12][14][16][32] dans la formulation la plus générale, pour des contacts
(approximativement) en forme de bande, et des vitesses tangentielles ayant des directions différentes (comme pour
les engrenages hypoïdes), voir annexe A, s'écrit
abs()vv�
���XX�w
g1 g2
mJ� Bn
���11, 1 � (3)
fl
()2� b Bv��(sin��)�B�(v�sin )
H M1 g1 12Mg22
Pour des engrenages cylindriques ou coniques, avec des contacts en forme de bande et des vitesses tangentielles
parallèles, la formulation générale, voir annexe A, s'écrit
abs()vv�
���XX � w
g1 g2
mJ� Bn
���11, 1 � (4)
fl
()2� b Bv��()B�v
H M1 g1 M2 g2
ou dans une formulation équivalente
abs��� / u
ej
y1 y2
X n
F I
M 3 1
����25,(2 �� � ��� w)� � (5)
G J
fl m JB�t
H K
50 60 4
�
yrel
où
� est le coefficient de frottement moyen, voir l’article 6;
m
X est le facteur thermoélastique, voir l’annexe A;
M
–3/4 1/2 –1/2
X =50K�N �s �m �mm pour acier d'usage général;
M
X est le facteur d'approche, voir l'article 8;
J
X est le facteur de répartition de charge, voir l'article 9;
�
w est la charge unitaire apparente (voir 5.3), en newtons par millimètre (N/mm);
Bt
n est la vitesse de rotation du pignon, en tours par minute (tr/min);
� est le rayon de courbure relatif local, en millimètres (mm);
yrel
� ��
y1 y2
� � (6)
yrel
���
y1 y2
� est le rayon de courbure local du flanc du pignon, en millimètres (mm):
y1
1��
y
��� ��a sin (engrenages cylindriques) (7)
wt
y1
1� u
� est le rayon de courbure local du flanc de la roue, en millimètres (mm):
y2
u��
y
��� ��a sin (engrenages cylindriques) (8)
y2 wt
1� u
Pour les engrenages coniques, voir les équations (37) et (38).
Pour une formutation adaptée, voir l'annexe A.
Les deux nombres de Péclet doivent être suffisamment élevés, ce qui se produit dans la plupart des cas lorsqu'il y
a risque de grippage. Pour des nombres de Péclet plus faibles, le flux thermique de la bande de contact dans les
dents d'engrenage entraîne une répartition de température différente pour laquelle les formules (3) à (6) ne sont
pas valables.
vb��� �c
g1 H M1 M1
Pe � � 5 (9)
��� sin
M1 1
vb��� �c
g2 H M2 M2
Pe � � 5 (10)
��� sin
M2 2
10 © ISO 2000 – Tous droits réservés

où
� est la densité du matériau du pignon, en kilogrammes par mètre cube (kg/m );
M1
� est la densité du matériau de la roue, en kilogrammes par mètre cube (kg/m );
M2
c est la chaleur massique par unité de masse du pignon, en joules par kilogramme kelvin [J/(kg�K)];
M1
c est la chaleur massique par unité de masse de la roue, en joules par kilogramme kelvin [J/(kg�K)];
M2
� est la conductivité thermique du pignon, en newtons par seconde kelvin [N/(s�K)];
M1
� est la conductivité thermique de la roue, en newtons par seconde kelvin [N/(s�K)].
M2
Pour les engrenages cylindriques et coniques, sin� =sin� =1.
1 2
5.3 Charge unitaire apparente
La charge unitaire apparente pour les engrenages cylindriques, respectivement pour les engrenages coniques, est
F
t
wK��K�K �K �K � (engrenages cylindriques) (11)
Bt A v Bm� p
B�
b
F
t
wK��K�K �K �K � (engrenages coniques) (12)
Bt A v Bm� p
B�
b
eff
où
F est la force tangentielle nominale sur le cercle primitif, en newtons (N);
t
b est la largeur de denture, en millimètres (mm);
b =0,85b (13)
eff
K est le facteur d'application (voir l'ISO 6336-1 pour les engrenages cylindriques, l’ISO 10300-1 pour les
A
engrenages coniques);
K est le facteur dynamique (voir l'ISO 6336-1 pour les engrenages cylindriques, l’ISO 10300-1 pour les
v
engrenages coniques);
K est le facteur de distribution longitudinale de la charge:
B�
K = K (voir l'ISO 6336-1 pour les engrenages cylindriques, l’ISO 10300-1 pour les engrenages
B� H�
coniques) (14)
K est le facteur de distribution transversale de la charge:
B�
K = K (voir l'ISO 6336-1 pour les engrenages cylindriques, l’ISO 10300-1 pour les engrenages
B� H�
coniques) (15)
K est le facteur d'engrènement multiple.
mp
Le facteur d'engrènement multiple K prend en compte la distribution non uniforme de la charge dans les
mp
transmissions à plusieurs mobiles en prise dépendant de la précision et de la souplesse des branches. Si aucune
analyse précise n'est disponible, les formulations suivantes peuvent s'appliquer:
� pour les trains planétaires épicycliques à n satellites (n W 3)
p p
Kn��10,25 �3 (16)
mp p
� pour les engrenages en tandem avec des arbres torsibles en torsion de� degrés sous pleine charge
K ��10(,2/�) (17)
mp
� pour les engrenages à denture hélicoïdale double avec une force axiale extérieure F
ex
F
ex
K ��1 (18)
mp
F � tan�
t
� dans les autres cas
K � 1 (19)
mp
5.4 Répartition des températures de masse globales
La perte par frottement qui caractérise le mieux les transmissions par engrenages est celle induite par la zone
d'engrènement. Dans ce cas, la chaleur est principalement générée par le frottement des dents. Il est admis que
l'énergie «de pompage» mécanique dépensée pour l'expulsion latérale de l'huile superflue est parfois loin d'être
négligeable. L'autre source de perte par frottement inévitable provient des paliers, qu'ils soient du type roulement ou
palier lisse. Dans les transmissions par engrenages à grande vitesse, les paliers lisses peuvent parfaitement
générer plus de chaleur de frottement que les engrenages. Les autres sources thermiques sont les turbulences de
l'huile et le frottement des joints. L'ensemble des sources thermiques ci-dessus ont en commun les caractéristiques
suivantes:
� dans chacune de ces sources, le frottement du fluide dépend d'une certaine viscosité de l'huile représentative
des conditions de fonctionnement;
� l'ensemble des sources thermiques sont thermiquement corrélées par le biais des éléments de transmission
vers les dissipateurs, tels que l'air ambiant ou le système de refroidissement.
L'interconnexion thermique permet d'utiliser des concepts de calcul tels que:
� les méthodes des éléments finis pour les composants discrets;
� les méthodes de bondgraph;
� les méthodes analogiques de réseau thermique [18].
La température de masse interfaciale, � , peut être moyennée de manière correcte à partir de deux températures
Mi
de masse globales des dents en contact, � et � . La formule suivante est valable pour une bonne
M1 M2
approximation (pour des nombres de Péclet élevés):
Bv�����B�v�
M1 g1 M1 M2 g2 M2
� � (20)
Mi
Bv��B�v
M1 g1 M2 g2
Bv�
M1 g1
Dans une plage ra
...