ISO 10300-3:2001
(Main)Calculation of load capacity of bevel gears — Part 3: Calculation of tooth root strength
Calculation of load capacity of bevel gears — Part 3: Calculation of tooth root strength
Specifies the fundamental formulae for use in the tooth-bending stress calculation of straight and helical (skew), zerol- and spiral-bevel gears with a minimum rim thickness under the root 3,5 mmm. All load influences on tooth stress are included, insofar as they are the result of load transmitted by the gearing and able to be evaluiated quantitatively. (Stresses such as those caused by the shrink-fitting of gear rims, which are superposed on stresses due to tooth loading, are to be taken into consideration in the calculation of the tooth root stress ofp or the permissible tooth root stress ofp.)
Tragfähigkeit von Kegelrädern - Teil 3: Berechnung der Zahnfußtragfähigkeit
Calcul de la capacité de charge des engrenages coniques — Partie 3: Calcul de la résistance du pied de dent
Izračun nosilnosti stožčastih zobnikov - 3. del: Izračun nosilnosti zobnega korena
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INTERNATIONAL ISO
STANDARD 10300-3
First edition
2001-08-01
Corrected version
2003-06-15
Calculation of load capacity of bevel
gears —
Part 3:
Calculation of tooth root strength
Calcul de la capacité de charge des engrenages coniques —
Partie 3: Calcul de la résistance du pied de dent
Reference number
ISO 10300-3:2001(E)
©
ISO 2001
---------------------- Page: 1 ----------------------
ISO 10300-3:2001(E)
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ii © ISO 2001 – All rights reserved
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ISO 10300-3:2001(E)
Contents Page
Foreword . iv
Introduction. v
1 Scope. 1
2 Normative references. 1
3 Terms and definitions. 2
4 Symbols and abbreviated terms. 2
5 Tooth breakage and safety factors. 2
6 Gear-tooth rating formulae. 3
7 Tooth form, Y , and correction, Y , factors — Method B1 . 5
Fa Sa
8 Contact-ratio, Y , bevel-gear, Y , and load-sharing, Y , factors — Method B1. 14
e K LS
9 Bending-strength combined geometry factor, Y — Method B2. 15
P
10 Relative sensitivity factor for allowable stress number, Y . 21
d rel T
11 Relative surface condition factor, Y . 23
R rel T
12 Size factor, Y . 25
X
13 Life factor, Y . 26
NT
Annex A (normative) Bevel gear adjustment factor, Y — Method B2. 29
A
Annex B (informative) Graphs of geometry factor, Y — Method B2. 31
J
© ISO 2001 – All rights reserved iii
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ISO 10300-3:2001(E)
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies (ISO
member bodies). The work of preparing International Standards is normally carried out through ISO technical
committees. Each member body interested in a subject for which a technical committee has been established has
the right to be represented on that committee. International organizations, governmental and non-governmental, in
liaison with ISO, also take part in the work. ISO collaborates closely with the International Electrotechnical
Commission (IEC) on all matters of electrotechnical standardization.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 3.
Draft International Standards adopted by the technical committees are circulated to the member bodies for voting.
Publication as an International Standard requires approval by at least 75 % of the member bodies casting a vote.
Attention is drawn to the possibility that some of the elements of this part of ISO 10300 may be the subject of
patent rights. ISO shall not be held responsible for identifying any or all such patent rights.
International Standard ISO 10300-3 was prepared by Technical Committee ISO/TC 60, Gears, Subcommittee
SC 2, Gear capacity calculation.
ISO 10300 consists of the following parts, under the general title Calculation of load capacity of bevel gears:
Part 1: Introduction and general influence factors
Part 2: Calculation of surface durability (pitting)
Part 3: Calculation of tooth root strength
Annex A forms an integral part of this part of ISO 10300. Annex B is for information only.
In this corrected version of ISO 10300-3, Equation (57) has been corrected.
iv © ISO 2001 – All rights reserved
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ISO 10300-3:2001(E)
Introduction
Parts 1, 2 and 3 of ISO 10300, taken together with ISO 6336-5, are intended to establish general principles and
procedures for the calculation of the load capacity of bevel gears. Moreover, ISO 10300 has been designed to
facilitate the application of future knowledge and developments, as well as the exchange of information gained from
experience. This part of ISO 10300 gives formulae for bending-strength rating in calculations for the avoidance of
tooth breakage.
Failure of gear teeth by breakage can be brought about in many ways — severe instantaneous overloads,
excessive pitting, case crushing and bending fatigue are some. The strength ratings determined by the formulae in
this part of ISO 10300 are based on cantilever-projection theory modified to consider the following:
compressive stress at the tooth roots caused by the radial component of the tooth load;
non-uniform moment distribution of the load, resulting from the inclined contact lines on the teeth of spiral
bevel gears;
stress concentration at the tooth root fillet;
load-sharing between adjacent contacting teeth;
lack of smoothness due to a low contact ratio.
The formulae can be used for determining a load rating that will prevent tooth root fillet fracture during the design
life of the gear teeth. Nevertheless, if there is insufficient material under the teeth (in the rim), a fracture can occur
from the root through the rim of the gear blank or to the bore — a type of failure not covered by this part of
ISO 10300. Moreover, special applications could require additional blank material to support the load.
Occasionally, surface distress (pitting or wear) may limit the strength rating, due either to stress concentration
around large sharp-cornered pits, or to wear steps on the tooth surface. Neither of these effects are considered in
this part of ISO 10300.
In most cases, the maximum tensile stress at the tooth root (arising from bending at the root when the load is
applied to the tooth flank) can be used as the criterion for the assessment of the bending tooth root strength, as
when the allowable stress number is exceeded the teeth can experience breakage. When calculating the tooth root
stresses of straight bevel gears, this part of ISO 10300 starts from the assumption that the load is applied at the
tooth tip of the virtual cylindrical gear. The load is subsequently converted to the outer point of single-tooth contact
with the aid of the contact-ratio factor Y (see clause 8). The procedure thus corresponds to method C for the tooth
e
root stress of cylindrical gears (see ISO 6336-3).
For spiral bevel gears with a high overlap ratio (e > 1), the mid point in the contact zone is regarded as the critical
vb
point of load application. There is an interpolation for medium overlap ratio (0 < e < 1).
vb
The breakage of a tooth generally means the end of a gear's life. It is often the case that all gear teeth are
destroyed as a consequence of the breakage of a single tooth. An S , the safety factor against tooth breakage,
F
higher than the safety factor against damage due to pitting, is therefore generally to be preferred
(see ISO 10300-1).
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INTERNATIONAL STANDARD ISO 10300-3:2001(E)
Calculation of load capacity of bevel gears —
Part 3:
Calculation of tooth root strength
1 Scope
This part of ISO 10300 specifies the fundamental formulae for use in the tooth-bending stress calculation of straight
and helical (skew), zerol- and spiral-bevel gears with a minimum rim thickness under the root W 3,5 m . All load
mn
influences on tooth stress are included, insofar as they are the result of load transmitted by the gearing and able to
be evaluated quantitatively. (Stresses such as those caused by the shrink-fitting of gear rims, which are
superposed on stresses due to tooth loading, are to be taken into consideration in the calculation of the tooth root
stress s or the permissible tooth root stress s .)
F FP
The formulae in this part of ISO 10300 are valid for bevel gears with teeth with a transverse contact ratio of e < 2,
va
while the results are valid within the range of the applied factors given in ISO 10300-1 and ISO 6336-3.
3
This part of ISO 10300 does not apply to stress levels above those permitted for 10 cycles, as stresses in that
range could exceed the elastic limit of the gear tooth.
CAUTION — The user is cautioned that when the methods are used for large spiral and pressure angles,
and for large face width b > 10 m , the calculated results of ISO 10300 should be confirmed by experience.
mn
2 Normative references
The following normative documents contain provisions which, through reference in this text, constitute provisions of
this part of ISO 10300. 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 10300 are encouraged to investigate the
possibility of applying the most recent editions of the normative documents indicated below. For undated
references, the latest edition of the normative document referred to applies. Members of ISO and IEC maintain
registers of currently valid International Standards.
ISO 53:1998, Cylindrical gears for general and heavy engineering — Standard basic rack tooth profile.
ISO 1122-1:1998, Vocabulary of gear terms — Part 1: Definitions related to geometry.
ISO 6336-3, Calculation of load capacity of spur and helical gears — Part 3: Calculation of tooth bending strength.
ISO 6336-5, Calculation of load capacity of spur and helical gears — Part 5: Strength and quality of materials.
ISO 10300-1:2001, Calculation of load capacity of bevel gears — Part 1: Introduction and general influence factors.
ISO 10300-2:2001, Calculation of load capacity of bevel gears — Part 2: Calculation of surface durability (pitting).
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ISO 10300-3:2001(E)
3 Terms and definitions
For the purposes of this part of ISO 10300, the geometrical gear terms given in ISO 53 and ISO 1122-1, and the
following term and definition, apply.
3.1
tooth bending strength
load capacity determined on the basis of the permissible bending stress
4 Symbols and abbreviated terms
For the purposes of this part of ISO 10300, the symbols and abbreviated terms given in Table 1 of
ISO 10300-1:2001, and the following abbreviated terms, apply.
Table 1 — Abbreviated terms
Abbreviation Description
2
St steel (s <800 N/mm )
B
2
V
through-hardened steel (s W 800 N/mm )
B
GG grey cast iron
GGG (perl., bai., ferr.) spheroidal cast iron (perlitic, bainitic, ferritic structure)
GTS (perl.) black malleable cast iron (perlitic structure)
Eh case-hardening steel, case-hardened
IF (root) steel and GGG, flame or induction-hardened (including root fillet)
NT (nitr.) nitriding steels, nitrided
NV (nitr.) through-hardened and case-hardening steel, nitrided
NV (nitrocar.) through-hardened and case-hardening steels, nitro-carburized
5 Tooth breakage and safety factors
Tooth breakage usually ends transmission service life. Sometimes the destruction of all gears in a transmission is a
consequence of the breakage of one tooth, while in certain instances the transmission path between input and
output shafts is broken.
Because of this, the chosen value of the safety factor, S , against tooth breakage should be larger than the square
F
of the safety factor, S , against pitting (see ISO 10300-1 for general comments on the choice of safety factor).
H
The value of the minimum safety factor for bending stress, S , should be W 1,3 for spiral bevel gears. For
Fmin
straight bevel gears, or where b u 5°, S should be W 1,5. It is recommended that the manufacturer and
m Fmin
customer agree on the value of the minimum safety factor.
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ISO 10300-3:2001(E)
6 Gear-tooth rating formulae
6.1 General
The capacity of a gear tooth to resist bending shall be determined by the comparison of the following stress values:
bending stress, based on the geometry of the tooth, the accuracy of its manufacture, the rigidity of the gear
blanks, bearings and housing, and the operating torque, expressed by the bending stress formula (see 6.2);
allowable stress number, and the effect of the working conditions under which the gears operate, expressed
by the permissible bending stress formula (see 6.3).
The determined tooth root stress, s , shall be u s , which is the permissible tooth root stress.
F FP
NOTE In respect of the allowable stress, reference is made to a stress “number”, a designation adopted because pure
stress, as determined by laboratory testing, is not calculated by the formulae in this part of ISO 10300. Instead, an arbitrary
value is calculated and used throughout, with accompanying changes to the allowable stress number in order to maintain
consistency for design comparison.
6.2 Tooth root stress
6.2.1 General
The tooth root stress is determined separately for pinion and wheel:
s = s K K K K u s (1)
F FO A v Fb Fa FP
where
s is the local tooth root stress defined as the maximum tensile stress arising at the tooth root due to the
F0
nominal torque when a perfect gear is loaded.
See ISO 10300-1 for K , K , K , K
A v Fb Fa
6.2.2 Local tooth root stress, s — Method B1
F0-B1
The calculation of the local tooth root stress is based on the maximum tensile stress at the tooth root (30° tangent
to the tooth root fillet). The determinant position of load application is:
a) the outer limit of single tooth contact (e = 0);
vb
b) the mid-point of the zone of contact (e W 1);
vb
c) interpolation between a) and b) (0 < e < 1).
vb
The transformation from tip to this determinant position is done by Y :
e
F
mt
s = Y Y Y Y Y (2)
F0-B1 Fa Sa e K LS
bm
mn
where
F is the nominal tangential force at the reference cone at mid-face width (see ISO 10300-1);
mt
b is the face width;
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ISO 10300-3:2001(E)
Y is the tooth form factor (see clause 7), which accounts for the influence of the tooth form on the nominal
Fa
bending stress for load application at the tooth tip;
Y is the stress correction factor (see clause 7), which accounts for the conversion of the nominal bending
Sa
stress for load application at tooth tip to the corresponding local tooth root stress. Thus Y accounts for
Sa
the stress-increasing effect of the notch (in the root fillet), as well as for the fact that the stress condition in
the critical root section is complex, but not for the influence of the bending moment arm;
Y is the contact-ratio factor (see clause 8), which accounts for the conversion of the local stress determined
e
for the load application at the tooth tip to the determinant position;
Y is the bevel-gear factor, which accounts for smaller values for l ’ compared to total face width b and the
K b
inclined lines of contact;
Y is the load sharing factor, which accounts for load distribution between two or more pairs of teeth.
LS
6.2.3 Local tooth root stress, s — Method B2
F0-B2
When applying method B2, the combined geometry factor Y replaces the factors Y , Y , Y , Y and Y in the
P Fa Sa e LS K
local tooth root stress Equation such that Equation (2) becomes:
F
mt
= (3)
s Y
F0-B2 P
b
m
mn
The value of Y is determined by:
P
m
Y m
A mt mn
= (4)
Y
P
2
Y
J m
et
Substitution in Equation (3):
F m
Y
mt mt A
= (5)
s
F0-B2
2
b
Y
m J
et
where
Y is the bevel-gear adjustment factor for method B2, for standard carburized and case-hardened bevel
A
gears (see annex A);
Y is the bending strength geometry factor for method B2 (see 9.2).
J
The bending-strength geometry factor, Y , evaluates the shape of the tooth, the position at which the most
J
damaging load is applied, the stress concentration due to the geometric shape of the root fillet, the sharing of load
between adjacent pairs of teeth, the tooth-thickness balance between the wheel and mating pinion, the effective
face width due to lengthwise crowning of the teeth, and the buttressing effect of an extended face width on one
member of the pair. Both the tangential (bending) and radial (compressive) components of the tooth load are
included.
6.3 Permissible tooth root stress
6.3.1 General
The permissible tooth root stress, s , is determined separately for pinion and wheel. It should be calculated on
FP
the basis of the strength determined at an actual gear, as this way the reference value for geometrical similarity,
course of movement and manufacture will lie within the field of application.
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ISO 10300-3:2001(E)
s Y
FE NT
= (6)
s YY Y
FP d rel T R rel T X
S
Fmin
s YY
ST NT
Flim
= (7)
s YY Y
FP d rel T R rel T X
S
Fmin
where
s is the allowable stress number (bending),
FE
s = s Y , the basic bending strength of the un-notched specimen under the assumption
FE F lim ST
that the material (including heat treatment) is fully elastic;
s is the bending stress number for the nominal stress in bending of the test gear, which accounts for
F lim
material, heat treatment, and surface influence at test gear dimensions (see ISO 6336-5);
Y is the stress-correction factor for the dimensions of the standard test gear Y = 2,0;
ST ST
S is the minimum safety factor (see ISO 10300-1);
F min
Y is the relative sensitivity factor (see clause 10) for the allowable stress number, related to the
d rel T
conditions at the standard test gear (Y = Y /Y accounts for the notch sensitivity of the material);
d rel T d dT
Y is the relative surface condition factor (see clause 11) (Y = Y /Y accounts for the surface
R rel T R rel T R RT
condition at the root fillet, related to the conditions at the test gear);
Y is the size factor for tooth root strength (see clause 12), which accounts for the influence of the module
X
on the tooth root strength;
Y is the life factor, which accounts for the influence of required numbers of cycles of operation
NT
(see clause 13).
6.3.2 Calculated safety factor
The calculated safety factor against tooth breakage shall be determined separately for pinion and wheel. On the
basis of the allowable stress number (bending), determined for the standard test gear:
YY Y
s Y d rel T R rel T X
FE NT
S= (8)
F
s KK K K
F0 A v FFba
NOTE This is the calculated safety factor with respect to the transmitted torque.
See ISO 10300-1 in reference to the safety factor and the risk (probability) of failure.
7 Tooth form, Y , and correction, Y , factors — Method B1
Fa Sa
7.1 General
The tooth form factor, Y , accounts for the influence of the tooth form on the nominal bending stress in the case of
Fa
load application at the tooth tip. It is determined separately for pinion and wheel.
NOTE In the case of gears with tip and root relief, the actual bending moment arm is slightly smaller, and the calculation is
on the safe side.
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ISO 10300-3:2001(E)
Bevel gears generally have octoid teeth and a tip and root relief. However, deviations from an involute profile are
small, especially in view of the tooth root chord and bending moment arm, and thus they may be neglected when
calculating the tooth-form factor.
The distance between the contact points of the 30° tangents at the root fillets of the tooth profile of the virtual
cylindrical gear is taken as a cross-section for calculation (see Figure 1).
In this part of ISO 10300, Y and Y are determined for the nominal gear without tolerances. The slight reduction
Fa Sa
in tooth thickness for backlash between teeth may be neglected for the load capacity calculation. The size
reduction shall be taken into account when the outer tooth thickness allowance A > 0,05 m .
sne mn
a
Base circle of virtual cylindrical gear
Figure 1 — Tooth root chord, s , and bending moment arm for load application at the tooth tip, h ,
Fn Fa
of the virtual cylindrical gear tooth profile
7.2 Y for generated gears
Fa
7.2.1 General
Equation (9) applies to virtual cylindrical gears in normal section with and without profile shift. The calculation is
valid under the following premises:
a) The contact point of the 30° tangent lies on the fillet curve generated by the tool tip radius.
b) The tool is manufactured with a finite tip radius (r ≠ 0).
a0
h
Fa
6cosa
Fan
m
mn
= (9)
Y
Fa
2
ʈ
s
Fn
cosa
n
Á˜
m
mn
˯
See Figure 1 for the respective definitions; see ISO 6336-3 for an evaluation of the decisive normal tooth load and
tooth form factor.
7.2.2 Auxiliary quantities
For the calculation of the tooth root chord, s , and bending moment arm, h , first the auxiliary quantities E, G, H
Fn Fa
and J need to be determined:
1s--ina
r ( )
p s
ʈ n pr
a0
E= -- tana- (10)
xm h
sm mn a0 n
Á˜
˯4cosa
n
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ISO 10300-3:2001(E)
r
h
a0 a0
G= - + x (11)
hm
mm
mn mn
ʈ
2ppE
H= -- (12)
Á˜
23m
z˯
vn
mn
2G
JJ =Htan - (13)
z
vn
For the solution of the transcendent Equation (13), J = p/6 may be inserted as the initial value. In most cases, the
Equation already converges after a few iteration steps.
7.2.3 Tooth root chord, s
Fn
ʈr
ʈp G
s
Fn a0
=+sin--J 3 (14)
z
vn Á˜
Á˜
m
mn ˯3cosJ m
˯mn
7.2.4 Fillet radius, r , at contact point of 30°°°° tangent
F
2
r r
2G
F a0
=+ (15)
2
m
m
mn mn
cosJJ z cos - 2G
()vn
7.2.5 Bending moment arm, h
Fa
ʈ
d
vbn
a = arc cos (16)
an
Á˜
d
˯
van
1 È p ˘
g = +x2 ( tanaa+x )+ inv - inva (17)
ahmnsm nan
Í ˙
z 2
Î ˚
vn
aa = -g (18)
Fan an a
È r ˘
1 ʈp G
hd
Fa van a0
= --sin tanaJ z cos -- + (19)
cosgg
()
Í Fan vn Á˜ ˙
aa
˯
mm23cosJm
mn mn mn
Î ˚
See annex A of ISO 10300-1:2001 for data of the virtual cylindrical gear in normal section. Dimensions at the basic
rack profile of the tooth are shown in Figure 2 of this part of ISO 10300, while Y may be taken from Figure 3 for a
Fa
basic rack profile of the tool with data a = 20°, h /m = 1,25, and r /m = 0,25 for x = 0. Diagrams for other
n a0 mn a0 mn sm
basic rack profiles are given in ISO 6336-3.
See Figures 4 to 6 for the combined tooth form factor Y = Y Y for generated bevel gears.
FS Fa Sa
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ISO 10300-3:2001(E)
Figure 2 — Dimensions at the basic rack profile of the tooth
7.3 Y for gears made by form cutting
Fa
Crown gears can be partly manufactured by the form cutting method (especially for larger gear ratios), by which the
profile of the space width of the measured gear is identical to the tool profile (rack profile). Here, the tooth form
factor for the crown gear can be directly determined at the tool profile.
Tooth root thickness:
s = p m − 2E − 2 ρ cos 30° (20)
Fn2 mn a02
with E according to Equation (10)
fillet radius at contact point of 30° tangent
ρ = ρ (21)
F2 a02
bending moment arm
r ʈp
ao2
hh=+-- +m x-tanaamtan (22)
Fa2 a02 mnÁ˜sm2 n mn n
˯
24
tooth form factor according to Equation (9), with a = a
Fan n
6 h
Fa2
m
mn
Y = (23)
Fa2
2
ʈ
s
Fn2
Á˜
m
˯mn
The tooth form factor of the pertaining bevel pinion manufactured by the generating method may be approximated
according to 7.2 in the case of gear ratios u > 3.
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ISO 10300-3:2001(E)
Figure 3 — Tooth form factor, Y , for generated gears
Fa
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ISO 10300-3:2001(E)
Figure 4 — Combined tooth form factor Y == Y Y for generated gears (r = 0,2 m )
==
FS Fa Sa a0 mn
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ISO 10300-3:2001(E)
Figure 5 — Combined tooth form factor Y == Y Y for gears generated by basic rack (r = 0,25 m )
==
FS Fa Sa a0 mn
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ISO 10300-3:2001(E)
Figure 6 — Combined tooth form factor Y == Y Y for gears generated by basic rack (r = 0,3 m )
==
FS Fa Sa a0 mn
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ISO 10300-3:2001(E)
7.4 Correction factor, Y
Sa
By means of the stress correction factor, Y , the nominal bending stress is converted to the local tooth root stress.
Sa
Y accounts for the stress increasing effect of the notch (= root fillet) as well as for other stress components that
Sa
arise beside the bending stress. (See ISO 6336-3 for further remarks.)
ʈ
1
Á˜
1,21 + 2,3
˯L
a
=q1,2 + 0,13 (24)
( )
YL
Sa a s
s
Fn
L = (25)
a
h
Fa
s
Fn
q = (26)
s
2 r
F
where
s is according to Equation (14) and Equation (20) respectively;
Fn
h is according to Equation (19) and Equation (22) respectively;
Fa
r is according to Equation (15) and Equation (21) respectively.
F
The range of validity of Equation (24) is 1 u q < 8.
s
The stress correction factor, Y , may be read from Figure 7 for the basic rack profile of the tool with a = 20°,
Sa n
h /m = 1,25, r /m = 0,25 for x = 0. See ISO 6336-3 for the influence of grinding notches.
a0 mn a0 mn sm
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ISO 10300-3:2001(E)
Figure 7 — Stress correction factor for load application at tooth tip, Y
sa
8 Contact-ratio, Y , bevel-gear, Y , and load-sharing, Y , factors — Method B1
e K LS
8.1 Contact-ratio factor, Y
e
The contact-ratio factor, Y , converts the load application at the tooth tip (here the tooth form factor, Y , and stress-
e Fa
correction factor, Y , apply) to the decisive point of load application.
Sa
Y is also used for the determination of the transverse load distribution factor K (see ISO 10300-1):
e Fa
0,75
=+0,25 W 0,625 ( e = 0) (27)
Y nb
ε
e
vα
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ISO 10300-3:2001(E)
ʈ
0,75 0,75
=+0,25 -- 0,375 W 0,625 (0 < e u 1) (28)
Y e
e vb nb
Á˜
ee
vvaa˯
Y = 0,625 (e > 1) (29)
e nb
8.2 Bevel-gear factor, Y
K
The bevel gear factor, Y , accounts for differences between bevel and cylindrical gears (smaller values of l′
K bm
because of inclined lines of contact):
2
ʈ11 l¢ b
bm
= + ◊ (30)
Y
K
Á˜
˯22 bl¢
bm
where
l′ is the projected length of the middle line of contact [see Equation (A.44) in ISO 10300-1:2001].
bm
8.3 Load sharing factor, Y
LS
The load sharing factor, Y , accounts for load sharing between two or more pair of teeth.
LS
2
Y=Z (31)
LS LS
(See ISO 10300-2 for Z .)
LS
9 Bending-strength combined geometry factor, Y — Method B2
P
9.1 Graphs and general
Figures in annex B contain graphs for the bevel geometry factor, Y , for straight-, zerol- and spiral-bevel gears for a
J
series of gear designs, based on the smaller of the face widths b = 0,3 R and 10 m . These may be used
e et
whenever the tooth proportions and thickness, face widths, tool edge radii, pressure and spiral angles of the
design, and driving with the
...
SLOVENSKI STANDARD
SIST ISO 10300-3:2008
01-julij-2008
,]UDþXQQRVLOQRVWLVWRåþDVWLK]REQLNRYGHO ,]UDþXQQRVLOQRVWL]REQHJDNRUHQD
Calculation of load capacity of bevel gears - Part 3: Calculation of tooth root strength
Tragfähigkeit von Kegelrädern - Teil 3: Berechnung der Zahnfußtragfähigkeit
Calcul de la capacité de charge des engrenages coniques - Partie 3: Calcul de la
résistance du pied de dent
Ta slovenski standard je istoveten z: ISO 10300-3:2001
ICS:
21.200 Gonila Gears
SIST ISO 10300-3:2008 en,fr
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.
---------------------- Page: 1 ----------------------
INTERNATIONAL ISO
STANDARD 10300-3
First edition
2001-08-01
Corrected version
2003-06-15
Calculation of load capacity of bevel
gears —
Part 3:
Calculation of tooth root strength
Calcul de la capacité de charge des engrenages coniques —
Partie 3: Calcul de la résistance du pied de dent
Reference number
ISO 10300-3:2001(E)
©
ISO 2001
---------------------- Page: 2 ----------------------
ISO 10300-3:2001(E)
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Published in Switzerland
ii © ISO 2001 – All rights reserved
---------------------- Page: 3 ----------------------
ISO 10300-3:2001(E)
Contents Page
Foreword . iv
Introduction. v
1 Scope. 1
2 Normative references. 1
3 Terms and definitions. 2
4 Symbols and abbreviated terms. 2
5 Tooth breakage and safety factors. 2
6 Gear-tooth rating formulae. 3
7 Tooth form, Y , and correction, Y , factors — Method B1 . 5
Fa Sa
8 Contact-ratio, Y , bevel-gear, Y , and load-sharing, Y , factors — Method B1. 14
e K LS
9 Bending-strength combined geometry factor, Y — Method B2. 15
P
10 Relative sensitivity factor for allowable stress number, Y . 21
d rel T
11 Relative surface condition factor, Y . 23
R rel T
12 Size factor, Y . 25
X
13 Life factor, Y . 26
NT
Annex A (normative) Bevel gear adjustment factor, Y — Method B2. 29
A
Annex B (informative) Graphs of geometry factor, Y — Method B2. 31
J
© ISO 2001 – All rights reserved iii
---------------------- Page: 4 ----------------------
ISO 10300-3:2001(E)
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies (ISO
member bodies). The work of preparing International Standards is normally carried out through ISO technical
committees. Each member body interested in a subject for which a technical committee has been established has
the right to be represented on that committee. International organizations, governmental and non-governmental, in
liaison with ISO, also take part in the work. ISO collaborates closely with the International Electrotechnical
Commission (IEC) on all matters of electrotechnical standardization.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 3.
Draft International Standards adopted by the technical committees are circulated to the member bodies for voting.
Publication as an International Standard requires approval by at least 75 % of the member bodies casting a vote.
Attention is drawn to the possibility that some of the elements of this part of ISO 10300 may be the subject of
patent rights. ISO shall not be held responsible for identifying any or all such patent rights.
International Standard ISO 10300-3 was prepared by Technical Committee ISO/TC 60, Gears, Subcommittee
SC 2, Gear capacity calculation.
ISO 10300 consists of the following parts, under the general title Calculation of load capacity of bevel gears:
Part 1: Introduction and general influence factors
Part 2: Calculation of surface durability (pitting)
Part 3: Calculation of tooth root strength
Annex A forms an integral part of this part of ISO 10300. Annex B is for information only.
In this corrected version of ISO 10300-3, Equation (57) has been corrected.
iv © ISO 2001 – All rights reserved
---------------------- Page: 5 ----------------------
ISO 10300-3:2001(E)
Introduction
Parts 1, 2 and 3 of ISO 10300, taken together with ISO 6336-5, are intended to establish general principles and
procedures for the calculation of the load capacity of bevel gears. Moreover, ISO 10300 has been designed to
facilitate the application of future knowledge and developments, as well as the exchange of information gained from
experience. This part of ISO 10300 gives formulae for bending-strength rating in calculations for the avoidance of
tooth breakage.
Failure of gear teeth by breakage can be brought about in many ways — severe instantaneous overloads,
excessive pitting, case crushing and bending fatigue are some. The strength ratings determined by the formulae in
this part of ISO 10300 are based on cantilever-projection theory modified to consider the following:
compressive stress at the tooth roots caused by the radial component of the tooth load;
non-uniform moment distribution of the load, resulting from the inclined contact lines on the teeth of spiral
bevel gears;
stress concentration at the tooth root fillet;
load-sharing between adjacent contacting teeth;
lack of smoothness due to a low contact ratio.
The formulae can be used for determining a load rating that will prevent tooth root fillet fracture during the design
life of the gear teeth. Nevertheless, if there is insufficient material under the teeth (in the rim), a fracture can occur
from the root through the rim of the gear blank or to the bore — a type of failure not covered by this part of
ISO 10300. Moreover, special applications could require additional blank material to support the load.
Occasionally, surface distress (pitting or wear) may limit the strength rating, due either to stress concentration
around large sharp-cornered pits, or to wear steps on the tooth surface. Neither of these effects are considered in
this part of ISO 10300.
In most cases, the maximum tensile stress at the tooth root (arising from bending at the root when the load is
applied to the tooth flank) can be used as the criterion for the assessment of the bending tooth root strength, as
when the allowable stress number is exceeded the teeth can experience breakage. When calculating the tooth root
stresses of straight bevel gears, this part of ISO 10300 starts from the assumption that the load is applied at the
tooth tip of the virtual cylindrical gear. The load is subsequently converted to the outer point of single-tooth contact
with the aid of the contact-ratio factor Y (see clause 8). The procedure thus corresponds to method C for the tooth
e
root stress of cylindrical gears (see ISO 6336-3).
For spiral bevel gears with a high overlap ratio (e > 1), the mid point in the contact zone is regarded as the critical
vb
point of load application. There is an interpolation for medium overlap ratio (0 < e < 1).
vb
The breakage of a tooth generally means the end of a gear's life. It is often the case that all gear teeth are
destroyed as a consequence of the breakage of a single tooth. An S , the safety factor against tooth breakage,
F
higher than the safety factor against damage due to pitting, is therefore generally to be preferred
(see ISO 10300-1).
© ISO 2001 – All rights reserved v
---------------------- Page: 6 ----------------------
INTERNATIONAL STANDARD ISO 10300-3:2001(E)
Calculation of load capacity of bevel gears —
Part 3:
Calculation of tooth root strength
1 Scope
This part of ISO 10300 specifies the fundamental formulae for use in the tooth-bending stress calculation of straight
and helical (skew), zerol- and spiral-bevel gears with a minimum rim thickness under the root W 3,5 m . All load
mn
influences on tooth stress are included, insofar as they are the result of load transmitted by the gearing and able to
be evaluated quantitatively. (Stresses such as those caused by the shrink-fitting of gear rims, which are
superposed on stresses due to tooth loading, are to be taken into consideration in the calculation of the tooth root
stress s or the permissible tooth root stress s .)
F FP
The formulae in this part of ISO 10300 are valid for bevel gears with teeth with a transverse contact ratio of e < 2,
va
while the results are valid within the range of the applied factors given in ISO 10300-1 and ISO 6336-3.
3
This part of ISO 10300 does not apply to stress levels above those permitted for 10 cycles, as stresses in that
range could exceed the elastic limit of the gear tooth.
CAUTION — The user is cautioned that when the methods are used for large spiral and pressure angles,
and for large face width b > 10 m , the calculated results of ISO 10300 should be confirmed by experience.
mn
2 Normative references
The following normative documents contain provisions which, through reference in this text, constitute provisions of
this part of ISO 10300. 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 10300 are encouraged to investigate the
possibility of applying the most recent editions of the normative documents indicated below. For undated
references, the latest edition of the normative document referred to applies. Members of ISO and IEC maintain
registers of currently valid International Standards.
ISO 53:1998, Cylindrical gears for general and heavy engineering — Standard basic rack tooth profile.
ISO 1122-1:1998, Vocabulary of gear terms — Part 1: Definitions related to geometry.
ISO 6336-3, Calculation of load capacity of spur and helical gears — Part 3: Calculation of tooth bending strength.
ISO 6336-5, Calculation of load capacity of spur and helical gears — Part 5: Strength and quality of materials.
ISO 10300-1:2001, Calculation of load capacity of bevel gears — Part 1: Introduction and general influence factors.
ISO 10300-2:2001, Calculation of load capacity of bevel gears — Part 2: Calculation of surface durability (pitting).
© ISO 2001 – All rights reserved 1
---------------------- Page: 7 ----------------------
ISO 10300-3:2001(E)
3 Terms and definitions
For the purposes of this part of ISO 10300, the geometrical gear terms given in ISO 53 and ISO 1122-1, and the
following term and definition, apply.
3.1
tooth bending strength
load capacity determined on the basis of the permissible bending stress
4 Symbols and abbreviated terms
For the purposes of this part of ISO 10300, the symbols and abbreviated terms given in Table 1 of
ISO 10300-1:2001, and the following abbreviated terms, apply.
Table 1 — Abbreviated terms
Abbreviation Description
2
St steel (s <800 N/mm )
B
2
V
through-hardened steel (s W 800 N/mm )
B
GG grey cast iron
GGG (perl., bai., ferr.) spheroidal cast iron (perlitic, bainitic, ferritic structure)
GTS (perl.) black malleable cast iron (perlitic structure)
Eh case-hardening steel, case-hardened
IF (root) steel and GGG, flame or induction-hardened (including root fillet)
NT (nitr.) nitriding steels, nitrided
NV (nitr.) through-hardened and case-hardening steel, nitrided
NV (nitrocar.) through-hardened and case-hardening steels, nitro-carburized
5 Tooth breakage and safety factors
Tooth breakage usually ends transmission service life. Sometimes the destruction of all gears in a transmission is a
consequence of the breakage of one tooth, while in certain instances the transmission path between input and
output shafts is broken.
Because of this, the chosen value of the safety factor, S , against tooth breakage should be larger than the square
F
of the safety factor, S , against pitting (see ISO 10300-1 for general comments on the choice of safety factor).
H
The value of the minimum safety factor for bending stress, S , should be W 1,3 for spiral bevel gears. For
Fmin
straight bevel gears, or where b u 5°, S should be W 1,5. It is recommended that the manufacturer and
m Fmin
customer agree on the value of the minimum safety factor.
2 © ISO 2001 – All rights reserved
---------------------- Page: 8 ----------------------
ISO 10300-3:2001(E)
6 Gear-tooth rating formulae
6.1 General
The capacity of a gear tooth to resist bending shall be determined by the comparison of the following stress values:
bending stress, based on the geometry of the tooth, the accuracy of its manufacture, the rigidity of the gear
blanks, bearings and housing, and the operating torque, expressed by the bending stress formula (see 6.2);
allowable stress number, and the effect of the working conditions under which the gears operate, expressed
by the permissible bending stress formula (see 6.3).
The determined tooth root stress, s , shall be u s , which is the permissible tooth root stress.
F FP
NOTE In respect of the allowable stress, reference is made to a stress “number”, a designation adopted because pure
stress, as determined by laboratory testing, is not calculated by the formulae in this part of ISO 10300. Instead, an arbitrary
value is calculated and used throughout, with accompanying changes to the allowable stress number in order to maintain
consistency for design comparison.
6.2 Tooth root stress
6.2.1 General
The tooth root stress is determined separately for pinion and wheel:
s = s K K K K u s (1)
F FO A v Fb Fa FP
where
s is the local tooth root stress defined as the maximum tensile stress arising at the tooth root due to the
F0
nominal torque when a perfect gear is loaded.
See ISO 10300-1 for K , K , K , K
A v Fb Fa
6.2.2 Local tooth root stress, s — Method B1
F0-B1
The calculation of the local tooth root stress is based on the maximum tensile stress at the tooth root (30° tangent
to the tooth root fillet). The determinant position of load application is:
a) the outer limit of single tooth contact (e = 0);
vb
b) the mid-point of the zone of contact (e W 1);
vb
c) interpolation between a) and b) (0 < e < 1).
vb
The transformation from tip to this determinant position is done by Y :
e
F
mt
s = Y Y Y Y Y (2)
F0-B1 Fa Sa e K LS
bm
mn
where
F is the nominal tangential force at the reference cone at mid-face width (see ISO 10300-1);
mt
b is the face width;
© ISO 2001 – All rights reserved 3
---------------------- Page: 9 ----------------------
ISO 10300-3:2001(E)
Y is the tooth form factor (see clause 7), which accounts for the influence of the tooth form on the nominal
Fa
bending stress for load application at the tooth tip;
Y is the stress correction factor (see clause 7), which accounts for the conversion of the nominal bending
Sa
stress for load application at tooth tip to the corresponding local tooth root stress. Thus Y accounts for
Sa
the stress-increasing effect of the notch (in the root fillet), as well as for the fact that the stress condition in
the critical root section is complex, but not for the influence of the bending moment arm;
Y is the contact-ratio factor (see clause 8), which accounts for the conversion of the local stress determined
e
for the load application at the tooth tip to the determinant position;
Y is the bevel-gear factor, which accounts for smaller values for l ’ compared to total face width b and the
K b
inclined lines of contact;
Y is the load sharing factor, which accounts for load distribution between two or more pairs of teeth.
LS
6.2.3 Local tooth root stress, s — Method B2
F0-B2
When applying method B2, the combined geometry factor Y replaces the factors Y , Y , Y , Y and Y in the
P Fa Sa e LS K
local tooth root stress Equation such that Equation (2) becomes:
F
mt
= (3)
s Y
F0-B2 P
b
m
mn
The value of Y is determined by:
P
m
Y m
A mt mn
= (4)
Y
P
2
Y
J m
et
Substitution in Equation (3):
F m
Y
mt mt A
= (5)
s
F0-B2
2
b
Y
m J
et
where
Y is the bevel-gear adjustment factor for method B2, for standard carburized and case-hardened bevel
A
gears (see annex A);
Y is the bending strength geometry factor for method B2 (see 9.2).
J
The bending-strength geometry factor, Y , evaluates the shape of the tooth, the position at which the most
J
damaging load is applied, the stress concentration due to the geometric shape of the root fillet, the sharing of load
between adjacent pairs of teeth, the tooth-thickness balance between the wheel and mating pinion, the effective
face width due to lengthwise crowning of the teeth, and the buttressing effect of an extended face width on one
member of the pair. Both the tangential (bending) and radial (compressive) components of the tooth load are
included.
6.3 Permissible tooth root stress
6.3.1 General
The permissible tooth root stress, s , is determined separately for pinion and wheel. It should be calculated on
FP
the basis of the strength determined at an actual gear, as this way the reference value for geometrical similarity,
course of movement and manufacture will lie within the field of application.
4 © ISO 2001 – All rights reserved
---------------------- Page: 10 ----------------------
ISO 10300-3:2001(E)
s Y
FE NT
= (6)
s YY Y
FP d rel T R rel T X
S
Fmin
s YY
ST NT
Flim
= (7)
s YY Y
FP d rel T R rel T X
S
Fmin
where
s is the allowable stress number (bending),
FE
s = s Y , the basic bending strength of the un-notched specimen under the assumption
FE F lim ST
that the material (including heat treatment) is fully elastic;
s is the bending stress number for the nominal stress in bending of the test gear, which accounts for
F lim
material, heat treatment, and surface influence at test gear dimensions (see ISO 6336-5);
Y is the stress-correction factor for the dimensions of the standard test gear Y = 2,0;
ST ST
S is the minimum safety factor (see ISO 10300-1);
F min
Y is the relative sensitivity factor (see clause 10) for the allowable stress number, related to the
d rel T
conditions at the standard test gear (Y = Y /Y accounts for the notch sensitivity of the material);
d rel T d dT
Y is the relative surface condition factor (see clause 11) (Y = Y /Y accounts for the surface
R rel T R rel T R RT
condition at the root fillet, related to the conditions at the test gear);
Y is the size factor for tooth root strength (see clause 12), which accounts for the influence of the module
X
on the tooth root strength;
Y is the life factor, which accounts for the influence of required numbers of cycles of operation
NT
(see clause 13).
6.3.2 Calculated safety factor
The calculated safety factor against tooth breakage shall be determined separately for pinion and wheel. On the
basis of the allowable stress number (bending), determined for the standard test gear:
YY Y
s Y d rel T R rel T X
FE NT
S= (8)
F
s KK K K
F0 A v FFba
NOTE This is the calculated safety factor with respect to the transmitted torque.
See ISO 10300-1 in reference to the safety factor and the risk (probability) of failure.
7 Tooth form, Y , and correction, Y , factors — Method B1
Fa Sa
7.1 General
The tooth form factor, Y , accounts for the influence of the tooth form on the nominal bending stress in the case of
Fa
load application at the tooth tip. It is determined separately for pinion and wheel.
NOTE In the case of gears with tip and root relief, the actual bending moment arm is slightly smaller, and the calculation is
on the safe side.
© ISO 2001 – All rights reserved 5
---------------------- Page: 11 ----------------------
ISO 10300-3:2001(E)
Bevel gears generally have octoid teeth and a tip and root relief. However, deviations from an involute profile are
small, especially in view of the tooth root chord and bending moment arm, and thus they may be neglected when
calculating the tooth-form factor.
The distance between the contact points of the 30° tangents at the root fillets of the tooth profile of the virtual
cylindrical gear is taken as a cross-section for calculation (see Figure 1).
In this part of ISO 10300, Y and Y are determined for the nominal gear without tolerances. The slight reduction
Fa Sa
in tooth thickness for backlash between teeth may be neglected for the load capacity calculation. The size
reduction shall be taken into account when the outer tooth thickness allowance A > 0,05 m .
sne mn
a
Base circle of virtual cylindrical gear
Figure 1 — Tooth root chord, s , and bending moment arm for load application at the tooth tip, h ,
Fn Fa
of the virtual cylindrical gear tooth profile
7.2 Y for generated gears
Fa
7.2.1 General
Equation (9) applies to virtual cylindrical gears in normal section with and without profile shift. The calculation is
valid under the following premises:
a) The contact point of the 30° tangent lies on the fillet curve generated by the tool tip radius.
b) The tool is manufactured with a finite tip radius (r ≠ 0).
a0
h
Fa
6cosa
Fan
m
mn
= (9)
Y
Fa
2
ʈ
s
Fn
cosa
n
Á˜
m
mn
˯
See Figure 1 for the respective definitions; see ISO 6336-3 for an evaluation of the decisive normal tooth load and
tooth form factor.
7.2.2 Auxiliary quantities
For the calculation of the tooth root chord, s , and bending moment arm, h , first the auxiliary quantities E, G, H
Fn Fa
and J need to be determined:
1s--ina
r ( )
p s
ʈ n pr
a0
E= -- tana- (10)
xm h
sm mn a0 n
Á˜
˯4cosa
n
6 © ISO 2001 – All rights reserved
---------------------- Page: 12 ----------------------
ISO 10300-3:2001(E)
r
h
a0 a0
G= - + x (11)
hm
mm
mn mn
ʈ
2ppE
H= -- (12)
Á˜
23m
z˯
vn
mn
2G
JJ =Htan - (13)
z
vn
For the solution of the transcendent Equation (13), J = p/6 may be inserted as the initial value. In most cases, the
Equation already converges after a few iteration steps.
7.2.3 Tooth root chord, s
Fn
ʈr
ʈp G
s
Fn a0
=+sin--J 3 (14)
z
vn Á˜
Á˜
m
mn ˯3cosJ m
˯mn
7.2.4 Fillet radius, r , at contact point of 30°°°° tangent
F
2
r r
2G
F a0
=+ (15)
2
m
m
mn mn
cosJJ z cos - 2G
()vn
7.2.5 Bending moment arm, h
Fa
ʈ
d
vbn
a = arc cos (16)
an
Á˜
d
˯
van
1 È p ˘
g = +x2 ( tanaa+x )+ inv - inva (17)
ahmnsm nan
Í ˙
z 2
Î ˚
vn
aa = -g (18)
Fan an a
È r ˘
1 ʈp G
hd
Fa van a0
= --sin tanaJ z cos -- + (19)
cosgg
()
Í Fan vn Á˜ ˙
aa
˯
mm23cosJm
mn mn mn
Î ˚
See annex A of ISO 10300-1:2001 for data of the virtual cylindrical gear in normal section. Dimensions at the basic
rack profile of the tooth are shown in Figure 2 of this part of ISO 10300, while Y may be taken from Figure 3 for a
Fa
basic rack profile of the tool with data a = 20°, h /m = 1,25, and r /m = 0,25 for x = 0. Diagrams for other
n a0 mn a0 mn sm
basic rack profiles are given in ISO 6336-3.
See Figures 4 to 6 for the combined tooth form factor Y = Y Y for generated bevel gears.
FS Fa Sa
© ISO 2001 – All rights reserved 7
---------------------- Page: 13 ----------------------
ISO 10300-3:2001(E)
Figure 2 — Dimensions at the basic rack profile of the tooth
7.3 Y for gears made by form cutting
Fa
Crown gears can be partly manufactured by the form cutting method (especially for larger gear ratios), by which the
profile of the space width of the measured gear is identical to the tool profile (rack profile). Here, the tooth form
factor for the crown gear can be directly determined at the tool profile.
Tooth root thickness:
s = p m − 2E − 2 ρ cos 30° (20)
Fn2 mn a02
with E according to Equation (10)
fillet radius at contact point of 30° tangent
ρ = ρ (21)
F2 a02
bending moment arm
r ʈp
ao2
hh=+-- +m x-tanaamtan (22)
Fa2 a02 mnÁ˜sm2 n mn n
˯
24
tooth form factor according to Equation (9), with a = a
Fan n
6 h
Fa2
m
mn
Y = (23)
Fa2
2
ʈ
s
Fn2
Á˜
m
˯mn
The tooth form factor of the pertaining bevel pinion manufactured by the generating method may be approximated
according to 7.2 in the case of gear ratios u > 3.
8 © ISO 2001 – All rights reserved
---------------------- Page: 14 ----------------------
ISO 10300-3:2001(E)
Figure 3 — Tooth form factor, Y , for generated gears
Fa
© ISO 2001 – All rights reserved 9
---------------------- Page: 15 ----------------------
ISO 10300-3:2001(E)
Figure 4 — Combined tooth form factor Y == Y Y for generated gears (r = 0,2 m )
==
FS Fa Sa a0 mn
10 © ISO 2001 – All rights reserved
---------------------- Page: 16 ----------------------
ISO 10300-3:2001(E)
Figure 5 — Combined tooth form factor Y == Y Y for gears generated by basic rack (r = 0,25 m )
==
FS Fa Sa a0 mn
© ISO 2001 – All rights reserved 11
---------------------- Page: 17 ----------------------
ISO 10300-3:2001(E)
Figure 6 — Combined tooth form factor Y == Y Y for gears generated by basic rack (r = 0,3 m )
==
FS Fa Sa a0 mn
12 © ISO 2001 – All rights reserved
---------------------- Page: 18 ----------------------
ISO 10300-3:2001(E)
7.4 Correction factor, Y
Sa
By means of the stress correction factor, Y , the nominal bending stress is converted to the local tooth root stress.
Sa
Y accounts for the stress increasing effect of the notch (= root fillet) as well as for other stress components that
Sa
arise beside the bending stress. (See ISO 6336-3 for further remarks.)
ʈ
1
Á˜
1,21 + 2,3
˯L
a
=q1,2 + 0,13 (24)
( )
YL
Sa a s
s
Fn
L = (25)
a
h
Fa
s
Fn
q = (26)
s
2 r
F
where
s is according to Equation (14) and Equation (20) respectively;
Fn
h is according to Equation (19) and Equation (22) respectively;
Fa
r is according to Equation (15) and Equation (21) respectively.
F
The range of validity of Equation (24) is 1 u q < 8.
s
The stress correction factor, Y , may be read from Figure 7 for the basic rack profile of the tool with a = 20°,
Sa n
h /m = 1,25, r /m = 0,25 for x = 0. See ISO 6336-3 for the influence of grinding notches.
a0 mn a0 mn sm
© ISO 2001 – All rights reserved 13
---------------------- Page: 19 ----------------------
ISO 10300-3:2001(E)
Figure 7 — Stress correction factor for load application at tooth tip, Y
sa
8 Contact-ratio, Y , bevel-gear, Y , and load-sharing, Y , factors — Method B1
e K LS
8.1 Contact-ratio factor, Y
e
The contact-ratio factor, Y , converts the load application at the tooth tip (here the tooth form factor, Y , and stress-
e Fa
correction factor, Y , apply) to the decisive point of load application.
Sa
Y is also used for the determination of the transverse load distribution factor K (see ISO 10300-1):
e Fa
0,75
=+0,25 W 0,625 ( e = 0) (27)
Y nb
ε
e
vα
14 © ISO 2001 – All rights reserved
---------------------- Page: 20 ----------------------
ISO 10300-3:2001(E)
ʈ
0,75 0,75
=+0,25 -- 0,375 W 0,625 (0 < e u 1) (28)
Y e
e vb nb
Á˜
ee
vvaa˯
Y = 0,625 (e > 1) (29)
e nb
8.2 Bevel-gear factor, Y
K
The bevel gear factor, Y , accounts for differences between bevel and cylindrical gears (smaller values of l′
K bm
because of inclined lines of contact):
2
ʈ11 l¢ b
bm
= + ◊ (30)
Y
K
Á˜
˯22 bl¢
bm
where
l′ is th
...
NORME ISO
INTERNATIONALE 10300-3
Première édition
2001-08-01
Version corrigée
2003-06-15
Calcul de la capacité de charge des
engrenages coniques —
Partie 3:
Calcul de la résistance du pied de dent
Calculation of load capacity of bevel gears —
Part 3: Calculation of tooth root strength
Numéro de référence
ISO 10300-3:2001(F)
©
ISO 2001
---------------------- Page: 1 ----------------------
ISO 10300-3:2001(F)
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Publié en Suisse
ii © ISO 2001 – Tous droits réservés
---------------------- Page: 2 ----------------------
ISO 10300-3:2001(F)
Sommaire Page
Avant-propos . iv
Introduction. v
1 Domaine d'application. 1
2 Références normatives. 1
3 Termes et définitions . 2
4 Symboles et abréviations . 2
5 Rupture de dent et coefficients de sécurité. 2
6 Formules de calcul de la capacité de charge . 3
7 Facteur de forme, Y , et facteur de concentration de contrainte, Y — Méthode B1 . 6
Fa Sa
8 Facteur de rapport de conduite, Y , facteur d'engrenage conique, Y , facteur de répartition de
e K
charge, Y — Méthode B1 . 15
LS
9 Facteur géométrique combiné de résistance à la flexion, Y — Méthode B2 . 15
P
10 Facteur de sensibilité relative pour la contrainte admissible, Y . 21
d rel T
11 Facteur de rugosité relative, Y . 24
R rel T
12 Facteur de dimension, Y . 25
X
13 Facteur de durée de vie, Y . 27
NT
Annexe A (normative) Facteur d'ajustement d'engrenage conique, Y — Méthode B2 . 30
A
Annexe B (informative) Graphiques du facteur, Y — Méthode B2. 32
J
© ISO 2001 – Tous droits réservés iii
---------------------- Page: 3 ----------------------
ISO 10300-3:2001(F)
Avant-propos
L'ISO (Organisation internationale de normalisation) est une fédération mondiale d'organismes nationaux de
normalisation (comités membres de l'ISO). L'élaboration des Normes internationales est en général confiée aux
comités techniques de l'ISO. Chaque comité membre intéressé par une étude a le droit de faire partie du comité
technique créé à cet effet. Les organisations internationales, gouvernementales et non gouvernementales, en
liaison avec l'ISO participent également aux travaux. L'ISO collabore étroitement avec la Commission
électrotechnique internationale (CEI) en ce qui concerne la normalisation électrotechnique.
Les Normes internationales sont rédigées conformément aux règles données dans les Directives ISO/CEI, Partie 3.
Les projets de Normes internationales adoptés par les comités techniques sont soumis aux comités membres pour
vote. Leur publication comme Normes internationales requiert l'approbation de 75 % au moins des comités
membres votants.
L’attention est appelée sur le fait que certains des éléments de la présente partie de l’ISO 10300 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.
La Norme internationale ISO 10300-3 a été élaborée par le comité technique ISO/TC 60, Engrenages, sous-comité
SC 2, Calcul de la capacité des engrenages.
L'ISO 10300 comprend les parties suivantes, présentées sous le titre général Calcul de la capacité de charge des
engrenages coniques:
Partie 1: Introduction et facteurs généraux d'influence
Partie 2: Calcul de la résistance à la pression superficielle (formation des piqûres)
Partie 3: Calcul de la résistance du pied de dent
L'annexe A constitue un élément normatif de la présente partie de l'ISO 10300. L’annexe B est donnée uniquement
à titre d'information.
Dans la présente version corrigée de l'ISO 10300-3:2001, l'équation (57) a été corrigée.
iv © ISO 2001 – Tous droits réservés
---------------------- Page: 4 ----------------------
ISO 10300-3:2001(F)
Introduction
La présente partie de l'ISO 10300, l'ISO 10300-1 et l'ISO 10300-3, ainsi que l'ISO 6336-5, établissent les principes
généraux et les procédures pour le calcul de la capacité de charge des engrenages coniques. Ainsi, l'ISO 10300 a
été conçue pour faciliter l'application des connaissances et du développement futurs, ainsi que les échanges
d'informations acquises par expérience. La présente partie de l'ISO 10300 donne les formules de détermination de
la résistance à la flexion utilisées dans les calculs permettant d'éviter la rupture de dent.
La détérioration des dents d'engrenage par rupture peut survenir de différente façons, telles que celles dues à des
surcharges instantanées sévères, à une formation de piqûres excessive, à l’effondrement de la couche cémentée
ou à la fatigue de flexion. Les valeurs de la résistance à la flexion, déterminées par les formules de la présente
partie de l'ISO 10300, sont basées sur une théorie de poutre encastrée en porte-à-faux modifiée de façon à tenir
compte:
de la contrainte de compression aux pieds de dent causée par le composant radial de la charge;
de la répartition non uniforme du moment résultant de lignes de contact inclinées sur les dents des engrenages
spiroconiques;
de la concentration de contrainte au profil de raccordement du pied de dent;
de la répartition de charge entre les dents voisines en contact;
du manque de douceur d’engrènement dû à un faible rapport de conduite.
Les formules de détermination de la résistance à la flexion peuvent être utilisées pour déterminer la capacité de
charge permettant d'empêcher la rupture au niveau du profil de raccordement en pied de dent, pendant la durée de
vie souhaitée des dentures. Cependant, lorsque l’épaisseur de matière sous les dents (dans la jante) est
insuffisante, une rupture peut se produire à partir du pied des dents à travers la jante du corps de roue ou jusqu'à
l'alésage. Ce type de dégradation n'est pas pris en compte par la présente partie de l'ISO 10300. Des applications
particulières peuvent exiger un corps de roue renforcé pour supporter la charge.
Parfois, une dégradation superficielle (formation de piqûres ou usure) peut limiter la détermination de la résistance
due soit à la concentration de contrainte autour d'importantes piqûres aux bords anguleux, soit à des marches
d’escalier d'usure à la surface de la dent. Aucun de ces effets n’est pris en considération dans la présente partie de
l'ISO 10300.
Dans la plupart des cas, la contrainte maximale de traction au pied de dent (contrainte se produisant au pied de
dent en raison de la flexion lorsque la charge est appliquée au flanc de la dent) peut être utilisée comme critère
pour évaluer la résistance à la flexion du pied de dent, puisque lorsque la contrainte nominale de référence est
dépassée, les dents peuvent casser. Lors du calcul des contraintes au pied de dent des engrenages coniques à
denture droite, la présente partie de l'ISO 10300 part de l'hypothèse que la charge est appliquée au sommet de
dent (de l'engrenage cylindrique équivalent). La charge est par la suite appliquée au point le plus haut de contact
unique à l'aide du facteur de rapport de conduite Y (voir article 8). La méthode correspond ainsi à la méthode C
e
pour la contrainte au pied de dent des engrenages cylindriques, voir l'ISO 6336-3.
Pour les engrenages spiroconiques avec un rapport de recouvrement élevé (e > 1), le point central de la zone de
vb
contact est considéré comme le point critique d’une application de la charge. Il y a une interpolation dans le cas
d’un rapport de recouvrement moyen (0 < e < 1).
vb
La rupture d'une dent signifie généralement la fin de la durée de vie de l'engrenage. Souvent, toutes les dents d'un
engrenage sont détruites en raison de la rupture d'une seule dent. Le coefficient de sécurité contre la rupture de
dent, S , plus élevé que le coefficient de sécurité contre la détérioration due à la formation de piqûres, est donc
F
généralement préféré (voir ISO 10300-1).
© ISO 2001 – Tous droits réservés v
---------------------- Page: 5 ----------------------
NORME INTERNATIONALE ISO 10300-3:2001(F)
Calcul de la capacité de charge des engrenages coniques —
Partie 3:
Calcul de la résistance du pied de dent
1 Domaine d'application
La présente partie de l'ISO 10300 spécifie les formules fondamentales à utiliser dans le calcul de la contrainte de
flexion de la dent d'engrenages coniques droits, coniques hélicoïdaux, coniques «zerol» et spiroconiques, avec une
épaisseur de jante minimale sous pied W 3,5 m . Toutes les influences de la charge sur la contrainte sont inclues,
mn
dans la mesure où elles sont le résultat des charges transmises par l’engrenage et à même d’être évaluées
quantitativement. (Les contraintes, telles que celles provoquées par le frettage de la jante, en plus de celles dues à
la charge sur les dents, sont à prendre en compte dans le calcul de la contrainte au pied de dent s , ou de la
F
contrainte admissible au pied de dent s .)
FP
Les formules données dans la présente partie de l'ISO 10300 sont valables pour les engrenages coniques pour
lesquels le rapport de conduite apparent des dents est e < 2. Les résultats sont valables dans le domaine ou les
va
facteurs s’appliquent, comme indiqué dans l'ISO 10300-1 et dans l'ISO 6336-3.
La présente partie de l'ISO 10300 ne s'applique pas aux niveaux de contrainte au-dessus de ceux permis pour
3
10 cycles, puisque les contraintes dans ce domaine peuvent dépasser la limite élastique de la dent.
AVERTISSEMENT — L'utilisateur est mis en garde sur le fait qu’il convient que, lorsque ces méthodes sont
utilisées pour des angles de spirale et de pression importants, et pour de grandes largeurs de denture
b > m , les résultats des calculs effectués conformément à l’ISO 10300 soient confirmés par l'expérience.
mn
2 Références 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 10300. 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 10300 sont invitées à rechercher la possibilité d’appliquer les
éditions les plus récentes des documents normatifs indiqués ci-après. Pour les références non datées, la dernière
édition du document normatif en référence s'applique. Les membres de l’ISO et de la CEI possèdent le registre des
Normes internationales en vigueur.
ISO 53:1998, Engrenages cylindriques de mécanique générale et de grosse mécanique — Tracé de référence.
ISO 1122-1:1998, Vocabulaire des engrenages — Partie 1: Définitions géométriques.
ISO 6336-3, Calcul de la capacité de charge des engrenages cylindriques à denture droite et hélicoïdale —
Partie 3: Calcul de la résistance à la flexion en pied de dent.
ISO 6336-5, Calcul de la capacité de charge des engrenages cylindriques à denture droite et hélicoïdale —
Partie 5: Résistance et qualité des matériaux.
ISO 10300-1:2001, Calcul de la capacité de charge des engrenages coniques — Partie 1: Introduction et facteurs
généraux d'influence.
© ISO 2001 – Tous droits réservés 1
---------------------- Page: 6 ----------------------
ISO 10300-3:2001(F)
ISO 10300-2, Calcul de la capacité de charge des engrenages coniques — Partie 2: Calcul de la résistance à la
pression superficielle (formation des piqûres).
3 Termes et définitions
Pour les besoins de la présente partie de l'ISO 10300, les termes et définitions donnés dans l’ISO 53 et dans
l’ISO 1122-1, et le terme et la définition suivants, s’appliquent.
3.1
contrainte de flexion de dent
capacité de charge déterminée sur la base de la contrainte de flexion admissible
4 Symboles et abréviations
Pour les besoins de la présente partie de l'ISO 10300, les symboles et abréviations donnés dans le Tableau 1 de
l’ISO 10300-1:2001, et les abréviations données dans le Tableau 1, s’appliquent.
Tableau 1 — Abréviations
Abréviation Description
2
St
acier (s < 800 N/mm )
B
2
V
acier traité dans la masse (s W 800 N/mm )
B
GG fonte grise
GGG (perl., bai., ferr.) fonte à graphite sphéroïdale (structure perlitique, bainitique, ferritique)
GTS (perl.) fontes malléables (structure perlitique)
Eh acier de cémentation, cémenté
IF (pied) acier et GGG, durcis superficiellement à la flamme ou par induction (y compris
le profil de raccordement en pied)
NT (nitr.) acier de nitruration, nitruré
NV (nitr.) acier traité dans la masse et acier cémenté, nitrurés
NV (nitrocar.) acier traité dans la masse et acier cémenté, nitrocarburés
5 Rupture de dent et coefficients de sécurité
La rupture de dent termine habituellement la durée de vie d'une transmission. Parfois, la destruction de tous les
engrenages dans une transmission est due à la rupture d'une seule dent, tandis que, dans certains cas, la
circulation de puissance entre les arbres d'entrée et de sortie est rompue.
Par conséquent, il convient de choisir une valeur du coefficient de sécurité, S , contre la rupture de dent plus grande
F
que le carré du coefficient de sécurité, S , contre la formation de piqûres (voir l’ISO 10300-1 pour les commentaires
H
généraux sur le choix du coefficient de sécurité).
Il convient que la valeur du coefficient de sécurité minimum pour la contrainte de flexion, S , soit W 1,3 pour les
Fmin
engrenages spiroconiques. Pour les engrenages à denture droite, ou lorsque b u 5°, il convient que S soit
m Fmin
W 1,5. Il est recommandé que le fabricant et l'acheteur se mettent d'accord sur les valeurs du coefficient de sécurité
minimum.
2 © ISO 2001 – Tous droits réservés
---------------------- Page: 7 ----------------------
ISO 10300-3:2001(F)
6 Formules de calcul de la capacité de charge
6.1 Généralités
La capacité d'une denture à résister à la flexion doit être déterminée par la comparaison des deux valeurs
suivantes:
a) contrainte de flexion, basée sur la géométrie de la dent, la précision de sa fabrication, la rigidité des corps de
roue, des paliers et du carter, et le couple de service, exprimée par la formule de contrainte de flexion (voir
6.2);
b) contrainte admissible de référence, et l'effet des conditions de fonctionnement dans lesquelles les
engrenages fonctionnent, exprimée par la formule de la pression de contrainte de flexion admissible (voir 6.3).
La contrainte effective au pied de dent, s , doit être u s , qui est la contrainte admissible au pied de dent.
F FP
NOTE Eu égard à la contrainte admissible, il est fait mention d'une «contrainte de référence», désignation adoptée parce
qu'une contrainte pure, telle que déterminée par des essais de laboratoire, n'est pas calculée par les formules de la présente
partie de l’ISO 10300. À la place, une valeur arbitraire est calculée et utilisée partout, avec les changements adéquats dans la
contrainte admissible de référence pour le maintien de la cohésion dans les comparaisons de conceptions.
6.2 Contrainte au pied de dent
6.2.1 Généralités
La contrainte au pied de dent est déterminée séparément pour le pignon et la roue.
s = s K K K K u s (1)
F FO A v Fb Fa FP
où
s est la contrainte de base au pied de dent définie comme la contrainte maximale de traction se produisant
FO
au pied de dent sous le couple nominal lorsqu'une roue parfaite est chargée.
Pour K , K , K , K , voir l'ISO 10300-1
A v Fb Fa
6.2.2 Contrainte de base au pied de dent, s — Méthode B1
F0-B1
Le calcul de la contrainte de base au pied de dent est basé sur la contrainte maximale de traction au pied de dent
(tangente à 30° au profil de raccordement du pied de dent). La position critique de l'application de la charge est:
a) le point le plus haut de contact unique (e = 0);
vb
b) le point central de la zone de contact (e W 1);
vb
c) l'interpolation entre a) et b) (0 < e < 1).
vb
La transformation depuis le sommet jusqu’à cette position critique se fait avec Y .
e
F
mt
s = Y Y Y Y Y (2)
F0-B1 Fa Sa e K LS
bm
mn
où
F est la force tangentielle nominale sur le cône de référence à mi-largeur de denture (voir l'ISO 10300-1);
m
© ISO 2001 – Tous droits réservés 3
---------------------- Page: 8 ----------------------
ISO 10300-3:2001(F)
b est la largeur de denture;
Y est le facteur de forme de dent (voir article 7), qui tient compte de l'influence de la forme de dent sur la
Fa
contrainte nominale de flexion pour l'application de charge au sommet de dent;
Y est le facteur de concentration de contrainte (voir article 7), qui tient compte de la conversion de la
Sa
contrainte nominale de flexion pour l'application de la charge au sommet de dent, en contrainte locale
correspondante au pied de dent. Ainsi Y tient compte de l'effet d'augmentation de la contrainte du fait de
Sa
l’effet d’entaille (au profil de raccordement du pied), ainsi que du fait que l’état de contrainte à la section
critique du pied est complexe, cependant, il ne tient pas compte de l'influence du bras de levier du
moment de flexion;
Y est le facteur du rapport de conduite (voir article 8), qui tient compte de la conversion de la contrainte
e
locale déterminée pour l'application de charge au sommet de dent pour la position critique;
Y est le facteur d'engrenage conique, qui tient compte de valeurs inférieures pour l ’ par comparaison avec
K b
la largeur totale de denture b et les lignes de contact inclinées;
Y est le facteur de répartition de charge, qui tient compte de la répartition de charge entre deux ou plusieurs
LS
paires de dents.
6.2.3 Contrainte de base au pied de dent, s — Méthode B2
F0-B2
En appliquant la méthode B2, le facteur géométrique combiné Y remplace les facteurs Y , Y , Y , Y et Y dans
P Fa Sa e LS K
l'équation de la contrainte de base au pied de dent, de façon que l'équation (2) devienne:
F
mt
= (3)
s Y
F0-B2 P
bm
mn
La valeur de Y est déterminée par:
P
mm
Y
A mt mn
= (4)
Y
P
2
Y
m
J
et
En remplaçant dans l'équation (3):
m
F Y
mt mt A
= (5)
s
F0-B2
2
b
Y
m J
et
où
Y est le facteur d'ajustement de l'engrenage conique pour la méthode B2, pour des engrenages coniques de
A
référence cémentés et durcis superficiellement (voir annexe A);
Y est le facteur géométrique de résistance à la flexion pour la méthode B2 (voir 9.2).
J
Le facteur géométrique de résistance à la flexion, Y , évalue la forme de la dent, la position à laquelle la charge la
J
plus endommageante est appliquée, la concentration de contrainte due à la forme géométrique du profil de
raccordement du pied de dent, la répartition de charge entre des paires de dents voisines, la répartition d'épaisseur
de dent entre la roue et le pignon conjugué, la largeur de denture effective due au bombé longitudinal des dents, et
l'effet de contrefort d'une largeur de denture plus grande sur un membre de la paire. Les deux composantes
tangentielle (flexion) et radiale (compression) de la charge de dent sont introduites.
4 © ISO 2001 – Tous droits réservés
---------------------- Page: 9 ----------------------
ISO 10300-3:2001(F)
6.3 Contrainte admissible au pied de dent
6.3.1 Généralités
La contrainte admissible au pied de dent, s , est déterminée séparément pour le pignon et la roue. Elle est
FP
calculée, de préférence, sur la base de la résistance déterminée sur un engrenage réel car, dans ce cas, la valeur
de référence de la similitude géométrique et de la fabrication se trouve dans le domaine d'application.
s Y
FE NT
= (6)
s YY Y
FP d rel T R rel T X
S
Fmin
s YY
Flim ST NT
= (7)
s YY Y
FP d rel T R rel T X
S
Fmin
où
s est la contrainte admissible de référence (flexion);
FE
s = s Y , la résistance à la flexion de base d'une éprouvette non entaillée, dans l'hypothèse
FE F lim ST
où le matériau (avec son traitement thermique) est totalement élastique;
s est la contrainte de référence à la fatigue de flexion pour la contrainte nominale en flexion des roues
F lim
d'essai (tient compte du matériau, du traitement thermique et de l'influence de la surface sur les
dimensions de la roue d'essai), voir l'ISO 6336-5;
Y est le facteur de concentration de contrainte pour les dimensions de la roue d'essai de référence
ST
Y = 2,0;
ST
S est le coefficient de sécurité minimal (voir l'ISO 10300-1);
F min
Y est le facteur de sensibilité relative (voir article 10) (pour la contrainte admissible de référence) par
d rel T
rapport aux conditions à l'engrenage d'essai de référence, (Y = Y /Y tient compte de la
d rel T d dT
sensibilité à l'entaille du matériau);
Y est le facteur de surface relatif (voir article 11) (Y = Y /Y tient compte de l'état de surface du
R rel T R rel T R RT
profil de raccordement du pied, par rapport aux conditions de l'engrenage d'essai);
Y est le facteur de dimension pour la résistance du pied de dent (voir article 12), qui tient compte de
X
l'influence du module sur la résistance du pied de dent;
Y est le facteur de durée de vie (tient compte de l'influence d'un nombre de cycles de mise en charge
NT
exigé (voir article 13).
6.3.2 Coefficient de sécurité calculé pour la contrainte au pied de dent
Le coefficient de sécurité calculé contre la rupture de dent doit être déterminé séparément pour le pignon et la roue.
Sur la base de la contrainte admissible de référence (flexion), le coefficient de sécurité déterminé pour la roue
d'essai de référence est:
YY Y
s Y d rel T R rel T X
FE NT
S= (8)
F
s KK K K
F0 A v FFba
NOTE Ceci est le coefficient de sécurité calculé vis-à-vis du couple transmis.
Pour des références sur le coefficient de sécurité ou du risque de dégradation (probabilité de dégradation), voir
l'ISO 10300-1.
© ISO 2001 – Tous droits réservés 5
---------------------- Page: 10 ----------------------
ISO 10300-3:2001(F)
7 Facteur de forme, Y , et facteur de concentration de contrainte, Y — Méthode B1
Fa Sa
7.1 Généralités
Le facteur de forme de dent, Y , tient compte de l'influence de la forme de dent sur la contrainte de flexion
Fa
nominale dans le cas de l'application de la charge au sommet de la dent. Il est déterminé séparément pour le
pignon et la roue.
NOTE Dans le cas de roue avec dépouille de tête et de pied, le bras de levier du moment de flexion réel est légèrement
inférieur; dans ce cas, le calcul est du côté de la sécurité.
Les roues coniques ont généralement des dents octoïdes et une dépouille de tête et de pied. Cependant, les
différences avec un profil en développante sont minimes, particulièrement en ce qui concerne la corde au pied de
dent et le bras de levier du moment de flexion. Par conséquent, elles peuvent être négligées lors du calcul du
facteur de forme.
La distance entre les points de contact des tangentes à 30° aux profils de raccordement du pied de dent de la roue
cylindrique équivalente est prise comme section de calcul (voir Figure 1).
Dans la présente partie de l’ISO 10300, les facteurs Y et Y sont déterminés pour la roue nominale sans
Fa Sa
tolérance. La légère réduction d'épaisseur de dent pour le jeu entre les dents peut être négligée pour le calcul de la
capacité de charge. La réduction de dimension est à prendre en compte uniquement lorsque la tolérance
d'épaisseur de dent extérieure est A > 0,05 m .
sne mn
a
Cercle de base de la roue cylindrique équivalente
Figure 1 — Corde au pied de dent, s , et bras de levier du moment de flexion pour l'application de la
Fn
charge au sommet de dent, h , d'un profil de dent d'une roue cylindrique équivalente
Fa
7.2 Y pour les roues obtenues par génération
Fa
7.2.1 Généralités
L'équation (9) s'applique à la roue cylindrique équivalente dans un plan normal avec et sans déport. Le calcul est
valable avec les hypothèses suivantes:
a) le point de contact des tangentes à 30° a lieu sur la courbe du profil de raccordement générée par le rayon de
tête d'outil;
b) l'outil est fabriqué avec un rayon de tête fini (r ≠ 0).
a0
6 © ISO 2001 – Tous droits réservés
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ISO 10300-3:2001(F)
h
Fa
6cosa
Fan
m
mn
= (9)
Y
Fa
2
ʈ
s
Fn
cosa
n
Á˜
m
mn
˯
Voir la Figure 1 pour les définitions; voir l'ISO 6336-3 pour l'évaluation de la charge normale critique et le facteur de
forme.
7.2.2 Valeurs auxiliaires
Pour le calcul de la corde au pied de dent, s , et du bras de levier du moment de flexion, h , les valeurs auxiliaires
Fn Fa
E, G, H et J doivent d'abord être déterminées.
r 1s--ina
( ) s
ʈp
n pr
a0
E= -- tana- (10)
xm h
Á˜sm mn a0 n
˯
4cosa
n
r
h
a0 a0
G= - + x (11)
hm
mm
mn mn
ʈ
2ppE
H= -- (12)
Á˜
z
vn23m
˯
mn
2G
JJ =Htan - (13)
z
vn
Pour la solution de l'équation transcendante (13), J p/6 peut être introduite comme valeur initiale. Dans la plupart
=
des cas, l'équation converge déjà après quelques itérations.
7.2.3 Corde au pied de dent, s
Fn
ʈr
p G
s ʈ
Fn a0
=+sin--J 3 (14)
z
vn
Á˜
Á˜
m
mn ˯ m
3c˯osJ
mn
7.2.4 Rayon de raccordement, r , au point de contact de la tangente à 30°
F
2
r r
2G
F a0
=+ (15)
2
mm
mn mn cosJJ z cos - 2G
()vn
7.2.5 Bras de levier du moment de flexion, h
Fa
ʈ
d
vbn
a = arc cos (16)
an
Á˜
d
˯
van
1 p
È ˘
g=2+x( ◊tanaa+x)+inv -inva (17)
ahmnsm nan
Í ˙
z 2
Î ˚
vn
aa = -g (18)
Fan an a
© ISO 2001 – Tous droits réservés 7
---------------------- Page: 12 ----------------------
ISO 10300-3:2001(F)
È r ˘
1 ʈp G
hd
Fa van a0
= --sin tanaJ z cos -- + (19)
cosgg
()
Í Fan vn Á˜ ˙
aa
mm23˯cosJm
mn mn mn
Î ˚
Voir l'annexe A de l'ISO 10300-1:2001 pour les données concernant les roues cylindriques équivalentes dans un
plan normal. Les dimensions du tracé de référence de l'outil sont représentées à la Figure 2, tandis que Y peut
Fa
être pris de la Figure 3 pour le tracé de référence de l'outil avec les données a = 20°, h /m = 1,25 et
n a0 mn
r /m = 0,25. Les diagrammes pour d'autres tracés de référence sont donnés dans l'ISO 6336-3.
a0 mn
Voir les Figures 4 à 6 pour le facteur de forme combiné pour les engrenages coniques obtenus par génération
Y = Y Y .
FS Fa Sa
Figure 2 — Dimensions du tracé de référence de l'outil
8 © ISO 2001 – Tous droits réservés
---------------------- Page: 13 ----------------------
ISO 10300-3:2001(F)
Figure 3 — Facteur de forme, Y , pour les roues obtenues par génération
Fa
© ISO 2001 – Tous droits réservés 9
---------------------- Page: 14 ----------------------
ISO 10300-3:2001(F)
Figure 4 — Facteur de forme combiné Y ==== Y Y pour les roues obtenues par génération (r = 0,2 m )
FS Fa Sa a0 mn
10 © ISO 2001 – Tous droits réservés
---------------------- Page: 15 ----------------------
ISO 10300-3:2001(F)
Figure 5 — Facteur de forme combiné Y == Y Y pour les roues obtenues par génération
==
FS Fa S
...
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