ISO/TR 15144-2:2014
(Main)Calculation of micropitting load capacity of cylindrical spur and helical gears — Part 2: Examples of calculation for micropitting
Calculation of micropitting load capacity of cylindrical spur and helical gears — Part 2: Examples of calculation for micropitting
ISO/TR 15144-2:2014 gives example calculations presented for guidance on the application of ISO/TR 15144-1 only. Any of the values or the data presented are not intended to be used as material or lubricant allowables or as recommendations for micro-geometry in real applications when applying this procedure. The necessary parameters and allowable film thickness values are intended to be determined for a given application in accordance with the procedures defined in ISO/TR 15144-1.
Calcul de la capacité de charge aux micropiqûres des engrenages cylindriques à dentures droite et hélicoïdale — Partie 2: Exemples de calcul pour micropiqûres
L ISO/TR 15144-2:2014 exemples de calcul présentés ici sont uniquement destinés à servir de guide pour l'application du rapport technique ISO/TR 15144-1. Il convient de n'utiliser, dans des applications réelles, aucune des valeurs ou données présentées ici comme des valeurs admissibles pour les matériaux ou les lubrifiants ou des recommandations pour la micro-géométrie lors de l'application de cette méthode. Il convient que les paramètres nécessaires et les valeurs admissibles d'épaisseur de film, λGFP, soient déterminés pour une application donnée conformément aux méthodes définies dans l'ISO/TR 15144-1.
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TECHNICAL ISO/TR
REPORT 15144-2
First edition
2014-10-01
Corrected version
2015-01-15
Calculation of micropitting load
capacity of cylindrical spur and
helical gears —
Part 2:
Examples of calculation for micropitting
Calcul de la capacité de charge aux micropiqûres des engrenages
cylindriques à dentures droite et hélicoïdale —
Partie 2: Exemples de calcul pour micropiqûres
Reference number
ISO/TR 15144-2:2014(E)
©
ISO 2014
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ISO/TR 15144-2:2014(E)
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ii © ISO 2014 – All rights reserved
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ISO/TR 15144-2:2014(E)
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 Example calculation . 4
4.1 Example 1 — Spur gear . 5
4.1.1 Input data . 6
4.1.2 Calculation according to method B . 7
4.1.3 Calculation according to method A .12
4.1.4 Calculation of the permissible lubricant film thickness.13
4.2 Example 2 — Spur gear .19
4.2.1 Input data .20
4.2.2 Calculation according to method B .21
4.3 Example 3 — Helical gear .28
4.3.1 Input data .29
4.3.2 Calculation according to method B .30
4.3.3 Calculation according to method A .36
4.4 Example 4 — Speed increaser .37
4.4.1 Input data .38
4.4.2 Calculation according to method B .39
4.4.3 Calculation according to method A .45
Bibliography .47
© ISO 2014 – All rights reserved iii
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ISO/TR 15144-2:2014(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.
The procedures used to develop this document and those intended for its further maintenance are
described in the ISO/IEC Directives, Part 1. In particular the different approval criteria needed for the
different types of ISO documents should be noted. This document was drafted in accordance with the
editorial rules of the ISO/IEC Directives, Part 2 (see www.iso.org/directives).
Attention is drawn to the possibility that some of the elements of this document may be the subject of
patent rights. ISO shall not be held responsible for identifying any or all such patent rights. Details of
any patent rights identified during the development of the document will be in the Introduction and/or
on the ISO list of patent declarations received (see www.iso.org/patents).
Any trade name used in this document is information given for the convenience of users and does not
constitute an endorsement.
For an explanation on the meaning of ISO specific terms and expressions related to conformity
assessment, as well as information about ISO’s adherence to the WTO principles in the Technical Barriers
to Trade (TBT) see the following URL: Foreword - Supplementary information
The committee responsible for this document is ISO/TC 60, Gears, Subcommittee SC 2, Gear capacity
calculation.
This corrected version of ISO/TR 15144-2:2014 incorporates the following corrections: errors in symbols
and equations have been corrected.
ISO/TR 15144 consists of the following parts, under the general title Calculation of micropitting load
capacity of cylindrical spur and helical gears:
— Part 1: Introduction and basic principles
— Part 2: Examples of calculation for micropitting
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ISO/TR 15144-2:2014(E)
Introduction
This part of ISO/TR 15144 provides worked examples for the application of the calculation procedures
defined in ISO/TR 15144-1. The example calculations cover the application to spur and helical cyclindrical
involute gears for both high-speed and low-speed operating conditions, determining the micropitting
safety factor for each gear pair. The calculation procedures used are consistent with those presented in
ISO/TR 15144-1. No additional calculations are presented here that are outside of the technical report.
Four worked examples are presented with the necessary input data for each gear set provided at the
beginning of the calculation. The worked examples are based on real gear pairs where either laboratory
or operational field performance data has been established, with the examples covering several
applications. When available, pictures and measurements are provided of the micropitting wear,
experienced on the gear sets when run under the conditions used in the worked examples. Calculation
details are presented in full for several of the initial calculations after which only summarized results
data are included. For better applicability, the numbering of the formulae follows ISO/TR 15144-1.
Several of the worked examples are presented with the calculation procedures performed in accordance
with the application of both methods A and B.
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TECHNICAL REPORT ISO/TR 15144-2:2014(E)
Calculation of micropitting load capacity of cylindrical
spur and helical gears —
Part 2:
Examples of calculation for micropitting
1 Scope
The example calculations presented here are provided for guidance on the application of the technical
report ISO/TR 15144-1 only. Any of the values or the data presented should not be used as material or
lubricant allowables or as recommendations for micro-geometry in real applications when applying this
procedure. The necessary parameters and allowable film thickness values, λ , should be determined
GFP
for a given application in accordance with the procedures defined in ISO/TR 15144-1.
2 Normative references
The following documents, in whole or in part, are normatively referenced in this document and are
indispensable for its application. For dated references, only the edition cited applies. For undated
references, the latest edition of the referenced document (including any amendments) applies.
ISO 1122-1:1998, Vocabulary of gear terms — Part 1: Definitions related to geometry
ISO 6336-1:2006, Calculation of load capacity of spur and helical gears — Part 1: Basic principles,
introduction and general influence factors
ISO 6336-2:2006, Calculation of load capacity of spur and helical gears — Part 2: Calculation of surface
durability (pitting)
ISO 21771:2007, Gears — Cylindrical involute gears and gear pairs — Concepts and geometry
ISO/TR 15144-1:2014, Calculation of micropitting load capacity of cylindrical spur and helical gears — Part
1: Introduction and basic principles
3 Terms, definitions, symbols, and units
3.1 Terms and definitions
For the purpose of this document, the terms and definitions given in ISO 1122-1, ISO 6336-1, and
ISO 6336-2 apply.
3.2 Symbols and units
The symbols used in this technical report are given in Table 1. The units of length metre, millimetre, and
micrometre are chosen in accordance with common practice. The conversions of the units are already
included in the given formulae.
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ISO/TR 15144-2:2014(E)
Table 1 — Symbols and units
Symbol Description Unit
a centre distance mm
0,5
B thermal contact coefficient of pinion N/(m·s ·K)
M1
0,5
B thermal contact coefficient of wheel N/(m·s ·K)
M2
b face width mm
C tip relief of pinion µm
a1
C tip relief of wheel µm
a2
c specific heat per unit mass of pinion J/(kg·K)
M1
c specific heat per unit mass of wheel J/(kg·K)
M2
c’ maximum tooth stiffness per unit face width (single stiffness) of a tooth pair N/(mm·µm)
c mean value of mesh stiffness per unit face width N/(mm·µm)
γα
d tip diameter of pinion mm
a1
d tip diameter of wheel mm
a2
d base diameter of pinion mm
b1
d base diameter of wheel mm
b2
d pitch diameter of pinion mm
w1
d pitch diameter of wheel mm
w2
d Y-circle diameter of pinion mm
Y1
d Y-circle diameter of wheel mm
Y2
2
E reduced modulus of elasticity N/mm
r
2
E modulus of elasticity of pinion N/mm
1
2
E modulus of elasticity of wheel N/mm
2
F nominal transverse load in plane of action (base tangent plane) N
bt
F (nominal) transverse tangential load at reference cylinder per mesh N
t
G material parameter -
M
g parameter on the path of contact (distance of point Y from point A) mm
Y
g length of path of contact mm
α
H load losses factor -
v
h local lubricant film thickness µm
Y
K application factor -
A
K transverse load factor -
Hα
K face load factor -
Hβ
K dynamic factor -
v
−1
n rotation speed of pinion min
1
P transmitted power kW
p transverse base pitch on the path of contact Mm
et
2
p local Hertzian contact stress including the load factors K N/mm
dyn,Y
2
p local nominal Hertzian contact stress N/mm
H,Y
Ra effective arithmetic mean roughness value µm
Ra arithmetic mean roughness value of pinion µm
1
Ra arithmetic mean roughness value of wheel µm
2
S local sliding parameter -
GF,Y
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ISO/TR 15144-2:2014(E)
Table 1 (continued)
Symbol Description Unit
S safety factor against micropitting -
λ
S minimum required safety factor against micropitting -
λ,min
T nominal torque at the pinion Nm
1
U local velocity parameter -
Y
u gear ratio -
v local sliding velocity m/s
g,Y
v local tangential velocity on pinion m/s
r1,Y
v local tangential velocity on wheel m/s
r2,Y
v sum of tangential velocities at pitch point m/s
Σ,C
v sum of tangential velocities at point Y m/s
Σ,Y
W material factor -
W
W local load parameter -
Y
X local buttressing factor -
but,Y
X tip relief factor -
Ca
X lubricant factor -
L
X roughness factor -
R
X lubrication factor -
S
X local load sharing factor -
Y
2 0,5
Z elasticity factor (N/mm )
E
z number of teeth of pinion -
1
z number of teeth of wheel -
2
α transverse pressure angle °
t
α pressure angle at the pitch cylinder °
wt
2
α pressure-viscosity coefficient at local contact temperature m /N
θB,Y
2
α pressure-viscosity coefficient at bulk temperature m /N
θM
2
α pressure-viscosity coefficient at 38 °C m /N
38
β base helix angle °
b
ε maximum addendum contact ratio -
max
ε transverse contact ratio -
α
ε virtual transverse contact ratio -
αn
ε overlap ratio -
β
ε total contact ratio -
γ
ε addendum contact ratio of the pinion -
1
ε addendum contact ratio of the wheel -
2
2
η dynamic viscosity at local contact temperature N·s/m
θB,Y
2
η dynamic viscosity at bulk temperature N·s/m
θM
2
η dynamic viscosity at oil inlet/sump temperature N·s/m
θoil
2
η dynamic viscosity at 38 °C N·s/m
38
θ local contact temperature °C
B,Y
θ local flash temperature °C
fl,Y
θ bulk temperature °C
M
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ISO/TR 15144-2:2014(E)
Table 1 (continued)
Symbol Description Unit
θ oil inlet/sump temperature °C
oil
λ minimum specific lubricant film thickness in the contact area -
GF,min
λ local specific lubricant film thickness -
GF,Y
λ permissible specific lubricant film thickness -
GFP
λ limiting specific lubricant film thickness of the test gears -
GFT
λ specific heat conductivity of pinion W/(m·K)
M1
λ specific heat conductivity of wheel W/(m·K)
M2
µ mean coefficient of friction -
m
2
ν kinematic viscosity at local contact temperature mm /s
θB,Y
2
ν kinematic viscosity at bulk temperature mm /s
θM
ν Poisson’s ratio of pinion -
1
ν Poisson’s ratio of wheel -
2
2
ν kinematic viscosity at 100 °C mm /s
100
2
ν kinematic viscosity at 40 °C mm /s
40
3
ρ density of pinion kg/m
M1
3
ρ density of wheel kg/m
M2
ρ normal radius of relative curvature at pitch diameter mm
n,C
ρ normal radius of relative curvature at point Y mm
n,Y
ρ transverse radius of relative curvature at point Y mm
t,Y
ρ transverse radius of curvature of pinion at point Y mm
t1,Y
ρ transverse radius of curvature of wheel at point Y mm
t2,Y
3
ρ density of lubricant at local contact temperature kg/m
θB,Y
3
ρ density of lubricant at bulk temperature kg/m
θM
3
ρ density of lubricant at 15 °C kg/m
15
Subscripts to symbols
Y Parameter for any contact point Y in the contact area for method A and on the path of contact for
method B (all parameters subscript Y has to be calculated with local values).
4 Example calculation
The following presents examples for the calculation of the safety factor against micropitting, S . Each
λ
example is first calculated according to method B and examples 1, 3, and 4 subsequently calculated
according to method A. The calculation sequence for method B has been provided to follow a logical
approach in relation to the input data. Beside the formulae itself, the formula numbers related to
ISO/TR 15144-1 are given.
The examples calculate the safety factor S of a specific gear set when compared to an allowable λ
λ GFP
value. For the examples 1, 2, and 4, the permissible specific oil film thickness, λ , was determined
GFP
[1]
from the test result of the lubricant in the FZG-FVA micropitting test. For these calculations medium
values for the standard FZG back-to-back test rig and standard test conditions for K and K were
Hβ v
used (K = 1,10 and K = 1,05). The calculation of the λ value from the test result of the FZG-FVA
Hβ v GFP
[1]
micropitting test (method B) is shown exemplary on the basis of the first example. For example 3, the
permissible specific oil film thickness, λ , was determined from a bench test.
GFP
NOTE The calculations were performed computer-based. If the calculations are performed manually, small
differences between the results can appear.
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ISO/TR 15144-2:2014(E)
4.1 Example 1 — Spur gear
The result of this example is confirmed by experimental investigations. The gears were obviously
micropitted and had profile deviations of approximately 8 to 10 µm. Figure 1 shows a diagram of the
observed location and severity of micropitting for pinion and wheel of example 1.
a) pinion b) wheel
Key
1 tip
2 root
Figure 1 — Diagram of schematic profile deviations of pinion and wheel for example 1
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ISO/TR 15144-2:2014(E)
4.1.1 Input data
Table 2 — Input data for Example 1
Example 1
Symbol Description Unit pinion wheel
comb.
z number of teeth - 18 18
- driving gear - x
m
normal module mm 10,93
n
α normal pressure angle ° 20
n
β helix angle ° 0
b face width mm 21,4
Geometry
a centre distance mm 200
x addendum modification factor - 0,158 0,158
d tip diameter of pinion mm 221,4 221,4
a
- tooth flank modifications - no modifications
Q gear quality - 5 5
Ra arithmetic mean roughness value µm 0,90 0,90
- material - Eh Eh
2
E modulus of elasticity N/mm 206 000 206 000
ν Poisson’s ratio - 0,3 0,3
λ specific heat conductivity W/(m·K) 45 45
M
Material
c specific heat per unit mass J/(kg·K) 440 440
M
3
ρ density kg/m 7 800 7 800
M
material factor according to ISO/
W TR 15144-1:2014, Table A.1 (for matching - 1,0
w
case carburised/case carburised)
K application factor - 1,0
A
K dynamic factor - 1,15
v
Application
K transverse load factor - 1,0
Hα
K face load factor - 1,10
Hβ
T nominal torque at the pinion Nm 1 878
1
Load
−1
n rotation speed of the pinion min 3 000
1
oil inlet temperature (injection lubrica-
θ °C 90
oil
tion)
2
ν kinematic viscosity at 40 °C mm /s 210
40
2
ν kinematic viscosity at 100 °C mm /s 18,5
100
3
ρ density of the lubricant at 15 °C kg/m 895
15
Lubricant
- oil type - mineral oil
failure load stage at test temperature
- - SKS 8
(90 °C) according to FVA 54/7
permissible lubricant film thickness
λ - 0,211
GFP
(see 4.1.4 for calculation)
6 © ISO 2014 – All rights reserved
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ISO/TR 15144-2:2014(E)
4.1.2 Calculation according to method B
4.1.2.1 Calculation of gear geometry (according to ISO 21771)
Basic values:
m = 10,93 mm
m t
n
m =
t
cosβ
d = z ∙ m d = 196,74 mm
1 1 t 1
d = z ∙ m d = 196,74 mm
2 2 t 2
u = 1
z
2
u=
z
1
α = 20 °
t
tanα
n
α =arctan
t
cosβ
d = d cosα d = 184,875 mm
b1 1 t b1
d = d cosα d = 184,875 mm
b2 2 t b2
d = 200 mm
w1
2⋅a
d =
w1
u+1
d = 200 mm
w2
da=⋅2 −d
w2 w1
α = 22,426 °
wt
zz+ ⋅⋅m cosα
()
12 tt
α =arccos
wt
2⋅a
ββ=⋅arcsin(sin cos)α
bn β = 0 °β =0
b
b
p = 32,267 mm
et
pm=⋅πα⋅cos
et tt
ε = 0,705
1
2
d
z
a1
1
ε = ⋅ −−1 tanα
1 wt
2⋅π
d
b1
ε = 0,705
2
2
z d
2 a2
ε = ⋅ −−1 tanα
2 wt
2⋅π d
b2
ε = 1,411
α
2 22 2
1 dd dd
ab1 1 a2 b2
εα=⋅ −+ −−a⋅sin
α wt
p 44 44
et
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ISO/TR 15144-2:2014(E)
ε =0
β
b⋅sinβ
ε =
β
m ⋅π
n
ε = ε + ε ε = 1,411
γ α β γ
g = 45,519 mm
α
2 2 22
gd=⋅05,s−+dd −da−⋅ inα
α ab1 1 ab2 2wt
Coordinates of the basic points (A, AB, B, C, D, DE, E) on the line of action:
g =0mm
(34) g = 0 mm
A
A
gp−
α et
g =
(35) g = 6,626 mm
AB
AB
2
g = g − p (36) g = 13,253 mm
B α et B
22
dd d
b1 a1 b1
(37) g = 22,760 mm
C
g =⋅tanα −− +g
C wt α
24 4
gp=
(38) g = 32,267 mm
Det
D
gp−
α et
g = +p
(39) g = 38,893 mm
DE
DE et
2
g = g (40) g = 45,519 mm
E α E
2
22 2
dd d
b1 a1 b1
(41) d = 187,419 mm
d =⋅2 +− −+gg
A1
A1 α A
44 4
d = 190,046 mm d = 193,546 mm d = 200,000 mm
AB1 B1 C1
d = 207,998 mm d = 214,394 mm d = 221,400 mm
D1 DE1 E1
2
22 2
dd d
b2 a2 b2
d =⋅2 +− −g (42) d = 221,400 mm
A2
A2 A
44 4
d = 214,394 mm d = 207,998 mm d = 200,000 mm
AB2 B2 C2
d = 193,546 mm d = 190, 046 mm d = 187,419 mm
D2 DE2 E2
Normal radius of relative curvature:
ρ
t,A
ρ =
(45) ρ = 12,285 mm
n,A n,A
cosβ
b
ρ = 15,663 mm ρ = 17,890 mm ρ = 19,074 mm
n,AB n,B n,C
ρ = 17,890 ρ = 15,663 mm ρ = 12,285 mm
n,D n,DE n,E
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ISO/TR 15144-2:2014(E)
4.1.2.2 Calculation of material data
−1
2 2
11−νν−
1 2
2
(6) E = 226 374 N/mm
E =⋅2 +
r
r
EE
1 2
0,5
Bc=⋅λρ ⋅
(82) B = 12 427,4 N/(ms K)
M1 M1 M1 M1 M1
0,5
Bc=⋅λρ ⋅
(83) B = 12 427,4 N/(ms K)
M2 M2 M2 M2 M2
4.1.2.3 Calculation of operating conditions
Loading:
nT
1 1
P =⋅2 π⋅⋅
(85) P = 590 kW
60 1000
T
1
F =⋅2000
F = 19 091 N
t
t
d
1
T
1
F =⋅2000
bt F = 20 316 N
bt
d
b1
Local load sharing factor:
NOTE No tooth flank modifications, spur gears, gear quality ≤7 (see ISO/TR 15144-1:2014, Figure 2).
Q2− 1 g
A
X = +⋅
A (46) X = 0,333
A
15 3 g
B
X = 0,500 X = 1,000 X = 1,000
AB B C
X = 1,000 X = 0,500 X = 0,333
D DE E
Elasticity factor:
E
r
2 0,5
Z = (26) Z = 189,812 (N/mm )
E
E
2⋅π
Local Hertzian contact stress:
FX⋅
tA
pZ=⋅ 2
(25) p = 963 N/mm
H,A,BE
H,A,B
b⋅⋅ραcosc⋅ osβ
n,At b
2 2 2
p = 1 045 N/mm p = 1 383 N/mm p = 1 339 N/mm
H,AB,B H,B,B H,C,B
2 2 2
p = 1 383 N/mm p = 1 045 N/mm p = 963 N/mm
H,D,B H,DE,B H,E,B
pp=⋅ KK⋅⋅KK⋅ 2
(24) p = 1 084 N/mm
dynA,,BH,A,B AV HHαβ
dyn,A,B
2 2 2
p = 1 175 N/mm p = 1 555 N/mm p = 1 506 N/mm
dyn,AB,B dyn,B,B dyn,C,B
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ISO/TR 15144-2:2014(E)
2 2 2
p = 1 555 N/mm p = 1 175 N/mm p = 1 084 N/mm
dyn,D,B dyn,DE,B dyn,E,B
Velocity:
ν = ν − ν (81) ν = −14,300 m/s
g,A r1,A r2,A g,A
ν = −10,137 m/s ν = −5,974 m/s ν = 0 m/s
g,AB g,B g,C
ν = 5,974 m/s ν = 10,137 m/s ν = 14,300 m/s
g,D g,DE g,E
ν = ν + ν (13) ν = 23,969 m/s
Σ,A r1,A r2,A Σ,A
ν = 23,969 m/s ν = 23,969 m/s ν =23,969 m/s
Σ,AB Σ,B Σ,C
ν = 23,969 m/s ν = 23,969 m/s ν =23,969 m/s
Σ,D Σ,DE Σ,E
Effective arithmetic mean roughness value:
RRaa=⋅05, + Ra
()
(3) Ra = 0,90 μm
12
4.1.2.4 Calculation of lubricant data
X = 1,0 for mineral oil (see ISO/TR 15144-1:2014, Table 3)
L
−8 0,1348 −82
αη=⋅2,657 10 ⋅ (9) α =⋅21,/510 mN
38 38 38
X = 1,2 for injection lubrication
S
4.1.2.5 Calculation of the material parameter
Mean coefficient of friction:
02, 5
Ra
X =⋅22, (87) X = 1,025
R
R
ρ
n,C
K = 1,0 for ε <2
Bγ γ
02,
−00, 5
KK⋅⋅KK⋅⋅FK⋅
Av HHαβ bt Bγ
3
μ =⋅0,045 ⋅⋅10 η ⋅⋅XX
(86) μ = 0,048
m ()θθoilR L m
bv⋅⋅ρ
Σ,C n,C
Bulk temperature:
11 π
2 2
H =+εε +−1 ε ⋅+ ⋅ for ε < 2
(91) H = 0,204
α
v ()1 2 α v
zz cosβ
12 b
ε = ε = ε
max 1 2
X = 1,0 for no profile modification (method B) (101)
CA
07, 2
PH⋅⋅μ X
mv S
(84) θ = 153,6° C
θθ=+7400⋅ ⋅
M
Moil
ab⋅ 1,2⋅X
Ca
10 © ISO 2014 – All rights reserved
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ISO/TR 15144-2:2014(E)
Material parameter:
6
G = 10 ∙ α ∙ E (5) G = 2 678,6
M θM r M
4.1.2.6 Calculation of the velocity parameter
v
Σ,A
U =⋅η -11
(12) U = 2,005 ∙ 10
AMθ
A
2000⋅⋅E ρ
rn,A
-11 -11 -11
U = 1,572 ∙ 10 U = 1,377 ∙ 10 U = 1,291 ∙ 10
AB B C
-11 -11 -11
U = 1,377 ∙ 10 U = 1,572 ∙ 10 U = 2,005 ∙ 10
D DE E
4.1.2.7 Calculation of the load parameter
2
2⋅⋅π p
dyn,A
-4
W =
(22) W = 1,440 ∙ 10
A A
2
E
r
-4 -4 -4
W = 1,694 ∙ 10 W = 2,966 ∙ 10 W = 2,781 ∙ 10
AB B C
-4 -4 -4
W = 2,966 ∙ 10 W = 1,694 ∙ 10 W = 1,440 ∙ 10
D DE E
4.1.2.8 Calculation of the sliding parameter
Local flash temperature:
6
10 ⋅⋅μ pv⋅
p
π mdyn,A g,A
dyn,A
θ =⋅ ⋅⋅8 ρ ⋅ (80) θ = 175,3° C
fl,A
fl,A n,A
2 11000⋅E
Bv +Bv
r
M1 r1,A M2 r2,A
θ = 154,1° C θ = 145,4° C θ = 0° C
fl,AB fl,B fl,C
θ = 145,4° C θ = 154,1° C θ = 175,3° C
fl,D fl,DE fl,E
Local contact temperature as sum of bulk and local flash temperature:
θ = θ + θ (79) θ = 328,9° C
B,A M fl,A B,A
θ = 307,7° C θ = 299,0° C θ = 153,6° C
B,AB B,B B,C
θ = 299,0° C θ = 307,7° C θ = 328,9° C
B,D B,DE B,E
Local sliding parameter:
αη⋅
θθB,AB,A
S =
(27) S = 0,057
GF,A
GF,A
αη⋅
θθMM
S = 0,076 S = 0,086 S = 1,000
GF,AB GF,B GF,C
S = 0,086 S = 0,076 S = 0,057
GF,D GF,DE GF,E
4.1.2.9 Calculation of the lubricant film thickness
06,,07 −01,,3022
hG=⋅1600 ρ ⋅⋅UW⋅⋅S (4) h = 0,122 μm
A
An,A MA AGF,A
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ISO/TR 15144-2:2014(E)
h = 0,137 μm h = 0,136 μm h = 0,241 μm
AB B C
h = 0,136 μm h = 0,137 μm h = 0,122 μm
D DE E
4.1.2.10 Calculation of the specific lubricant film thickness
h
A
λ =
(2) λ = 0,136
GF,A
GF,A
Ra
λ =0,153 λ = 0,152 λ = 0,267
GF,AB GF,B GF,C
λ = 0,152 λ = 0,153 λ = 0,136
GF,D GF,DE GF,E
λ = λ = λ λ = 0,136
GF,min GF,A GF,E GF,min
4.1.2.11 Calculation of the micropitting safety factor
λ
GF,min
S =
(1) S = 0,644
λ λ
λ
GFP
The calculation of the permissible specific lubricant film thickness, λ , for example 1 is shown
GFP
exemplary in 4.1.4.
The final results for the calculation of the safety factor against micropitting, S , for example 1 are
λ
shown in Table 3.
Table 3 — Results of calculation according to method B — Example 1
Point A AB B C D DE E
λ 0,136 0,153 0,152 0,267 0,152 0,153 0,136
GF,Y
λ 0,136
GF,min
λ 0,211
GFP
S 0,644
λ
4.1.3 Calculation according to method A
The calculation of example 1 according to method A was carried out by a 3D-calculation programme.
Calculated results during method A will vary depending on the method of determining load distribution.
The load distribution, on which the following calculation according to method A is based, is shown in
Table 4. The maximum values are printed in bold.
2
Table 4 — Matrix of pressure distribution — p in N/mm
H,Y,A
Width in mm
0,0 7,6 13,8 21,4
A 1 115 1 110 1 110 1 114
AB 1 048 1 044 1 044 1 047
B 1 375 1 373 1 373 1 375
C 1 342 1 339 1 339 1 342
D 1 048 1 045 1 045 1 048
DE 1 050 1 046 1 046 1 050
E 1 099 1 094 1 094 1 099
12 © ISO 2014 – All rights reserved
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ISO/TR 15144-2:2014(E)
The resulting matrix of specific lubricant film thickness according to method A is shown in Table 5. The
minimum value is printed in bold.
Table 5 — Matrix of resulting specific lubricant film thickness λ
GF,Y
Width in mm
0,0 7,6 13,8 21,4
A 0,122 0,123 0,123 0,122
AB 0,159 0,160 0,160 0,159
B 0,159 0,159 0,159 0,159
C 0,270 0,271 0,271 0,270
D 0,197 0,198 0,198 0,197
DE 0,159 0,159 0,159 0,159
E 0,124 0,125 0,125 0,124
For the calculation of the micropitting safety factor according to method A, the minimum value of the
matrix of resulting specific lubricant film thickness, shown in Table 5, was used.
λ
GF,min
S =
(1) S = 0,577
λ λ
λ
GFP
NOTE The difference in safety factor calculated between methods A and B in the above example 1 results
from the simplified calculation of load distribution according to method B.
4.1.4 Calculation of the permissible lubricant film thickness
Calculation of the permissible specific lubricant film thickness from the test result of the FZG-FVA
[1]
micropitting test (Method B) with the reference test gears type C-GF.
The calculation of the reference value, λ , is done for point A because the minimum specific lubricant
GFT
film thickness for gear type C is always at point A. All data of the reference test gears type C-GF have the
subscript “Ref”.
© ISO 2014 – All rights reserved 13
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ISO/TR 15144-2:2014(E)
Table 6 — Input data for calculation of the permissible lubricant film thickness
C-GF
Symbol Description Unit pinion whe
...
TECHNICAL ISO/TR
REPORT 15144-2
First edition
Calculation of micro-pitting load
capacity of cylindrical spur and helical
gears —
Part 2:
Examples of calculation for
micropitting
Calcul de la capacité de charge aux micropiqûres des engrenages
cylindriques à dentures droite et hélicoïdale —
Partie 2: Exemples de calculation pour micropiqûres
PROOF/ÉPREUVE
Reference number
ISO/TR 15144-2:2014(E)
©
ISO 2014
---------------------- Page: 1 ----------------------
ISO/TR 15144-2:2014(E)
COPYRIGHT PROTECTED DOCUMENT
© ISO 2014
All rights reserved. Unless otherwise specified, no part of this publication may be reproduced or utilized otherwise in any form
or by any means, electronic or mechanical, including photocopying, or posting on the internet or an intranet, without prior
written permission. Permission can be requested from either ISO at the address below or ISO’s member body in the country of
the requester.
ISO copyright office
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Tel. + 41 22 749 01 11
Fax + 41 22 749 09 47
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Published in Switzerland
ii PROOF/ÉPREUVE © ISO 2014 – All rights reserved
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ISO/TR 15144-2:2014(E)
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 Example calculation . 4
4.1 Example 1 — Spur gear . 5
4.2 Example 2 — Spur gear .19
4.3 Example 3 — Helical gear .28
4.4 Example 4 — Speed increaser .37
Bibliography .47
© ISO 2014 – All rights reserved PROOF/ÉPREUVE iii
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ISO/TR 15144-2:2014(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.
The procedures used to develop this document and those intended for its further maintenance are
described in the ISO/IEC Directives, Part 1. In particular the different approval criteria needed for the
different types of ISO documents should be noted. This document was drafted in accordance with the
editorial rules of the ISO/IEC Directives, Part 2 (see www.iso.org/directives).
Attention is drawn to the possibility that some of the elements of this document may be the subject of
patent rights. ISO shall not be held responsible for identifying any or all such patent rights. Details of
any patent rights identified during the development of the document will be in the Introduction and/or
on the ISO list of patent declarations received (see www.iso.org/patents).
Any trade name used in this document is information given for the convenience of users and does not
constitute an endorsement.
For an explanation on the meaning of ISO specific terms and expressions related to conformity
assessment, as well as information about ISO’s adherence to the WTO principles in the Technical Barriers
to Trade (TBT) see the following URL: Foreword - Supplementary information
The committee responsible for this document is ISO/TC 60, Gears, Subcommittee SC 2, Gear capacity
calculation.
ISO/TR 15144 consists of the following parts, under the general title Calculation of micropitting load
capacity of cylindrical spur and helical gears:
— Part 1: Introduction and basic principles
— Part 2: Examples of calculation for micropitting
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ISO/TR 15144-2:2014(E)
Introduction
This part of ISO/TR 15144 provides worked examples for the application of the calculation procedures
defined in ISO/TR 15144-1. The example calculations cover the application to spur and helical cyclindrical
involute gears for both high-speed and low-speed operating conditions, determining the micropitting
safety factor for each gear pair. The calculation procedures used are consistent with those presented in
ISO/TR 15144-1. No additional calculations are presented here that are outside of the technical report.
Four worked examples are presented with the necessary input data for each gear set provided at the
beginning of the calculation. The worked examples are based on real gear pairs where either laboratory
or operational field performance data has been established, with the examples covering several
applications. When available, pictures and measurements are provided of the micropitting wear,
experienced on the gear sets when run under the conditions used in the worked examples. Calculation
details are presented in full for several of the initial calculations after which only summarized results
data are included. For better applicability, the numbering of the formulae follows ISO/TR 15144-1.
Several of the worked examples are presented with the calculation procedures performed in accordance
with the application of both methods A and B.
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TECHNICAL REPORT ISO/TR 15144-2:2014(E)
Calculation of micro-pitting load capacity of cylindrical
spur and helical gears —
Part 2:
Examples of calculation for micropitting
1 Scope
The example calculations presented here are provided for guidance on the application of the technical
report ISO/TR 15144-1 only. Any of the values or the data presented should not be used as material or
lubricant allowables or as recommendations for micro-geometry in real applications when applying this
procedure. The necessary parameters and allowable film thickness values, λ , should be determined
GFP
for a given application in accordance with the procedures defined in ISO/TR 15144-1.
2 Normative references
The following documents, in whole or in part, are normatively referenced in this document and are
indispensable for its application. For dated references, only the edition cited applies. For undated
references, the latest edition of the referenced document (including any amendments) applies.
ISO 1122-1:1998, Vocabulary of gear terms — Part 1: Definitions related to geometry
ISO 6336-1:2006, Calculation of load capacity of spur and helical gears — Part 1: Basic principles,
introduction and general influence factors
ISO 6336-2:2006, Calculation of load capacity of spur and helical gears — Part 2: Calculation of surface
durability (pitting)
ISO 21771:2007, Gears — Cylindrical involute gears and gear pairs — Concepts and geometry
ISO/TR 15144-1:2014, Calculation of micropitting load capacity of cylindrical spur and helical gears — Part
1: Introduction and basic principles
3 Terms, definitions, symbols, and units
3.1 Terms and definitions
For the purpose of this document, the terms and definitions given in ISO 1122-1, ISO 6336-1, and
ISO 6336-2 apply.
3.2 Symbols and units
The symbols used in this technical report are given in Table 1. The units of length metre, millimetre, and
micrometre are chosen in accordance with common practice. The conversions of the units are already
included in the given formulae.
Table 1 — Symbols and units
Symbol Description Unit
a centre distance mm
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ISO/TR 15144-2:2014(E)
Table 1 (continued)
Symbol Description Unit
0,5
B thermal contact coefficient of pinion N/(m·s ·K)
M1
0,5
B thermal contact coefficient of wheel N/(m·s ·K)
M2
b face width mm
C tip relief of pinion µm
a1
C tip relief of wheel µm
a2
c specific heat per unit mass of pinion J/(kg·K)
M1
c specific heat per unit mass of wheel J/(kg·K)
M2
c’ maximum tooth stiffness per unit face width (single stiffness) of a tooth pair N/(mm·µm)
c mean value of mesh stiffness per unit face width N/(mm·µm)
γα
d tip diameter of pinion mm
a1
d tip diameter of wheel mm
a2
d base diameter of pinion mm
b1
d base diameter of wheel mm
b2
d pitch diameter of pinion mm
w1
d pitch diameter of wheel mm
w2
d Y-circle diameter of pinion mm
Y1
d Y-circle diameter of wheel mm
Y2
2
E reduced modulus of elasticity N/mm
r
2
E modulus of elasticity of pinion N/mm
1
2
E modulus of elasticity of wheel N/mm
2
F nominal transverse load in plane of action (base tangent plane) N
bt
F (nominal) transverse tangential load at reference cylinder per mesh N
t
G material parameter -
M
g parameter on the path of contact (distance of point Y from point A) mm
Y
g length of path of contact mm
α
H load losses factor -
v
h local lubricant film thickness µm
Y
K application factor -
A
K transverse load factor -
Hα
K face load factor -
Hβ
K dynamic factor -
v
−1
n rotation speed of pinion min
1
P transmitted power kW
p transverse base pitch on the path of contact Mm
et
2
p local Hertzian contact stress including the load factors K N/mm
dyn,Y
2
p local nominal Hertzian contact stress N/mm
H,Y
Ra effective arithmetic mean roughness value µm
Ra arithmetic mean roughness value of pinion µm
1
Ra arithmetic mean roughness value of wheel µm
2
S local sliding parameter -
GF,Y
S safety factor against micropitting -
λ
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ISO/TR 15144-2:2014(E)
Table 1 (continued)
Symbol Description Unit
S minimum required safety factor against micropitting -
λ,min
T nominal torque at the pinion Nm
1
U local velocity parameter -
Y
u gear ratio -
v local sliding velocity m/s
g,Y
v local tangential velocity on pinion m/s
r1,Y
v local tangential velocity on wheel m/s
r2,Y
v sum of tangential velocities at pitch point m/s
Σ,C
v sum of tangential velocities at point Y m/s
Σ,Y
W material factor -
W
W local load parameter -
Y
X local buttressing factor -
but,Y
X tip relief factor -
Ca
X lubricant factor -
L
X roughness factor -
R
X lubrication factor -
S
X local load sharing factor -
Y
2 0,5
Z elasticity factor (N/mm )
E
z number of teeth of pinion -
1
z number of teeth of wheel -
2
α transverse pressure angle °
t
α pressure angle at the pitch cylinder °
wt
2
α pressure-viscosity coefficient at local contact temperature m /N
θB,Y
2
α pressure-viscosity coefficient at bulk temperature m /N
θM
2
α pressure-viscosity coefficient at 38 °C m /N
38
β base helix angle °
b
ε maximum addendum contact ratio -
max
ε transverse contact ratio -
α
ε virtual transverse contact ratio -
αn
ε overlap ratio -
β
ε total contact ratio -
γ
ε addendum contact ratio of the pinion -
1
ε addendum contact ratio of the wheel -
2
2
η dynamic viscosity at local contact temperature N·s/m
θB,Y
2
η dynamic viscosity at bulk temperature N·s/m
θM
2
η dynamic viscosity at oil inlet/sump temperature N·s/m
θoil
2
η dynamic viscosity at 38 °C N·s/m
38
θ local contact temperature °C
B,Y
θ local flash temperature °C
fl,Y
θ bulk temperature °C
M
θ oil inlet/sump temperature °C
oil
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ISO/TR 15144-2:2014(E)
Table 1 (continued)
Symbol Description Unit
λ minimum specific lubricant film thickness in the contact area -
GF,min
λ local specific lubricant film thickness -
GF,Y
λ permissible specific lubricant film thickness -
GFP
λ limiting specific lubricant film thickness of the test gears -
GFT
λ specific heat conductivity of pinion W/(m·K)
M1
λ specific heat conductivity of wheel W/(m·K)
M2
µ mean coefficient of friction -
m
2
ν kinematic viscosity at local contact temperature mm /s
θB,Y
2
ν kinematic viscosity at bulk temperature mm /s
θM
ν Poisson’s ratio of pinion -
1
ν Poisson’s ratio of wheel -
2
2
ν kinematic viscosity at 100 °C mm /s
100
2
ν kinematic viscosity at 40 °C mm /s
40
3
ρ density of pinion kg/m
M1
3
ρ density of wheel kg/m
M2
ρ normal radius of relative curvature at pitch diameter mm
n,C
ρ normal radius of relative curvature at point Y mm
n,Y
ρ transverse radius of relative curvature at point Y mm
t,Y
ρ transverse radius of curvature of pinion at point Y mm
t1,Y
ρ transverse radius of curvature of wheel at point Y mm
t2,Y
3
ρ density of lubricant at local contact temperature kg/m
θB,Y
3
ρ density of lubricant at bulk temperature kg/m
θM
3
ρ density of lubricant at 15 °C kg/m
15
Subscripts to symbols: parameter for any contact point Y in the contact area for method A and on the path of
contact for method B (all parameters subscript Y has to be calculated with local values).
4 Example calculation
The following presents examples for the calculation of the safety factor against micropitting, S . Each
λ
example is first calculated according to method B and examples 1, 3, and 4 subsequently calculated
according to method A. The calculation sequence for method B has been provided to follow a logical
approach in relation to the input data. Beside the formulae itself, the formula numbers related to
ISO/TR 15144-1 are given.
The examples calculate the safety factor S of a specific gear set when compared to an allowable λ
λ GFP
value. For the examples 1, 2, and 4, the permissible specific oil film thickness, λ , was determined
GFP
from the test result of the lubricant in the FZG-FVA micropitting test (1). For these calculations medium
values for the standard FZG back-to-back test rig and standard test conditions for K and K were
Hβ v
used (K = 1,10 and K = 1,05). The calculation of the λ value from the test result of the FZG-FVA
Hβ v GFP
micropitting test (1) (method B) is shown exemplary on the basis of the first example. For example 3, the
permissible specific oil film thickness, λ , was determined from a bench test.
GFP
NOTE The calculations were performed computer-based. If the calculations are performed manually, small
differences between the results can appear.
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ISO/TR 15144-2:2014(E)
4.1 Example 1 — Spur gear
The result of this example is confirmed by experimental investigations. The gears were obviously
micropitted and had profile deviations of approximately 8 to 10 µm. Figure 1 shows a diagram of the
observed location and severity of micropitting for pinion and wheel of example 1.
a) pinion b) wheel
Key
1 tip
2 root
Figure 1 — Diagram of schematic profile deviations of pinion and wheel for example 1
© ISO 2014 – All rights reserved PROOF/ÉPREUVE 5
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ISO/TR 15144-2:2014(E)
4.1.1 Input data
Table 2 — Input data for Example 1
Example 1
Symbol Description Unit pinion wheel
comb.
z number of teeth - 18 18
- driving gear - x
m
normal module mm 10,93
n
α normal pressure angle ° 20
n
β helix angle ° 0
b face width mm 21,4
Geometry
a centre distance mm 200
x addendum modification factor - 0,158 0,158
d tip diameter of pinion mm 221,4 221,4
a
- tooth flank modifications - no modifications
Q gear quality - 5 5
Ra arithmetic mean roughness value µm 0,90 0,90
- material - Eh Eh
2
E modulus of elasticity N/mm 206 000 206 000
ν Poisson’s ratio - 0,3 0,3
λ specific heat conductivity W/(m·K) 45 45
M
Material
c specific heat per unit mass J/(kg·K) 440 440
M
3
ρ density kg/m 7 800 7 800
M
material factor according to ISO/
W TR 15144-1, Table A.1 (for matching case - 1,0
w
carburised/case carburised)
K application factor - 1,0
A
K dynamic factor - 1,15
v
Application
K transverse load factor - 1,0
Hα
K face load factor - 1,10
Hβ
T nominal torque at the pinion Nm 187 8
1
Load
−1
n rotation speed of the pinion min 300 0
1
oil inlet temperature (injection lubrica-
ϑ °C 90
oil
tion)
2
ν kinematic viscosity at 40 °C mm /s 210
40
2
ν kinematic viscosity at 100 °C mm /s 18,5
100
3
ρ density of the lubricant at 15 °C kg/m 895
15
Lubricant
- oil type - mineral oil
failure load stage at test temperature
- - SKS 8
(90 °C) according to FVA 54/7
permissible lubricant film thickness
λ - 0,211
GFP
(see 4.1.4 for calculation)
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ISO/TR 15144-2:2014(E)
4.1.2 Calculation according to method B
4.1.2.1 Calculation of gear geometry (according to ISO 21771)
Basic values:
mt = 10,93 mm
m
n
m =
t
cosβ
d = z ∙ m d = 196,74 mm
1 1 t 1
D = z ∙ m d = 196,74 mm
2 2 t 2
z u = 1
2
u=
z
1
α = 20 °
t
tanα
n
α =arctan
t
cosβ
d = d cos d = 184,875 mm
b1 1 αt b1
d = d cos d = 184,875 mm
b2 2 αt b2
d = 200 mm
2⋅a
w1
d =
w1
u+1
da=⋅2 −d d = 200 mm
w2
w2 w1
α = 22,426 °
w1
zz+ ⋅⋅m cosα
()
12 tt
α =arccos
wt
2⋅a
ββ=⋅arcsin(sin cos)α
bn β = 0 °β =0
b
b
p = 32,267 mm
pm=⋅πα⋅cos
et
et tt
ε = 0,705
1
2
z d
1 a1
ε = ⋅ −−1 tanα
1 wt
2⋅π d
b1
ε = 0,705
2
2
z d
2 a2
ε = ⋅ 1 tanα
−−
2 wt
2⋅π d
b2
ε = 1,411
22 22 α
dd dd
1
a1 b1 a2 b2
εα=⋅ −+ −−a⋅sin
± wt
p 44 44
et
ε = 0
b⋅sinβ
β
ε =
²
m ⋅π
n
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ISO/TR 15144-2:2014(E)
ε = ε + ε ε = 1,411
γ α β γ
g = 45,519 mm
α
22 22
gd=⋅05,s−+dd −da−⋅ inα
±a1b1a2b2wt
Coordinates of the basic points (A, AB, B, C, D, DE, E) on the line of action:
g = 0 mm (34) g = 0 mm
A A
gp−
g = 6,626
±et
AB
g =
(35)
AB
mm
2
g = 13,253
AB
g = gα − p (36)
AB et
mm
22
dd d g = 22,760
C
b1 a1 b1
(37)
g =⋅tanα −− +g
C wt ± mm
24 4
g = 45,519
E
g = gα (40)
E
mm
2
22 2
dd d
d = 187,419
b1 a1 b1 A1
(41)
d =⋅2 +− −+gg
A1 α A
mm
44 4
d = d = 200,000
AB1 C1
d = 193,546 mm
B1
190,046 mm mm
d = 207,998 d = 221,400
D1 E1
d = 214,394 mm
DE1
mm mm
2
22 2
dd d
d = 221,400
A2
b2 a2 b2
(42)
d =⋅2 +− −g
A2 A
mm
44 4
d = d = 200,000
AB2 C2
d = 207,998 mm
B2
214,394 mm mm
d = 193,546 d = 184,419
D2 E2
d = 190, 046 mm
DE2
mm mm
Normal radius of relative curvature:
ρ
t,A
ρ =
(45) ρ = 12,285 mm
n,A
n,A
cosβ
b
ρ = 15,663 mm ρ = 17,890 mm ρ = 19,074 mm
n,AB n,B n,C
ρ = 17,890 ρ = 15,663 mm ρ = 12,285 mm
n,D n,DE n,E
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ISO/TR 15144-2:2014(E)
4.1.2.2 Calculation of material data
−1
2 2
11−νν−
1 2
2
(6) E = 226 37,4 N/mm
E =⋅2 +
R
r
EE
1 2
0,5
Bc=⋅λρ ⋅
(82) B = 12 427,4 N/(ms K)
M1 M1 M1 M1 M1
0,5
Bc=⋅λρ ⋅
(83) B = 12 427,4 N/(ms K)
M2 M2 M2 M2 M2
4.1.2.3 Calculation of operating conditons
Loading:
nT
1 1
P =⋅2 π⋅⋅
(85) P = 590 kW
A
60 1000
T
1
F =⋅2000
F = 19 091 N
t t
d
1
T
1
F =⋅2000
F = 20 316 N
bt
bt
d
b1
Local load sharing factor:
NOTE No tooth flank modifications, spur gears, gear quality ≤7 (see ISO/TR 15144-1, Figure 2).
Q2− 1 g
A
X = +⋅
(46) X = 0,333
A
A
15 3 g
B
X = 0,500 mm X = 1,000 mm X = 1,000 mm
AB B C
X = 1,000 X = 0,500 X = 0,333
D DE E
Elasticity factor:
E
r
2 0,5
(26) Z = 189,812 (N/mm )
Z = E
E
2⋅π
Local Hertzian contact stress:
FX⋅
tA
2
pZ=⋅
(25) p = 963 N/mm
H,A,B
H,A,BE
b⋅⋅ραcosc⋅ osβ
n,At b
2 2 2
p = 104 5 N/mm p = 138 3 N/mm p = 1 339 N/mm
H,AB,B H,B,B H,C,B
2 2 2
p = 138 3 N/mm p = 104 5 N/mm p = 963 N/mm
H,D,B H,DE,B H,E,B
pp=⋅ KK⋅⋅KK⋅ 2
(24) p = 1 084 N/mm
dyn,A,BH,A,B Av H± H² dyn,A,B
© ISO 2014 – All rights reserved PROOF/ÉPREUVE 9
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ISO/TR 15144-2:2014(E)
2 2 2
p = 117 5 N/mm p = 155 5 N/mm p = 1 506 N/mm
dyn,AB,B dyn,B,B dyn,C,B
2 2 2
p = 155 5 N/mm p = 117 5 N/mm p = 1 084 N/mm
dyn,D,B dyn,DE,B dyn,E,B
Velocity:
ν = −ν − ν (81) ν = −14,300 m/s
g,A r1,A r1,A g,A
ν = 10,137 m/s ν = 5,974 m/s ν = 0 m/s
g,AB g,B g,C
ν = 5,974 m/s ν = 10,137 m/s ν = 14,300 m/s
g,D g,DE g,E
ν = ν + ν (13) ν = 23,969 m/s
Σ,A r2,A r2,A Σ,A
ν = 23,969 m/s ν = 23,969 m/s ν =23,969 m/s
Σ,AB Σ,B Σ,C
ν = 23,969 m/s ν = 23,969 m/s ν =23,969 m/s
Σ,D Σ,DE Σ,E
Effective arithmetic mean roughness value:
RRaa=⋅05, + Ra
()
(3) Ra = 0,90 μm
12
4.1.2.4 Calculation of lubricant data
X = 1,0 for mineral oil (see Table 3 in ISO/TR 15144-1)
L
−8 0,1348
−82
α =⋅2,657 10 ⋅
(9) α =⋅21,/510 mN
38 η38
38
X = 1,2 for injection lubrication
S
4.1.2.5 Calculation of the material parameter
Mean coefficient of friction:
02, 5
Ra
(87) X = 1,025
X =⋅22, R
R
ρ
n,C
K = 1,0 for ε <2
Bγ γ
02,
KK⋅⋅KK⋅⋅FK⋅ −00, 5
Av HHαβ bt Bγ
3
μ =⋅0,045 ⋅⋅10 η ⋅⋅XX (86) μ = 0,048
m
m ()θθoilR L
bv⋅⋅ρ
£,Cn,C
Bulk temperature:
11 π
2 2
H =+ε εε+−1 ⋅+ ⋅
(91) H = 0,204
v
v±()1 2 α
zz cosβ
12 b
ε = ε = ε
max 1 2
X = 1,0 for no profile modification (method B) (101)
CA
07, 2
PH⋅⋅μ X
mv S
(84) θ = 153,6° C
θθ=+7400⋅ ⋅
M
Moil
ab⋅ 1,2⋅X
Ca
10 PROOF/ÉPREUVE © ISO 2014 – All rights reserved
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ISO/TR 15144-2:2014(E)
Material parameter:
6
G = 10 ∙ α ∙ E (5) G = 2 678,6
M θM r M
4.1.2.6 Calculation of the velocity parameter
v
Σ,A
11
U =⋅η
(12) U = 2,005 ∙ 10
AMθ A
2000⋅⋅E ρ
rn,A
11 11 11
U = 1,572 ∙ 10 U = 1,377 ∙ 10 U = 1,291 ∙ 10
AB B C
11 11 11
U = 1,377 ∙ 10 U = 1,572 ∙ 10 U = 2,005 ∙ 10
D DE E
4.1.2.7 Calculation of the load parameter
2
2⋅⋅π p
dyn,A
-4
W = (22) W = 1,1440 ∙ 10
A
A
2
E
r
-4 -4 -4
W = 1,694 ∙ 10 W = 2,966 ∙ 10 W = 2,781 ∙ 10
AB B C
-4 -4 -4
W = 2,966 ∙ 10 W = 1,694 ∙ 10 W = 1,1440 ∙ 10
AD DE E
4.1.2.8 Calculation of the sliding parameter
Local flash temperature:
6
10 ⋅⋅μ pv⋅
p
π mdyn,A g,A
dyn,A
θ =⋅ ⋅⋅8 ρ ⋅
(80) θ = 175,3° C
fl,A
fl,A n,A
2 11000⋅E
Bv +Bv
r
M1 r1,A M2 r2,A
θ = 154,1° C θ = 145,4° C θ = 0° C
fl,AB fl,B fl,C
θ = 145,4° C θ = 154,1° C θ = 175,3° C
fl,D fl,DE fl,E
Local contact temperature as sum of bulk and local flash temperature:
θ = θ + θ (79) θ = 328,9° C
B,A M fl,A B,A
θ = 307,7° C θ = 299,0° C θ = 153,6° C
B,AB B,B B,C
θ = 299,0° C θ = 307,7° C θ = 328,9° C
B,D B,DE B,E
Local sliding parameter:
αη⋅
θθB,AB,A
S =
(27) S = 0,057
GF,A GF,A
αη⋅
θθMM
S = 0,076 S = 0,086 S = 1,000
GF,AB GF,B GF,C
S = 0,086 S = 0,076 S = 0,057
GF,D GF,DE GF,E
4.1.2.9 Calculation of the lubricant film thickness
06,,07 −01,,3022
hG=⋅1600 ρ ⋅⋅UW⋅⋅S (4) h = 0,122 μm
A
An,A MA AGF,A
© ISO 2014 – All rights reserved PROOF/ÉPREUVE 11
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ISO/TR 15144-2:2014(E)
h = 0,137 μm h = 0,136 μm h = 0,241 μm
AB B C
h = 0,136 μm h = 0,137 μm h = 0,122 μm
D DE E
4.1.2.10 Calculation of the specific lubricant film thickness
h
A
λ =
(2) λ = 0,136
GF,A
GF,A
Ra
λ =0,153 λ = 0,152 λ = 0,267
GF,AB GF,B GF,C
λ = 0,152 λ = 0,153 λ = 0,136
GF,D GF,DE GF,E
λ = λ = λ λ = 0,136
GF,min GF,A GF,E GF,min
4.1.2.11 Calculation of the micropitting safety factor
λ
GF,min
S =
(1) S = 0,644
λ λ
λ
GFP
The calculation of the permissible specific lubricant film thickness, λ , for example 1 is shown
GFP
exemplary in 4.1.4.
The final results for the calculation of the safety factor against micropitting, S , for example 1 are shown
λ
in Table 3.
Table 3 — Results of calculation according to method B — Example 1
Point A AB B C D DE E
λ 0,136 0,153 0,152 0,267 0,152 0,153 0,136
GF,Y
λ 0,136
F,min
λ 0,211
GFP
S 0,644
λ
4.1.3 Calculation according to method A
The calculation of example 1 according to method A was carried out by a 3D-calculation programme.
Calculated results during method A will vary depending on the method of determining load distribution.
The load distribution, on which the following calculation according to method A is based, is shown in
Table 4. The maximum values are printed in bold.
2
Table 4 — Matrix of pressure distribution — p in N/mm
H,Y,A
Width in mm
0,0 7,6 13,8 21,4
A 1 115 1 110 1 110 1 114
AB 1 048 1 044 1 044 1 047
B 1 375 1 373 1 373 1 375
C 1 342 1 339 1 339 1 342
D 1 048 1 045 1 045 1 048
DE 1 050 1 046 1 046 1 050
E 1 099 1 094 1 094 1 099
12 PROOF/ÉPREUVE © ISO 2014 – All rights reserved
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ISO/TR 15144-2:2014(E)
The resulting matrix of specific lubricant film thickness according to method A is shown in Table 5. The
minimum value is printed in bold.
Table 5 — Matrix of resulting specific lubricant film thickness λ
GF,Y
Width in mm
0,0 7,6 13,8 21,4
A 0,122 0,123 0,123 0,122
AB 0,159 0,160 0,160 0,159
B 0,159 0,159 0,159 0,159
C 0,270 0,271 0,271 0,270
D 0,197 0,198 0,198 0,197
DE 0,159 0,159 0,159 0,159
E 0,124 0,125 0,125 0,124
For the calculation of the micropitting safety factor according to method A, the minimum value of the
matrix of resulting specific lubricant film thickness, shown in Table 5, was used.
λ
GF,min
S =
(1) S = 0,577
λ
λ
λ
GFP
NOTE The difference in safety factor calculated between methods A and B in the above example 1 results
from the simplified calculation of load distribution according to method B.
4.1.4 Calculation of the permissible lubricant film thickness
Calculation of the permissible specific lubricant film thickness from the test result of the FZG-FVA
micropitting test (1) (Method B) with the reference test gears type C-GF.
The calculation of the reference value, λ , is done for point A because the minimum specific lubricant
GFT
film thickness for gear type C is always at point A. All data of the reference test gears type C-GF have the
subscript “Ref”.
© ISO 2014 – All rights reserved PROOF/ÉPREUVE 13
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ISO/TR 15144-2:2014(E)
Table 6 — Input data for calculation of the permissible lubricant film thickness
C-GF
Symbol Description Unit pinion wheel
comb.
z number of teeth - 16 24
Ref
m transverse module (m = m ) mm 4,5
tRef nRef tRef
α transverse pressure angle (α = α ) ° 20
nRef nRef tRef
β base helix angle (β = β ) ° 0
bRef bRef Ref
b face width mm 14
Ref
a centre distance mm 91,5
Ref
x addendum modification factor - 0,1817 0,1716
Ref
d tip diameter of pinion mm 82,45 118,35
aRef
- tooth flank modifications - no modifications
Geometry
Ra arithmetic mean roughness value µm 0,50 0,50
Ref
2
E modulus of elasticity N/mm 206 000 206 000
Ref
ν Poisson’s ratio - 0,3 0,3
Ref
λ specific heat conductivity W/(m·K) 45 45
MRef
c specific heat per unit mass J/(kg·K) 440 440
MRef
3
ρ density kg/m 7 800 7 800
MRef
material factor according to ISO/TR 15144-1,
Table A.1
W - 1,0
w
(for matching case carburised/case carbur-
ised)
K application factor - 1,0
ARef
K dynamic factor - 1,05
vRef
Application
K transverse load factor - 1,0
HαRef
K face load factor - 1,10
HβRef
T nominal torque at the pinion for SKS 8 Nm 171,6
1Ref
−1
n rotation speed of the pinion min 2 250
1Ref
Load
nominal Hertzian contact stress at point A
2
p N/mm 1 191
H,A,A
according to method A for SKS 8 (see Table 6)
− lubrication - injection lubrication
NOTE The used values for K and K are valid for the standard FZG back-to-back test rig and standard
vRef HβRef
conditions.
Table 7 gives the nominal Hertzian contact stress at point A for the reference test gears type C-GF as a
function of the reached failure load stage (SKS) in the FZG-FVA micropitting test (1).
Table 7 — Relation between failure load stage according to FZG-FVA micropitting test (1) and
nominal Hertzian contact stress at point A
Nominal torque at the Hertzian contact stress at
Nominal Hertzian contact stress
SKS pinion in point C in
at point A according to method A
2
Nm N/mm
5 70,0 795,1 764
6 98,9 945,1 906
14 PROOF/ÉPREUVE © ISO 2014 – All rights reserved
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ISO/TR 15144-2:2014(E
...
RAPPORT ISO/TR
TECHNIQUE 15144-2
Première édition
2014-10-01
Version corrigée
2015-01-15
Calcul de la capacité de charge
aux micropiqûres des engrenages
cylindriques à dentures droite et
hélicoïdale —
Partie 2:
Exemples de calcul pour micropiqûres
Calculation of micropitting load capacity of cylindrical spur and
helical gears —
Part 2: Examples of calculation for micropitting
Numéro de référence
ISO/TR 15144-2:2014(F)
©
ISO 2014
---------------------- Page: 1 ----------------------
ISO/TR 15144-2:2014(F)
DOCUMENT PROTÉGÉ PAR COPYRIGHT
© ISO 2014
Droits de reproduction réservés. Sauf indication contraire, aucune partie de cette publication ne peut être reproduite ni utilisée
sous quelque forme que ce soit et par aucun procédé, électronique ou mécanique, y compris la photocopie, l’affichage sur
l’internet ou sur un Intranet, sans autorisation écrite préalable. Les demandes d’autorisation peuvent être adressées à l’ISO à
l’adresse ci-après ou au comité membre de l’ISO dans le pays du demandeur.
ISO copyright office
Case postale 56 • CH-1211 Geneva 20
Tel. + 41 22 749 01 11
Fax + 41 22 749 09 47
E-mail copyright@iso.org
Web www.iso.org
Publié en Suisse
ii © ISO 2014 – Tous droits réservés
---------------------- Page: 2 ----------------------
ISO/TR 15144-2:2014(F)
Sommaire Page
Avant-propos .iv
Introduction .v
1 Domaine d’application . 1
2 Références normatives . 1
3 Termes, définitions, symboles et unités . 1
3.1 Termes et définitions . 1
3.2 Symboles et unités . 1
4 Exemple de calcul . 4
4.1 Exemple 1 — Dentures droites . 5
4.1.1 Données d’entrée . 6
4.1.2 Calcul selon la méthode B . 7
4.1.3 Calcul selon la méthode A .13
4.1.4 Calcul de l’épaisseur admissible du film lubrifiant .13
4.2 Exemple 2 — Dentures droites .19
4.2.1 Données d’entrée .20
4.2.2 Calcul selon la méthode B .21
4.3 Exemple 3 — Dentures hélicoïdales .27
4.3.1 Données d’entrée .29
4.3.2 Calcul selon la méthode B .30
4.3.3 Calcul selon la méthode A .36
4.4 Exemple 4 — Multiplicateur .37
4.4.1 Données d’entrée .38
4.4.2 Calcul selon la méthode B .39
4.4.3 Calcul selon la méthode A .45
Bibliographie .47
© ISO 2014 – Tous droits réservés iii
---------------------- Page: 3 ----------------------
ISO/TR 15144-2:2014(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 (IEC) en ce qui concerne
la normalisation électrotechnique.
Les procédures utilisées pour élaborer le présent document et celles destinées à sa mise à jour sont
décrites dans les Directives ISO/IEC, Partie 1. Il convient, en particulier de prendre note des différents
critères d’approbation requis pour les différents types de documents ISO. Le présent document a été
rédigé conformément aux règles de rédaction données dans les Directives ISO/IEC, Partie 2 (voir www.
iso.org/directives).
L’attention est appelée sur le fait que certains des éléments du présent document peuvent faire l’objet de
droits de propriété intellectuelle ou de droits analogues. L’ISO ne saurait être tenue pour responsable
de ne pas avoir identifié de tels droits de propriété et averti de leur existence. Les détails concernant les
références aux droits de propriété intellectuelle ou autres droits analogues identifiés lors de l’élaboration
du document sont indiqués dans l’Introduction et/ou dans la liste des déclarations de brevets reçues par
l’ISO (voir www.iso.org/brevets).
Les appellations commerciales éventuellement mentionnées dans le présent document sont données pour
information, par souci de commodité, à l’intention des utilisateurs et ne sauraient constituer un engagement.
Pour une explication de la signification des termes et expressions spécifiques de l’ISO liés à l’évaluation de
la conformité, ou pour toute information au sujet de l’adhésion de l’ISO aux principes de l’OMC concernant
les obstacles techniques au commerce (OTC), voir le lien suivant: Avant-propos — Informations
supplémentaires.
Le comité chargé de l’élaboration du présent document est l’ISO/TC 60, Engrenages, sous-comité SC 2,
Calcul de la capacité des engrenages.
La présente version corrigée de l’ISO/TR 15144-2:2014 inclut les corrections suivantes: des erreurs dans
les symboles et les équations ont été corrigées.
L’ISO/TR 15144 comprend les parties suivantes, présentées sous le titre général Calcul de la capacité de
charge aux micropiqûres des engrenages cylindriques à dentures droite et hélicoïdale:
— Partie 1: Introduction et principes fondamentaux
— Partie 2: Exemples de calcul pour les micropiqûres
iv © ISO 2014 – Tous droits réservés
---------------------- Page: 4 ----------------------
ISO/TR 15144-2:2014(F)
Introduction
La présente partie de l’ISO/TR 15144 fournit des exemples pratiques pour l’application des méthodes de
calcul définies dans l’ISO/TR 15144-1. Les exemples de calcul concernent l’application aux engrenages
cylindriques à dentures droite et hélicoïdale et à profil en développante de cercle, à la fois dans des
conditions de fonctionnement à grande vitesse et à faible vitesse, en déterminant le coefficient de
sécurité contre la formation de micropiqûres pour chaque engrenage. Les méthodes de calcul utilisées
sont cohérentes avec celles présentées dans l’ISO/TR 15144-1. Aucun des calculs supplémentaires
présentés ici n’est exclu du domaine d’application du rapport technique.
Quatre exemples pratiques sont présentés, les jeux de données d’entrée nécessaires pour chaque
engrenage sont indiqués au début de chaque calcul. Ces exemples pratiques sont fondés sur des
engrenages réels pour lesquels des données de performance en laboratoire ou sur le terrain ont été
établies, les exemples couvrant plusieurs types d’applications. Le cas échéant, des images et des mesures
de l’usure par micropiqûres rencontrée sur les trains d’engrenages utilisés dans les conditions des
exemples pratiques sont fournies. Les détails des calculs sont présentés en intégralité pour les premiers
exemples de calculs, puis par la suite seul un récapitulatif des résultats est donné. Pour une meilleure
applicabilité, la numérotation des formules suit celle de l’ISO/TR 15144-1. Pour plusieurs des exemples
pratiques présentés, les calculs sont effectués à la fois selon la méthode A et la méthode B en fonction de
l’application.
© ISO 2014 – Tous droits réservés v
---------------------- Page: 5 ----------------------
RAPPORT TECHNIQUE ISO/TR 15144-2:2014(F)
Calcul de la capacité de charge aux micropiqûres des
engrenages cylindriques à dentures droite et hélicoïdale —
Partie 2:
Exemples de calcul pour micropiqûres
1 Domaine d’application
Les exemples de calcul présentés ici sont uniquement destinés à servir de guide pour l’application du
rapport technique ISO/TR 15144-1. Il convient de n’utiliser, lors de l’application de cette méthode, dans
des cas d’applications réelles autres, aucune des valeurs ou données présentées ici comme des valeurs
admissibles pour les matériaux ou les lubrifiants ou des recommandations pour la micro-géométrie.
Il convient que les paramètres nécessaires et les valeurs admissibles d’épaisseur de film, λ , soient
GFP
déterminés pour une application donnée conformément aux méthodes définies dans l’ISO/TR 15144-1.
2 Références normatives
Les documents suivants, en tout ou partie, sont référencés de façon normative dans le présent document
et sont indispensables pour son application. Pour les références datées, seule l’édition citée s’applique.
Pour les références non datées, la dernière édition du document de référence s’applique (y compris les
éventuels amendements).
ISO 1122-1:1998, Vocabulaire des engrenages — Partie 1: Définitions géométriques
ISO 6336-1:2006, 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
ISO 6336-2:2006, Calcul de la capacité de charge des engrenages cylindriques à dentures droite et hélicoïdale
— Partie 2: Calcul de la résistance à la pression superficielle (piqûre)
ISO 21771:2007, Engrenages — Roues et engrenages cylindriques à développante — Concepts et géométrie
ISO/TR 15144-1:2014, Calcul de la capacité de charge aux micropiqûres des engrenages cylindriques à
dentures droite et hélicoïdale — Partie 1: Introduction et principes fondamentaux
3 Termes, définitions, symboles et unités
3.1 Termes et définitions
Pour les besoins du présent document, les termes et définitions donnés dans l’ISO 1122-1, l’ISO 6336-1
et l’ISO 6336-2 s’appliquent.
3.2 Symboles et unités
Les symboles utilisés dans le présent rapport technique sont donnés dans le Tableau 1. Les unités de
longueur mètre, millimètre et micromètre sont choisies conformément à l’usage en la matière. Les
conversions des unités sont déjà comprises dans les formules données.
© ISO 2014 – Tous droits réservés 1
---------------------- Page: 6 ----------------------
ISO/TR 15144-2:2014(F)
Tableau 1 — Symboles et unités
Symbole Description Unité
a entraxe mm
0,5
B coefficient de contact thermique du pignon N/(m·s ·K)
M1
0,5
B coefficient de contact thermique de la roue N/(m·s ·K)
M2
b largeur de denture mm
C dépouille de tête du pignon µm
a1
C dépouille de tête de la roue µm
a2
c chaleur spécifique par unité de masse du pignon J/(kg·K)
M1
c chaleur spécifique par unité de masse de la roue J/(kg·K)
M2
rigidité maximale par unité de largeur de denture (rigidité simple) d’une paire de
c’ N/(mm·µm)
dents
c valeur moyenne de la rigidité d’engrènement par unité de largeur de denture N/(mm·µm)
γα
d diamètre de tête du pignon mm
a1
d diamètre de tête de la roue mm
a2
d diamètre de base du pignon mm
b1
d diamètre de base de la roue mm
b2
d diamètre primitif de fonctionnement du pignon mm
w1
d diamètre primitif de fonctionnement de la roue mm
w2
d diamètre du cercle Y du pignon mm
Y1
d diamètre du cercle Y de la roue mm
Y2
2
E module d’élasticité réduit N/mm
r
2
E module d’élasticité du pignon N/mm
1
2
E module d’élasticité de la roue N/mm
2
force nominale apparente dans le plan d’action (plan tangent aux cylindres de
F N
bt
base)
F force tangentielle (nominale) sur le cylindre de référence par engrènement N
t
G paramètre de matériau -
M
g paramètre sur la ligne de conduite (distance du point Y au point A) mm
Y
g longueur de la ligne de conduite mm
α
H facteur de pertes de charge -
v
h épaisseur locale du film lubrifiant µm
Y
K facteur d’application -
A
K facteur de distribution transversale de la charge -
Hα
K facteur de distribution longitudinale de la charge -
Hβ
K facteur dynamique -
v
−1
n vitesse de rotation du pignon min
1
P puissance transmise kW
p pas de base apparent sur la ligne de conduite mm
et
2
p pression de contact hertzienne locale comprenant les facteurs de charge K N/mm
dyn,Y
2
p pression de contact hertzienne nominale locale N/mm
H,Y
Ra rugosité arithmétique moyenne effective µm
Ra rugosité arithmétique moyenne du pignon µm
1
2 © ISO 2014 – Tous droits réservés
---------------------- Page: 7 ----------------------
ISO/TR 15144-2:2014(F)
Tableau 1 (suite)
Symbole Description Unité
Ra rugosité arithmétique moyenne de la roue µm
2
S paramètre de glissement local -
GF,Y
S coefficient de sécurité contre la formation de micropiqûres -
λ
S coefficient de sécurité minimal requis contre la formation de micropiqûres -
λ,min
T couple nominal sur le pignon Nm
1
U paramètre de vitesse local -
Y
u rapport d’engrenage -
v vitesse de glissement locale m/s
g,Y
v vitesse tangentielle locale sur le pignon m/s
r1,Y
v vitesse tangentielle locale sur la roue m/s
r2,Y
v somme des vitesses tangentielles au point primitif m/s
Σ,C
v somme des vitesses tangentielles au point Y m/s
Σ,Y
W facteur de matériau -
W
W paramètre de charge local -
Y
X facteur de contrefort local -
but,Y
X facteur de dépouille de tête -
Ca
X facteur lubrifiant -
L
X facteur de rugosité -
R
X facteur de lubrification -
S
X facteur de répartition de charge local -
Y
2 0,5
Z facteur d’élasticité (N/mm )
E
z nombre de dents du pignon -
1
z nombre de dents de la roue -
2
α angle de pression apparent °
t
α angle de pression apparent sur le cylindre primitif de fonctionnement °
wt
2
α coefficient de piezoviscosité à la température locale de contact m /N
θB,Y
2
α coefficient de piezoviscosité à la température de masse m /N
θM
2
α coefficient de piezoviscosité à 38 °C m /N
38
β angle d’hélice de base °
b
ε rapport maximal de conduite de saillie -
max
ε rapport de conduite apparent -
α
ε rapport de conduite équivalent -
αn
ε rapport de recouvrement -
β
ε rapport de conduite total -
γ
ε rapport de conduite de saillie du pignon -
1
ε rapport de conduite de saillie de la roue -
2
2
η viscosité dynamique à la température locale de contact N·s/m
θB,Y
2
η viscosité dynamique à la température de masse N·s/m
θM
2
η viscosité dynamique à la température d’huile en entrée/au bain N·s/m
θoil
2
η viscosité dynamique à 38 °C N·s/m
38
θ température locale de contact °C
B,Y
© ISO 2014 – Tous droits réservés 3
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ISO/TR 15144-2:2014(F)
Tableau 1 (suite)
Symbole Description Unité
θ température-éclair locale °C
fl,Y
θ température de masse °C
M
θ température au bain d’huile °C
oil
λ épaisseur spécifique minimale du film lubrifiant dans la zone de contact -
GF,min
λ épaisseur spécifique locale du film lubrifiant -
GF,Y
λ épaisseur spécifique admissible du film lubrifiant -
GFP
λ épaisseur spécifique limite du film lubrifiant de l’engrenage d’essai -
GFT
λ conductivité thermique spécifique du pignon W/(m·K)
M1
λ conductivité thermique spécifique de la roue W/(m·K)
M2
µ coefficient de frottement moyen -
m
2
ν viscosité cinématique à la température locale de contact mm /s
θB,Y
2
ν viscosité cinématique à la température de masse mm /s
θM
ν coefficient de Poisson du pignon -
1
ν coefficient de Poisson de la roue -
2
2
ν viscosité cinématique à 100 °C mm /s
100
2
ν viscosité cinématique à 40 °C mm /s
40
3
ρ densité du pignon kg/m
M1
3
ρ densité de la roue kg/m
M2
ρ rayon de courbure équivalent normal au diamètre primitif mm
n,C
ρ rayon de courbure équivalent normal au point Y mm
n,Y
ρ rayon de courbure équivalent apparent au point Y mm
t,Y
ρ rayon de courbure apparent du pignon au point Y mm
t1,Y
ρ rayon de courbure apparent de la roue au point Y mm
t2,Y
3
ρ densité du lubrifiant à la température locale de contact kg/m
θB,Y
3
ρ densité du lubrifiant à la température de masse kg/m
θM
3
ρ densité du lubrifiant à 15 °C kg/m
15
Indices des symboles
Paramètre pour tout point de contact Y dans la zone de contact pour la méthode A et sur la ligne de
Y conduite pour la méthode B (tous les paramètres indicés Y doivent être calculés avec des valeurs
locales).
4 Exemple de calcul
Des exemples de calcul du coefficient de sécurité contre la formation de micropiqûres, S , sont présentés
λ
ci-après. Chaque exemple est d’abord calculé selon la méthode B et les exemples 1, 3 et 4 sont ensuite
calculés selon la méthode A. La séquence de calcul pour la méthode B suit une approche logique par
rapport aux données d’entrée. En regard de la formule, les numéros des formules de l’ISO/TR 15144-1
sont indiqués.
Les exemples calculent le coefficient de sécurité S d’un train d’engrenages spécifique lorsqu’il y a une
λ
comparaison avec une valeur admissible λ . Pour les exemples 1, 2 et 4, l’épaisseur spécifique admissible
GFP
du film lubrifiant, λ , a été déterminée à partir du résultat du lubrifiant lors de l’essai de micropiqûres
GFP
(1)
FZG-FVA . Pour ces calculs, les valeurs médianes obtenues avec le banc d’essai à circulation de puissance
FZG normalisé et dans les conditions d’essai normalisées pour K et K ont été utilisées (K = 1,10
Hβ v Hβ
(1)
et K = 1,05). Le calcul de la valeur de λ à partir du résultat de l’essai de micropiqûres FZG-FVA
v GFP
4 © ISO 2014 – Tous droits réservés
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ISO/TR 15144-2:2014(F)
(méthode B) est indiqué à titre d’exemple sur la base du premier exemple. Pour l’exemple 3, l’épaisseur
spécifique admissible du film lubrifiant, λ , a été déterminée à partir d’un essai sur banc.
GFP
NOTE Les calculs ont été effectués par des ordinateurs. Si les calculs sont effectués manuellement, de légers
écarts entre les résultats peuvent apparaître.
4.1 Exemple 1 — Dentures droites
Le résultat de cet exemple est confirmé par des études expérimentales. Les engrenages comportaient
des micropiqûres évidentes et avaient des écarts de profil d’environ 8 µm à 10 µm. La Figure 1 représente
un diagramme de l’emplacement observé et de la sévérité des micropiqûres pour le pignon et la roue
de l’exemple 1.
a) pignon b) roue
Légende
1 tête
2 pied
Figure 1 — Représentation schématique des écarts de profil du pignon et de la roue pour
l’exemple 1
© ISO 2014 – Tous droits réservés 5
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ISO/TR 15144-2:2014(F)
4.1.1 Données d’entrée
Tableau 2 — Données d’entrée pour l’exemple 1
Exemple 1
Symbole Description Unité pignon roue
comb.
z nombre de dents - 18 18
- roue menante - x
m
module normal mm 10,93
n
α angle de pression normal ° 20
n
β angle d’hélice ° 0
b largeur de denture mm 21,4
Géométrie
a entraxe mm 200
x coefficient de déport - 0,158 0,158
d diamètre de tête du pignon mm 221,4 221,4
a
- corrections de profil des dents - aucune correction
Q classe de tolérance de la denture - 5 5
Ra rugosité arithmétique moyenne µm 0,90 0,90
- matériau - Eh Eh
2
E module d’élasticité N/mm 206 000 206 000
ν coefficient de Poisson - 0,3 0,3
λ conductivité thermique spécifique W/(m·K) 45 45
M
Matériau
c chaleur spécifique par unité de masse J/(kg·K) 440 440
M
3
ρ densité kg/m 7 800 7 800
M
facteur matériau conformément à l’ISO/
W TR 15144-1:2014, Tableau A.1 (pour apparie- - 1,0
w
ment entre acier cémenté/acier cémenté)
K facteur d’application - 1,0
A
K facteur dynamique - 1,15
v
facteur de distribution transversale de la
Application
K - 1,0
Hα
charge
facteur de distribution longitudinale de la
K - 1,10
Hβ
charge
T couple nominal sur le pignon Nm 1 878
1
Charge
−1
n vitesse de rotation du pignon min 3 000
1
6 © ISO 2014 – Tous droits réservés
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ISO/TR 15144-2:2014(F)
Tableau 2 (suite)
Exemple 1
Symbole Description Unité pignon roue
comb.
température d’entrée de l’huile (lubrification
θ °C 90
oil
par injection)
2
ν viscosité cinématique à 40 °C mm /s 210
40
2
ν viscosité cinématique à 100 °C mm /s 18,5
100
3
ρ densité du lubrifiant à 15 °C kg/m 895
15
Lubrifiant
- type d’huile - huile minérale
niveau de la charge de rupture à la tempéra-
- - SKS 8
ture d’essai (90 °C) selon FVA 54/7
épaisseur spécifique admissible du film lubri-
fiant
λ - 0,211
GFP
(voir 4.1.4 pour le calcul)
4.1.2 Calcul selon la méthode B
4.1.2.1 Calcul des caractéristiques géométriques de l’engrenage (selon l’ISO 21771)
Valeurs de base:
m = 10,93 mm
t
m
n
m =
t
cosβ
d = z ∙ m d = 196,74 mm
1 1 t 1
d = z ∙ m d = 196,74 mm
2 2 t 2
u = 1
z
2
u=
z
1
α = 20°
t
tanα
n
α =arctan
t
cosβ
d = d cosα d = 184,875 mm
b1 1 t b1
d = d cosα d = 184,875 mm
b2 2 t b2
d = 200 mm
w1
2⋅a
d =
w1
u+1
d = 200 mm
w2
da=⋅2 −d
w2 w1
α = 22,426°
wt
zz+ ⋅⋅m cosα
()
12 tt
α =arccos
wt
2⋅a
β = 0°
b
ββ=⋅arcsin(sin cos)α
bn
© ISO 2014 – Tous droits réservés 7
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ISO/TR 15144-2:2014(F)
p = 32,267 mm
et
pm=⋅πα⋅cos
et tt
ε = 0,705
1
2
z d
1 a1
ε = ⋅ −−1 tanα
1 wt
2⋅π d
b1
ε = 0,705
2
2
z d
2 a2
ε = ⋅ −−1 tanα
2 wt
2⋅π d
b2
ε = 1,411
α
2 22 2
dd dd
1
ab1 1 a2 b2
εα=⋅ −+ −−a⋅sin
α wt
p 44 44
et
b⋅sinβ
ε =
ε = 0
β β
m ⋅π
n
ε = ε + ε ε = 1,411
γ α β γ
g = 45,519 mm
α
2 2 22
gd=⋅05,s−+dd −da−⋅ inα
α ab1 1 ab2 2wt
Coordonnées des points de base (A, AB, B, C, D, DE, E) sur la ligne de conduite:
g = 0 mm (34) g = 0 mm
A A
gp−
α et
g =
(35) g = 6,626 mm
AB
AB
2
g = g − p (36) g = 13,253 mm
B α et B
22
dd d
b1 a1 b1
(37) g = 22,760 mm
C
g =⋅tanα −− +g
C wt α
24 4
gp=
(38) g = 32,267 mm
D
Det
gp−
α et
g = +p
(39) g = 38,893 mm
DE
DE et
2
g = g (40) g = 45,519 mm
E α E
2
22 2
dd d
b1 a1 b1
(41) d = 187,419 mm
d =⋅2 +− −+gg A1
A1 α A
44 4
d = 190,046 mm d = 193,546 mm d = 200,000 mm
AB1 B1 C1
d = 207,998 mm d = 214,394 mm d = 221,400 mm
D1 DE1 E1
8 © ISO 2014 – Tous droits réservés
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ISO/TR 15144-2:2014(F)
2
22 2
dd d
b2 a2 b2
(42) d = 221,400 mm
d =⋅2 +− −g A2
A2 A
44 4
d = 214,394 mm d = 207,998 mm d = 200,000 mm
AB2 B2 C2
d = 193,546 mm d = 190, 046 mm d = 187,419 mm
D2 DE2 E2
Rayon de courbure équivalent normal:
ρ
t,A
ρ =
(45) ρ = 12,285 mm
n,A n,A
cosβ
b
ρ = 15,663 mm ρ = 17,890 mm ρ = 19,074 mm
n,AB n,B n,C
ρ = 17,890 ρ = 15,663 mm ρ = 12,285 mm
n,D n,DE n,E
4.1.2.2 Calcul des données relatives aux matériaux
−1
2 2
11−νν−
1
2
2
E =⋅2 + (6) E = 226 374 N/mm
r
r
EE
1 2
Bc=⋅λρ ⋅ 0,5
(82) B = 12 427,4 N/(ms K)
M1
M1 M1 M1 M1
0,5
Bc=⋅λρ ⋅
(83) B = 12 427,4 N/(ms K)
M2 M2 M2 M2 M2
4.1.2.3 Calcul des conditions de fonctionnement
Chargement:
nT
1 1
P =⋅2 π⋅⋅
(85) P = 590 kW
60 1000
T
1
F =⋅2000
F = 19 091 N
t
t
d
1
T
1
F =⋅2000
bt F = 20 316 N
bt
d
b1
Facteur de répartition de charge local:
NOTE Aucune correction de profil des dents, engrenages à denture droite, classe de tolérance de denture ≤ 7
(voir l’ISO/TR 15144-1:2004, Figure 2).
g
Q2− 1
A
X = +⋅
(46) X = 0,333
A A
15 3 g
B
X = 0,500 X = 1,000 X = 1,000
AB B C
X = 1,000 X = 0,500 X = 0,333
D DE E
© ISO 2014 – Tous droits réservés 9
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ISO/TR 15144-2:2014(F)
Facteur d’élasticité:
E
r
2 0,5
Z = (26) Z = 189,812 (N/mm )
E
E
2⋅π
Pression de contact hertzienne locale:
FX⋅
tA
pZ=⋅ 2
(25) p = 963 N/mm
H,A,BE H,A,B
b⋅⋅ραcosc⋅ osβ
n,At b
2 2 2
p = 104 5 N/mm p = 138 3 N/mm p = 1 339 N/mm
H,AB,B H,B,B H,C,B
2 2 2
p = 138 3 N/mm p = 104 5 N/mm p = 963 N/mm
H,D,B H,DE,B H,E,B
pp=⋅ KK⋅⋅KK⋅ 2
(24) p = 1 084 N/mm
dynA,,BH,A,B AV HHαβ
dyn,A,B
2 2 2
p = 117 5 N/mm p = 155 5 N/mm p = 1 506 N/mm
dyn,AB,B dyn,B,B dyn,C,B
2 2 2
p = 155 5 N/mm p = 117 5 N/mm p = 1 084 N/mm
dyn,D,B dyn,DE,B dyn,E,B
Vitesse:
ν = −ν − ν (81) ν = −14,300 m/s
g,A r1,A r2,A g,A
ν = 10,137 m/s ν = 5,974 m/s ν = 0 m/s
g,AB g,B g,C
ν = 5,974 m/s ν = 10,137 m/s ν = 14,300 m/s
g,D g,DE g,E
ν = ν + ν (13) ν = 23,969 m/s
Σ,A r1,A r2,A Σ,A
ν = 23,969 m/s ν = 23,969 m/s ν =23,969 m/s
Σ,AB Σ,B Σ,C
ν = 23,969 m/s ν = 23,969 m/s ν =23,969 m/s
Σ,D Σ,DE Σ,E
Rugosité arithmétique moyenne effective:
RRaa=⋅05, + Ra
()
(3) Ra = 0,90 µm
12
4.1.2.4 Calcul des données relatives au lubrifiant
X = 1,0 pour l’huile minérale
L
(voir l’ISO/TR 15144-1:2014, Tableau 3)
−8 0,1348 −82
αη=⋅2,657 10 ⋅ (9) α =⋅21,/510 mN
38 38 38
X = 1,2 pour la lubrification par injection
S
4.1.2.5 Calcul du paramètre matériau
Coefficient de frottement moyen:
02, 5
Ra
X =⋅22, (87) X = 1,025
R
R
ρ
n,C
10 © ISO 2014 – Tous droits réservés
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ISO/TR 15144-2:2014(F)
K = 1,0 pour ε < 2
Bγ γ
02,
−00, 5
KK⋅⋅KK⋅⋅FK⋅
Av HHαβ bt Bγ
3
μ =⋅0,045 ⋅⋅10 η ⋅⋅XX
(86) μ = 0,048
() m
m θθoilR L
bv⋅⋅ρ
Σ,C n,C
Température de masse:
11 π
2 2
H =+εε +−1 ε ⋅+ ⋅ pour ε < 2
(91) H = 0,204
α
v ()1 2 α v
zz cosβ
12 b
ε = ε = ε
max 1 2
X = 1,0 pour les engrenages sans correction de profil
CA
(101)
(méthode B)
07, 2
PH⋅⋅μ X
mv S
θθ=+7400⋅ ⋅ (84) θ = 153,6 °C
M
Moil
ab⋅ 1,2⋅X
Ca
Paramètre de matériau:
6
G = 10 ∙ α ∙ E (5) G = 2 678,6
M θM r M
4.1.2.6 Calcul du paramètre de vitesse local
v
Σ,A
-11
U =⋅η
(12) U = 2,005 ∙ 10
AMθ A
2000⋅⋅E ρ
rn,A
-11 -11 -11
U = 1,572 ∙ 10 U = 1,377 ∙ 10 U = 1,291 ∙ 10
AB B C
-11 -11 -11
U = 1,377 ∙ 10 U = 1,572 ∙ 10 U = 2,005 ∙ 10
D DE E
4.1.2.7 Calcul du paramètre de charge
2
2⋅⋅π p
dyn,A
-4
W =
(22) W = 1,440 ∙ 10
A
A
2
E
r
-4 -4 -4
W = 1,694 ∙ 10 W = 2,966 ∙ 10 W = 2,781 ∙ 10
AB B C
-4 -4 -4
W = 2,966 ∙ 10 W = 1,694 ∙ 10 W = 1,440 ∙ 10
D DE E
4.1.2.8 Calcul du paramètre de glissement
Température-éclair locale:
6
10 ⋅⋅μ pv⋅
p
π mdyn,A g,A
dyn,A
θ =⋅ ⋅⋅8 ρ ⋅
(80) θ = 175,3 °C
fl,A n,A fl,A
2 11000⋅E
Bv +Bv
r
M1 r1,A M2 r2,A
θ = 154,1 °C θ = 145,4 °C θ = 0 °C
fl,AB fl,B fl,C
θ = 145,4 °C θ = 154,1 °C θ = 175,3 °C
fl,D fl,DE fl,E
© ISO 2014 – Tous droits réservés 11
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ISO/TR 15144-2:2014(F)
Température locale de contact en tant que somme de la température de masse et de la température-éclair:
θ = θ + θ (79) θ = 328,9 °C
B,A M fl,A B,A
θ = 307,7 °C θ = 299,0 °C θ = 153,6 °C
B,AB B,B B,C
θ = 299,0 °C θ = 307,7 °C θ = 328,9 °C
B,D B,DE B,E
Paramètre de glissement local:
αη⋅
θθB,AB,A
S =
(27) S = 0,057
GF,A GF,A
αη⋅
θθMM
S = 0,076 S = 0,086 S = 1,000
GF,AB GF,B GF,C
S = 0,086 S = 0,076 S = 0,057
GF,D GF,DE GF,E
4.1.2.9 Calcul de l’épaisseur du film lubrifiant
06,,07 −01,,3022
hG=⋅1600 ρ ⋅⋅UW⋅⋅S
(4) h = 0,122 μm
An,A MA AGF,A A
h = 0,137 μm h = 0,136 μm h = 0,241 μm
AB B C
h = 0,136 μm h = 0,137 μm h = 0,122 μm
D DE E
4.1.2.10 Calcul de l’épaisseur spécifique du film lubrifiant
h
A
λ =
(2) λ = 0,136
GF,A GF,A
Ra
λ =0,153 λ = 0,152 λ = 0,267
GF,AB GF,B GF,C
λ = 0,152 λ = 0,153 λ = 0,136
GF,D GF,DE GF,E
λ = λ = λ λ = 0,136
GF,min GF,A GF,E GF,min
4.1.2.11 Calcul du coefficient de sécurité contre la formation de micropiqûres
λ
GF,min
S =
(1) S = 0,644
λ λ
λ
GFP
Le calcul de l’épaisseur spécifique admissible du film lubrifiant, λ , pour l’exemple 1 est indiqué à titre
GFP
d’exemple en 4.1.4.
Les résultats finaux du ca
...
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