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CLASSIFICATION NOTES
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No. 41.2
Calculation of Gear Rating for Marine Transmissions
MAY 2012DET NORSKE VERITAS AS
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FOREWORDDET NORSKE VERITAS (DNV) is an autonomous and
independent foundation with the objectives of safeguarding
life,property and the environment, at sea and onshore. DNV
undertakes classification, certification, and other verification
andconsultancy services relating to quality of ships, offshore
units and installations, and onshore industries worldwide,
andcarries out research in relation to these
functions.Classification NotesClassification Notes are publications
that give practical information on classification of ships and
other objects. Examplesof design solutions, calculation methods,
specifications of test procedures, as well as acceptable repair
methods for some
components are given as interpretations of the more general rule
requirements. Det Norske Veritas AS May 2012
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shall mean the Foundation Det Norske Veritas as well as all its
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Classification Notes - No.41.2, May 2012Changes Page 3
CHANGES
GeneralThis document supersedes CN 41.2, May 2003.
Text affected by the main changes in this edition is highlighted
in red colour. However, if the changes involvea whole chapter,
section or sub-section, normally only the title will be in red
colour.
Main Changes 2.6 Minor changes in one formula.DET NORSKE VERITAS
AS
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Classification Notes - No.41.2, May 2012Contents Page 4
CONTENTS
1. Basic Principles and General Influence Factors
.................................................................................
51.1 Scope and Basic Principles
.......................................................................................................................51.2
Symbols, Nomenclature and Units
...........................................................................................................51.3
Geometrical
Definitions............................................................................................................................71.4
Bevel Gear Conversion Formulae and Specific Formulae
.......................................................................81.5
Nominal Tangential Load, Ft, Fbt, Fmt and
Fmbt.......................................................................................91.6
Application Factors, KA and
KAP.............................................................................................................91.7
Load Sharing Factor, K
.........................................................................................................................111.8
Dynamic Factor, Kv
................................................................................................................................111.9
Face Load Factors, KH and KF
...........................................................................................................151.10
Transversal Load Distribution Factors, KH and KF
...........................................................................201.11
Tooth Stiffness Constants, c and c
.......................................................................................................201.12
Running-in Allowances
..........................................................................................................................222.
Calculation of Surface
Durability.......................................................................................................
232.1 Scope and General Remarks
...................................................................................................................232.2
Basic
Equations.......................................................................................................................................232.3
Zone Factors ZH, ZB,D and ZM
...............................................................................................................252.4
Elasticity Factor,
ZE................................................................................................................................252.5
Contact Ratio Factor, Z
.........................................................................................................................262.6
Helix Angle Factor, Z
...........................................................................................................................262.7
Bevel Gear Factor, ZK
............................................................................................................................262.8
Values of Endurance Limit, Hlim and Static Strength, ,
.......................................................................262.9
Life Factor,
ZN........................................................................................................................................272.10
Influence Factors on Lubrication Film, ZL, ZV and
ZR..........................................................................272.11
Work Hardening Factor,
ZW...................................................................................................................282.12
Size Factor,
ZX........................................................................................................................................292.13
Subsurface
Fatigue..................................................................................................................................293.
Calculation of Tooth Strength
............................................................................................................
303.1 Scope and General Remarks
...................................................................................................................303.2
Tooth Root Stresses
................................................................................................................................313.3
Tooth Form Factors YF,
YFa...................................................................................................................323.4
Stress Correction Factors YS, YSa
..........................................................................................................353.5
Contact Ratio Factor
Y..........................................................................................................................353.6
Helix Angle Factor
Y............................................................................................................................363.7
Values of Endurance Limit, FE
.............................................................................................................363.8
Mean stress influence Factor,
YM...........................................................................................................373.9
Life Factor, YN
.......................................................................................................................................383.10
Relative Notch Sensitivity Factor, YrelT
...............................................................................................393.11
Relative Surface Condition Factor, YRrelT
.............................................................................................403.12
Size Factor, YX
.......................................................................................................................................403.13
Case Depth Factor,
YC............................................................................................................................403.14
Thin rim factor YB
..................................................................................................................................413.15
Stresses in Thin
Rims..............................................................................................................................423.16
Permissible Stresses in Thin
Rims..........................................................................................................444.
Calculation of Scuffing Load Capacity
..............................................................................................
454.1
Introduction.............................................................................................................................................454.2
General
Criteria.......................................................................................................................................464.3
Influence
Factors.....................................................................................................................................474.4
The Flash Temperature fla
....................................................................................................................49Appendix
A.Fatigue Damage Accumulation
.....................................................................................................................
57Appendix B.Application Factors for Diesel Driven
Gears...............................................................................................
59Appendix C.Calculation of Pinion-Rack
...........................................................................................................................
61DET NORSKE VERITAS AS
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Classification Notes - No.41.2, May 2012Sec.1. Basic Principles
and General Influence Factors Page 5
1. Basic Principles and General Influence Factors1.1 Scope and
Basic PrinciplesThe gear rating procedures given in this
Classification Note are mainly based on the ISO6336 Part 1 to
5(cylindrical gears), and partly on ISO 10300 Part 1 to 3 (bevel
gears) and ISO Technical Reports on Scuffingand Fatigue Damage
Accumulation, but especially applied for marine purposes, such as
marine propulsion andimportant auxiliaries onboard ships and mobile
offshore units.The calculation procedures cover gear rating as
limited by contact stresses (pitting, spalling or case
crushing),tooth root stresses (fatigue breakage or overload
breakage), and scuffing resistance. Even though no
calculationprocedures for other damages such as wear, grey staining
(micropitting), etc. are given, such damages may limitthe gear
rating. The Classification Note applies to enclosed parallel shaft
gears, epicyclic gears and bevel gears (withintersecting axis).
However, open gear trains may be considered with regard to tooth
strength, i.e. part 1 and 3may apply. Even pinion-rack tooth
strength may be considered, but since such gear trains often are
designedwith non-involute pinions, the calculation procedure of
pinion-racks is described in Appendix C.Steel is the only material
considered.The methods applied throughout this document are only
valid for a transverse contact ratio 1 < < 2. If >
2,either special considerations are to be made, or suggested
simplification may be used.All influence factors are defined
regarding their physical interpretation. Some of the influence
factors aredetermined by the gear geometry or have been established
by conventions. These factors are to be calculatedin accordance
with the equations provided. Other factors are approximations,
which are clearly stated in thetext by terms as may be calculated
as. These approximations are substitutes for exact evaluations
where suchare lacking or too extensive for practical purposes, or
factors based on experience. In principle, any suitablemethod may
replace these approximations.Bevel gears are calculated on basis of
virtual (equivalent) cylindrical gears using the geometry of
themidsection. The virtual (helical) cylindrical gear is to be
calculated by using all the factors as a real cylindricalgear with
some exceptions. These exceptions are mentioned in connection with
the applicable factors.Wherever a factor or calculation procedure
has no reference to either cylindrical gears or bevel gears, it
isgenerally valid, i.e. combined for both cylindrical and bevel. In
order to minimise the volume of this Classification Note such
combinations are widely used, and everywhereit is necessary to
distinguish, it is clearly pointed out by local headings such
as:Cylindrical gearsBevel gearsThe permissible contact stresses,
tooth root stresses and scuffing load capacity depend on the safety
factors asrequired in the respective Rule sections.Terms as
endurance limit and static strength are used throughout this
Classification Note.Endurance limit is to be understood as the
fatigue strength in the range of cycles beyond the lower knee of
theN curves, regardless if it is constant or drops with higher
number of cycles.Static strength is to be understood as the fatigue
strength in the range of cycles less than at the upper knee ofthe N
curves.For gears that are subjected to a limited number of cycles
at different load levels, a cumulative fatiguecalculation applies.
Information on this is given in Appendix A.When the term infinite
life is used, it means number of cycles in the range 108 to
1010.
1.2 Symbols, Nomenclature and UnitsThe main symbols as influence
factors (K, Z, Y and X with indeces) etc. are presented in their
respectiveheadings. Symbols which are not explained in their
respective sections are as follows:
a = centre distance (mm).b = facewidth (mm).d = reference
diameter (mm).da = tip diameter (mm).db = base diameter (mm).dw =
working pitch diameter (mm).DET NORSKE VERITAS AS
ha = addendum (mm).
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Classification Notes - No.41.2, May 2012Sec.1. Basic Principles
and General Influence Factors Page 6
Index 1 refers to the pinion, 2 to the wheel.Index n refers to
normal section or virtual spur gear of a helical gear.Index w
refers to pitch point.Special additional symbols for bevel gears
are as follows:
Index v refers to the virtual (equivalent) helical cylindrical
gear.
ha0 = addendum of tool ref. to mn.hfp = dedendum of basic rack
ref. to mn (= ha0).hFe = bending moment arm (mm) for tooth root
stresses for application of load at the outer point of single tooth
pair
contact.hFa = bending moment arm (mm) for tooth root stresses
for application of load at tooth tip.HB = Brinell hardness.HV =
Vickers hardness.HRC = Rockwell C hardnessmn = normal module.n =
rev. per minute.NL = number of load cycles.qs = notch parameter.Ra
= average roughness value (m).Ry = peak to valley roughness (m).Rz
= mean peak to valley roughness (m).san = tooth top land thickness
(mm).sat = transverse top land thickness (mm).sFn = tooth root
chord (mm) in the critical section.spr = protuberance value of tool
minus grinding stock, equal residual undercut of basic rack, ref.
to mn.T = torque (Nm).u = gear ratio (per stage).v = linear speed
(m/s) at reference diameter.x = addendum modification coefficient.z
= number of teeth.zn = virtual number of spur teeth.n = normal
pressure angle at ref. cylinder.t = transverse pressure angle at
ref. cylinder.a = transverse pressure angle at tip cylinder.wt =
transverse pressure angle at pitch cylinder. = helix angle at ref.
cylinder.b = helix angle at base cylinder.a = helix angle at tip
cylinder. = transverse contact ratio. = overlap ratio. = total
contact ratio.a0 = tip radius of tool ref. to mn.fp = root radius
of basic rack ref. to mn ( = a0).C = effective radius (mm) of
curvature at pitch point.F = root fillet radius (mm) in the
critical section.B = ultimate tensile strength (N/mm2).y = yield
strength resp. 0.2% proof stress (N/mm2).
= angle between intersection axis.= angle modification
(Klingelnberg)
m0 = tool module (Klingelnberg) = pitch cone angle.xsm = tooth
thickness modification coefficient (midface).R = pitch cone
distance (mm).
KDET NORSKE VERITAS AS
Index m refers to the midsection of the bevel gear.
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Classification Notes - No.41.2, May 2012Sec.1. Basic Principles
and General Influence Factors Page 7
1.3 Geometrical DefinitionsFor internal gearing z2, a; da2, dw2,
d2 and db2 are negative, x2 is positive if da2 is increased, i.e.
the numericvalue is decreased.The pinion has the smaller number of
teeth, i.e.
For calculation of surface durability b is the common facewidth
on pitch diameter.For tooth strength calculations b1 or b2 are
facewidths at the respective tooth roots. If b1 or b2 differ much
fromb above, they are not to be taken more than 1 module on either
side of b.Cylindrical gears
tan t = tan n / cos tan b = tan cos ttan a = tan da / dcos a =
db/dad = z mn / cos mt = mn /cos db = d cos t = dw cos wta = 0.5
(dw1 + dw2)dw1/dw2 = z1 / z2inv = tan - (radians)inv wt = inv t + 2
tan n (x1 + x2)/(z1 + z2)zn = z / (cos2 b cos )
where fw1 is to be taken as the smaller of:
and
, where fw2 is calculated as fw1substituting the values for the
wheel by the values for the pinion and visa versa.
11
2 =zz
u
1
aw1fw1 T
+=
wtfw1 tan =
soi1
b1wtfw1 d
dacostan -tan =
1
2wt
a2
b2fw1 z
ztan
d
dacostan
=
2
1fw2aw1 z
z =
11 z
2T =
( ) +
+=
2sinxhm
2d2d nfpfp1fpnsoi1
21
t
nfpfplfpn2
tan)sinx(hm
+
nmsinb
=DET NORSKE VERITAS AS
(for double helix, b is to be taken as the width of one
helix).
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Classification Notes - No.41.2, May 2012Sec.1. Basic Principles
and General Influence Factors Page 8
1.4 Bevel Gear Conversion Formulae and Specific
FormulaeConversion of bevel gears to virtual equivalent helical
cylindrical gears is based on the bevel gear midsection.The
conversion formulae are:Number of teeth:
zv1.2 = z1,2/ cos 1,2(1 + 2 = )
Gear ratio:
tan vt = tan n/ cos mtan bm = tan m cos vt
Base pitch:
Reference, pitch, diameters:
Centre distance:av = 0.5 (dv1 + dv2)
Tip diameters:dva 1.2 = dv 1,2 + 2 ham 1,2
Addenda:for gears with constant addenda (Klingelnberg):
ham 1,2 = mmn (1 + xm 1,2)for gears with variable addenda
(Gleason):
ham 1,2 = ha 1,2 b/2 tan (a 1,2 1,2)
y =
C =
v =
pbt =
sat =
san =
+
( )2bwt
u1cossinua+
311 10dn60
coscosm tn
+
+
at
ninvinv
z
tanx22
d a
acossat
1
2
v
vv z
zu =
m
vtnmbtm cos
cosmp =
2,1
2,1m2.1v cos
dd
=DET NORSKE VERITAS AS
(when ha is addendum at outer end and a is the outer cone
angle).
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Classification Notes - No.41.2, May 2012Sec.1. Basic Principles
and General Influence Factors Page 9
Addendum modification coefficients:
Base circle:dvb 1,2=dv 1,2 cos vt
Transverse contact ratio:*)
Overlap ratio*) (theoretical value for bevel gears with no
crowning, but used as approximations in thecalculation
procedures):
Total contact ratio:*)
(* Note that index v is left out in order to combine formulae
for cylindrical and bevel gears.)Tangential speed at
midsection:
Effective radius of curvature (normal section):
Length of line of contact:
1.5 Nominal Tangential Load, Ft, Fbt, Fmt and FmbtThe nominal
tangential load (tangential to the reference cylinder with diameter
d and perpendicular to an axialplane) is calculated from the
nominal (rated) torque T transmitted by the gear set.Cylindrical
gears
Bevel gears
1.6 Application Factors, KA and KAPThe application factor KA
accounts for dynamic overloads from sources external to the
gearing.It is distinguished between the influence of repetitive
cyclic torques KA (1.6.1) and the influence of temporaryoccasional
peak torques KAP (1.6.2).Calculations are always to be made with
KA. In certain cases additional calculations with KAP may be
mn
1,2am2,1am2,1m m2
hhx
=
btm
vtv2
2vb2
2va2
1vb2
1va
P
sinadd0.5dd0.5 +=
nm
mmsinb
=
22 +=
3m11mt 10dn60
v =
( )2vbmvtvv
vcu1cossinua
+=
( )( )( )1if
12cosb
l 2
22bm
b 0.85, as the influence of higher modes hasto be considered,
see 1.8.2. In case of significant lateral shaft flexibility (e.g.
overhung mounted bevel gears),the influence of coupled bending and
torsional vibrations between pinion and wheel should be considered
if N 0.75 , see 1.8.2.
where:c is the actual mesh stiffness per unit facewidth, see
1.11.For gears with inactive ends of the facewidth, as e.g. due to
high crowning or end relief such as often appliedfor bevel gears,
the use of c in connection with determination of natural
frequencies may need correction. cis defined as stiffness per unit
facewidth, but when used in connection with the total mesh
stiffness, it is not assimple as c b, as only a part of the
facewidth is active. Such corrections are given in 1.11.mred is the
reduced mass of the gear pair, per unit facewidth and referred to
the plane of contact.For a single gear stage where no significant
inertias are closely connected to neither pinion nor wheel, mred
iscalculated as:
The individual masses per unit facewidth are calculated as
where I is the polar moment of inertia (kgmm2).The inertia of
bevel gears may be approximated as discs with diameter equal the
midface pitch diameter andwidth equal to b. However, if the shape
of the pinion or wheel body differs much from this idealised
cylinder,the inertia should be corrected accordingly.For all kind
of gears, if a significant inertia (e.g. a clutch) is very rigidly
connected to the pinion or wheel, itshould be added to that
particular inertia (pinion or wheel). If there is a shaft piece
between these inertias, thetorsional shaft stiffness alters the
system into a 3-mass (or more) system. This can be calculated as in
1.8.2, butalso simplified as a 2-mass system calculated with only
pinion and wheel masses.
1.8.1.2 Factors used for determination of KvNon-dimensional gear
accuracy dependent parameters:
1E
1nnN =
red
1
3
1E mc
z1030n =
21
21red mm
mmm+
=
22,1b
2,12,1
)2/d(b
Im =
( )b/KKF
yf'cB
At
pptp
=
( )yF'cB ff =DET NORSKE VERITAS AS
b/KKF At
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Classification Notes - No.41.2, May 2012Sec.1. Basic Principles
and General Influence Factors Page 13
Non-dimensional tip relief parameter:
For gears of quality grade (ISO 1328) Q = 7 or coarser, Bk =
1.For gears with Q 6 and excessive tip relief, Bk is limited to
max. 1.For gears (all quality grades) with tip relief of more than
2Ceff (see 4.3.2) the reduction of has to beconsidered (see
4.4.3).Where:
1.8.1.3 Kv in the subcritical range:
Kv = 1 + N KK = Cv1 Bp + Cv2 Bf + Cv3 Bk
Cv1 accounts for the pitch error influenceCv1 = 0.32Cv2 accounts
for profile error influence
Cv3 accounts for the cyclic mesh stiffness variation
1.8.1.4 Kv in the main resonance range:
Running in this range should preferably be avoided, and is only
allowed for high precision gears.Kv = 1 + Cv1 Bp + Cv2 Bf + Cv4
Bk
Cv4 accounts for the resonance condition with the cyclic mesh
stiffness variation.
fpt = the single pitch deviation (ISO 1328), max. of pinion or
wheel F = the total profile form deviation (ISO 1328), max. of
pinion or wheel (Note: F is p.t. not available for bevel
gears, thus use F = fpt)yp and yf = the respective running-in
allowances and may be calculated similarly to y in 1.12, i.e. the
value of fpt is
replaced by F for yf.c = the single tooth stiffness, see 1.11Ca
= the amount of tip relief, see 4.3.3. In case of different tip
relief on pinion and wheel, the value that results
in the greater value of Bk is to be used. If Ca is zero by
design, the value of running-in tip relief Cay (see 1.12) may be
used in the above formula.
Cylindrical gears: N 0.85Bevel gears: N 0.75
Cv2 = 0.34 for 2
for > 2
Cv3 = 0.23 for 2
for > 2
Cylindrical gears: 0.85 < N 1.15Bevel gears: 0.75 < N
1.25
Cv4 = 0.90 for 2
for > 2
/bKKFc'C
1BAt
ak
=
0.30.57C2v
=
1.560.096C3v
=
1.440.050.57
C
4v
=DET NORSKE VERITAS AS
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Classification Notes - No.41.2, May 2012Sec.1. Basic Principles
and General Influence Factors Page 14
1.8.1.5 Kv in the supercritical range:
Special care should be taken as to influence of higher vibration
modes, and/or influence of coupled bending(i.e. lateral shaft
vibrations) and torsional vibrations between pinion and wheel.
These influences are notcovered by the following approach.
Kv = Cv5 Bp + Cv6 Bf + Cv7 Cv5 accounts for the pitch error
influence.Cv5 = 0.47Cv6 accounts for the profile error
influence.
Cv7 relates the maximum externally applied tooth loading to the
maximum tooth loading of ideal, accurategears operating in the
supercritical speed sector, when the circumferential vibration
becomes very soft.
1.8.1.6 Kv in the intermediate range:
Comments raised in 1.8.1.4 and 1.8.1.5 should be observed.Kv is
determined by linear interpolation between Kv for N = 1.15
respectively 1.25 and N = 1.5 asCylindrical gears
Bevel gears
1.8.2 Multi-resonance methodFor high speed gear (v > 40 m/s),
for multimesh medium speed gears, for gears with significant
lateral shaftflexibility etc. it is advised to determine Kv on
basis of relevant dynamic analysis.Incorporating lateral shaft
compliance requires transformation of even a simple pinion-wheel
system into alumped multi-mass system. It is advised to incorporate
all relevant inertias and torsional shaft stiffnesses intoan
equivalent (to pinion speed) system. Thereby the mesh stiffness
appears as an equivalent torsional stiffness:
c b (db1/2)2 (Nm/rad)
The natural frequencies are found by solving the set of
differential equations (one equation per inertia). Notethat for a
gear put on a laterally flexible shaft, the coupling
bending-torsionals is arranged by introducing thegear mass and the
lateral stiffness with its relation to the torsional displacement
and torque in that shaft.Only the natural frequency (ies) having
high relative displacement and relative torque through the
actual
Cylindrical gears: N 1.5Bevel gears: N 1.5
Cv6 = 0.47 for 2
for > 2
Cv7 = 0.75 for 1.5
for 1.5 < 2.5
Cv7 = 1.0 for > 2.5
Cylindrical gears: 1.15 < N < 1.5Bevel gears: 1.25 < N
< 1.5
1.740.12C6v
=
[ ] 875.0)2(sin 0.125v7C +=
( ) ( ) ( )[ ]5.1Nv15.1Nv5.1Nvv KK35.0N5.1KK
===
+=
( ) ( ) ( )[ ]5.1Nv25.1Nv5.1Nvv KK25.0N5.1KK
===
+=DET NORSKE VERITAS AS
pinion-wheel flexible element, need(s) to be considered as
critical frequency (ies).
-
Classification Notes - No.41.2, May 2012Sec.1. Basic Principles
and General Influence Factors Page 15
Kv may be determined by means of the method mentioned in 1.8.1
thereby using N as the least favourable ratio(in case of more than
one pinion-wheel dominated natural frequency). I.e. the N-ratio
that results in the highestKv has to be considered.The level of the
dynamic factor may also be determined on basis of simulation
technique using numeric timeintegration with relevant tooth
stiffness variation and pitch/profile errors.
1.9 Face Load Factors, KH and KFThe face load factors, KH for
contact stresses and for scuffing, KF for tooth root stresses,
account for non-uniform load distribution across the facewidth.KH
is defined as the ratio between the maximum load per unit facewidth
and the mean load per unit facewidth.KF is defined as the ratio
between the maximum tooth root stress per unit facewidth and the
mean tooth rootstress per unit facewidth. The mean tooth root
stress relates to the considered facewidth b1 respectively b2.Note
that facewidth in this context is the design facewidth b, even if
the ends are unloaded as often applies toe.g. bevel gears.The plane
of contact is considered.
1.9.1 Relations between KH and KF
where h/b is the ratio tooth height/facewidth. The maximum of
h1/b1, and h2/b2 is to be used, but not higherthan 1/3. For double
helical gears, use only the facewidth of one helix.If the tooth
root facewidth (b1 or b2) is considerably wider than b, the value
of KF(1or2) is to be speciallyconsidered as it may even exceed
KH.E.g. in pinion-rack lifting systems for jack up rigs, where b =
b2 mn and b1 3 mn, the typical KH KF2 1 and KF1 1.3.1.9.2
Measurement of face load factorsPrimarily,KF may be determined by a
number of strain gauges distributed over the facewidth. Such strain
gauges mustbe put in exactly the same position relative to the root
fillet. Relations in 1.9.1 apply for conversion to
KH.Secondarily,KH may be evaluated by observed contact patterns on
various defined load levels. It is imperative that thevarious test
loads are well defined. Usually, it is also necessary to evaluate
the elastic deflections. Some teethat each 90 degrees are to be
painted with a suitable lacquer. Always consider the poorest of the
contact patterns.After having run the gear for a suitable time at
test load 1 (the lowest), observe the contact pattern with
respectto extension over the facewidth. Evaluate that KH by means
of the methods mentioned in this section. Proceedin the same way
for the next higher test load etc., until there is a full face
contact pattern. From these data, theinitial mesh misalignment
(i.e. without elastic deflections) can be found by extrapolation,
and then also the KHat design load can be found by calculation and
extrapolation. See example.
HF KK =2(h/b)h/b1
1++DET NORSKE VERITAS AS
Figure 1.1 Example of experimental determination of KHb
-
Classification Notes - No.41.2, May 2012Sec.1. Basic Principles
and General Influence Factors Page 16
It must be considered that inaccurate gears may accumulate a
larger observed contact pattern than the actualsingle mesh to mesh
contact patterns. This is particularly important for lapped bevel
gears. Ground or hardmetal hobbed bevel gears are assumed to
present an accumulated contact pattern that is practically equal
theactual single mesh to mesh contact patterns. As a rough guidance
the (observed) accumulated contact patternof lapped bevel gears may
be reduced by 10% in order to assess the single mesh to mesh
contact pattern whichis used in 1.9.9.
1.9.3 Theoretical determination of KHThe methods described in
1.9.3 to 1.9.8 may be used for cylindrical gears. The principles
may to some extentalso be used for bevel gears, but a more
practical approach is given in 1.9.9.General: For gears where the
tooth contact pattern cannot be verified during assembly or under
load, allassumptions are to be well on the safe side.KH is to be
determined in the plane of contact.The influence parameters
considered in this method are:
mean mesh stiffness c (see 1.11) (if necessary, also variable
stiffness over b) mean unit load Fm/b = Fbt KA K Kv/b (for double
helical gears, see 1.7 for use of K) misalignment fsh due to
elastic deflections of shafts and gear bodies (both pinion and
wheel) misalignment fdefl due to elastic deflections of and working
positions in bearings misalignment fbe due to bearing clearance
tolerances misalignment fma due to manufacturing tolerances helix
modifications as crowning, end relief, helix correction running in
amount y (see 1.12).In practice several other parameters such as
centrifugal expansion, thermal expansion, housing deflection,
etc.contribute to KH. However, these parameters are not taken into
account unless in special cases when beingconsidered as
particularly important.When all or most of the a.m. parameters are
to be considered, the most practical way to determine KH is bymeans
of a graphical approach, described in 1.9.3.1.If c can be
considered constant over the facewidth, and no helix modifications
apply, KH can be determinedanalytically as described in
1.9.3.2.
1.9.3.1 Graphical methodThe graphical method utilises the
superposition principle, and is as follows:
Calculate the mean mesh deflection M as a function of Fm /b and
c, see 1.11. Draw a base line with length b, and draw up a
rectangular with height M. (The area M b is proportional
to the transmitted force). Calculate the elastic deflection fsh
in the plane of contact. Balance this deflection curve around a
zero line,
so that the areas above and below this zero line are equal.
Figure 1.2 fsh balanced around zero line
Superimpose these ordinates of the fsh curve to the previous
load distribution curve. (The area under thisnew load distribution
curve is still M b.).
Calculate the bearing deflections and/or working positions in
the bearings and evaluate the influence fdeflin the plane of
contact. This is a straight line and is balanced around a zero line
as indicated in Fig. 1.4, butDET NORSKE VERITAS AS
with one distinct direction. Superimpose these ordinates to the
previous load distribution curve.
-
Classification Notes - No.41.2, May 2012Sec.1. Basic Principles
and General Influence Factors Page 17
The amount of crowning, end relief or helix correction (defined
in the plane of contact) is to be balancedaround a zero line
similarly to fsh.
Figure 1.3 Crowning Cc balanced around zero line
Superimpose these ordinates to the previous load distribution
curve. In case of high crowning etc. as e.g.often applied to bevel
gears, the new load distribution curve may cross the base line (the
real zero line). Theresult is areas with negative load that is not
real, as the load in those areas should be zero. Thus
correctiveactions must be made, but for practical reasons it may be
postponed to after next operation.
The amount of initial mesh misalignment, fma + fbe (defined in
the plane of contact), is to be balancedaround a zero line. If the
direction of fma + fbe is known (due to initial contact check), or
if the direction offbe is known due to design (e.g. overhang bevel
pinion), this should be taken into account. If directionunknown,
the influence of fma + fbe in both directions as well as equal
zero, should be considered.
Figure 1.4 fma+fbe in both directions, balanced around zero
line.Superimpose these ordinates to the previous load distribution
curve. This results in up to 3 different curves, ofwhich the one
with the highest peak is to be chosen for further evaluation.
If the chosen load distribution curve crosses the base line
(i.e. mathematically negative load), the curve isto be corrected by
adding the negative areas and dividing this with the active
facewidth. The (constant)ordinates of this rectangular correction
area are to be subtracted from the positive part of the
loaddistribution curve.It is advisable to check that the area
covered under this new load distribution curve is still equal M
b.
If c cannot be considered as constant over b, then correct the
ordinates of the load distribution curve withthe local (on various
positions over the facewidth) ratio between local mesh stiffness
and average meshstiffness c (average over the active facewidth
only).Note that the result is to be a curve that covers the same
area M b as before.
The influence of running in y is to be determined as in 1.12
whereby the value for Fx is to be taken astwice the distance
between the peak of the load distribution curve and M.
Determine
1.9.3.2 Simplified analytical method for cylindrical gearsThe
analytical approach is similar to 1.9.3.1 but has a more limited
application as c is assumed constant overthe facewidth and no helix
modification applies.
Calculate the elastic deflection fsh in the plane of contact.
Balance this deflection curve around a zero line,
M
H
ycurveofpeakK
=DET NORSKE VERITAS AS
so that the area above and below this zero line are equal, see
Fig. 1.2. The max. positive ordinate is fsh.
-
Classification Notes - No.41.2, May 2012Sec.1. Basic Principles
and General Influence Factors Page 18
Calculate the initial mesh alignment as
The negative signs may only be used if this is justified and/or
verified by a contact pattern test. Otherwise,always use positive
signs. If a negative sign is justified, the value of Fx is not to
be taken less than thelargest of each of these elements.
Calculate the effective mesh misalignment asFy = Fx - y (y see
1.12)
Determine
or
where c as used here is the effective mesh stiffness, see
1.11.
1.9.4 Determination of fshfsh is the mesh misalignment due to
elastic deflections. Usually it is sufficient to consider the
combined meshdeflection of the pinion body and shaft and the wheel
shaft. The calculation is to be made in the plane of contact(of the
considered gear mesh), and to consider all forces (incl. axial)
acting on the shafts. Forces from othermeshes can be parted into
components parallel respectively vertical to the considered plane
of contact. Forcesvertical to this plane of contact have no
influence on fsh.It is advised to use following diameters for
toothed elements:
Usually, fsh is calculated on basis of an evenly distributed
load. If the analysis of KH shows a considerablemaldistribution in
term of hard end contact, or if it is known by other reasons that
there exists a hard endcontact, the load should be correspondingly
distributed when calculating fsh. In fact, the whole KH
procedurecan be used iteratively. 2 to 3 iterations will be enough,
even for almost triangular load distributions.
1.9.5 Determination of fdeflfdefl is the mesh misalignment in
the plane of contact due to bearing deflections and working
positions (housingdeflection may be included if determined). First
the journal working positions in the bearings are to be determined.
The influence of external moments andforces must be considered.
This is of special importance for twin pinion single output gears
with all 3 shafts inone plane.For rolling bearings fdefl is further
determined on basis of the elastic deflection of the bearings. An
elasticbearing deflection depends on the bearing load and size and
number of rolling elements. Note that the bearingclearance
tolerances are not included here. For fluid film bearings fdefl is
further determined on basis of the lift and angular shift of the
shafts due tolubrication oil film thickness. Note that fbe takes
into account the influence of the bearing clearance tolerance. When
working positions, bearing deflections and oil film lift are
combined for all bearings, the angularmisalignment as projected
into the plane of the contact is to be determined. fdefl is this
angular misalignment(radians) times the facewidth.
1.9.6 Determination of fbefbe is the mesh misalignment in the
plane of contact due to tolerances in bearing clearances. In
principle fbe andfdefl could be combined. But as fdefl can be
determined by analysis and has a distinct direction, and fbe
isdependent on tolerances and in most cases has no distinct
direction (i.e. tolerance), it is practicable to separatethese two
influences.Due to different bearing clearance tolerances in both
pinion and wheel shafts the two shaft axis will have an
d + 2 x mn for bending and shear deflectiond + 2 mn (x ha0 +
0.2) for torsional deflection
deflbemashx ffffF =
2KforF2
bFc1K H
m
H +=
2KforF
bFc2K H
m
H >=DET NORSKE VERITAS AS
angular misalignment in the plane of contact that is
superimposed to the working positions determined in 1.9.5.
-
Classification Notes - No.41.2, May 2012Sec.1. Basic Principles
and General Influence Factors Page 19
fbe is the facewidth times this angular misalignment. Note that
fbe may have a distinct direction or be given asa tolerance, or a
combination of both. For combination of tolerance it is adviced to
use
fbe is particularly important for overhang designs, for gears
with widely different kinds of bearings on eachside, and when the
bearings have wide tolerances on clearances. In general it shall be
possible to replacestandard bearings without causing the real load
distribution to exceed the design premises. For slow speed
gearswith journal bearings, the expected wear should also be
considered.
1.9.7 Determination of fmafma is the mesh misalignment due to
manufacturing tolerances (helix slope deviation) of pinion fH1,
wheelfH2 and housing bore.For gear without specifically approved
requirements to assembly control, the value of fma is to be
determined as
For gears with specially approved assembly control, the value of
fma will depend on those specificrequirements.
1.9.8 Comments to various gear typesFor double helical gears, KH
is to be determined for both helices. Usually an even load share
between thehelices can be assumed. If not, the calculation is to be
made as described in 1.7.1.For planetary gears the free floating
sun pinion suffers only twist, no bending. It must be noted that
the totaltwist is the sum of the twist due to each mesh. If the
value of K 1, this must be taken into account whencalculating the
total sun pinion twist (i.e. twist calculated with the force per
mesh without K, and multipliedwith the number of planets).When
planets are mounted on spherical bearings, the mesh misalignments
sun-planet respectively planet-annulus will be balanced. I.e. the
misalignment will be the average between the two theoretical
individualmisalignments. The faceload distribution on the flanks of
the planets can take full advantage of this. However,as the sun and
annulus mesh with several planets with possibly different lead
errors, the sun and annulus cannotobtain the above mentioned
advantage to the full extent.
1.9.9 Determination of KH for bevel gearsIf a theoretical
approach similar to 1.9.3 to 1.9.8 is not documented, the following
may be used.
beff / b represents the relative active facewidth (regarding
lapped gears, see 1.9.2 last part).Higher values than beff / b =
0.90 are normally not to be used in the formula.For dual
directional gears it may be difficult to obtain a high beff / b in
both directions. In that case the smallerbeff / b is to be
used.Ktest represents the influence of the bearing arrangement,
shaft stiffness, bearing stiffness, housing stiffness etc.on the
faceload distribution and the verification thereof. Expected
variations in length- and height-wise toothprofile is also
accounted for to some extent.
a) Ktest = 1For ground or hard metal hobbed gears with the
specified contact pattern verified at full rating or at fulltorque
slow turning at a condition representative for the thermal
expansion at normal operation.It also applies when the bearing
arrangement/support has insignificant elastic deflections and
thermal axialexpansion. However, each initial mesh contact must be
verified to be within acceptance criteria that arecalibrated
against a type test at full load. Reproduction of the gear tooth
length- and height-wise profilemust also be verified. This can be
made through 3D measurements or by initial contact movements
causedby defined axial offsets of the pinion (tolerances to be
agreed upon).
b) Ktest = 1 + 0.4(beff/b0.6)For designs with possible influence
of thermal expansion in the axial direction of the pinion. The
initialmesh contact verified with low load or spin test where the
acceptance criteria are calibrated against a type
........fff 22be21bebe ++=
22H
21Hma fff +=
testeff
H Kbb85.185.1K
=DET NORSKE VERITAS AS
test at full load.
-
Classification Notes - No.41.2, May 2012Sec.1. Basic Principles
and General Influence Factors Page 20
c) Ktest = 1.2if mesh is only checked by toolmakers blue or by
spin test contact. For gears in this category beff./b > 0.85is
not to be used in the calculation.
1.10 Transversal Load Distribution Factors, KH and KFThe
transverse load distribution factors, KH for contact stresses and
for scuffing, KF for tooth root stressesaccount for the effects of
pitch and profile errors on the transversal load distribution
between 2 or more pairsof teeth in mesh.The following relations may
be used:Cylindrical gears:
valid for 2
valid for > 2 where:
Limitations of KH and KF:If the calculated values for
KF = KH < 1, use KF = KH = 1.0
Bevel gears:For ground or hard metal hobbed gears, KF = KH =
1For lapped gears, KF = KH = 1.1
1.11 Tooth Stiffness Constants, c and cThe tooth stiffness is
defined as the load which is necessary to deform one or several
meshing gear teeth having1 mm facewidth by an amount of 1 m, in the
plane of contact.c is the maximum stiffness of a single pair of
teeth.c is the mean value of the mesh stiffness in a transverse
plane (brief term: mesh stiffness).
FtH = Ft KA K Kv KHc = See 1.11 = See 1.12fpt = Maximum single
pitch deviation (m) of pinion or wheel, or maximum total profile
form deviation F of
pinion or wheel if this is larger than the maximum single pitch
deviation.Note: In case of adequate equivalent tip relief adapted
to the load, half of the above mentioned fpt can be introduced.
A tip relief is considered adequate when the average of Ca1 and
Ca2 is within 40% of the value of Ceff in 4.3.2:
If the calculated value of use
If the calculated value of use
where
(for n see 3.3.1.c)
( )
+==tH
ptHF F
byfc0.40.9
2
KK
( ) ( )tH
pt
HF F
byfc
120.40.9KK
+==
2
HZ
K > 2
HZ
K =
F Y
K >
F Y
K =
n0.750.25Y +=DET NORSKE VERITAS AS
Both valid for high unit load. (Unit load = Ft KA K/b).
-
Classification Notes - No.41.2, May 2012Sec.1. Basic Principles
and General Influence Factors Page 21
Cylindrical gearsThe real stiffness is a combination of the
progressive Hertzian contact stiffness and the linear tooth
bendingstiffnesses. For high unit loads the Hertzian stiffness has
little importance and can be disregarded. Thisapproach is on the
safe side for determination of KH and KH. However, for moderate or
low loads Kv maybe underestimated due to determination of a too
high resonance speed.The linear approach is described in A.An
optional approach for inclusion of the non-linear stiffness is
described in B.A. The linear approach.
and
where:
(for internal gears, use zn2 equal infinite and x2 = 0).ha0 =
hfp for all practical purposes.CR considers the increased
flexibility of the wheel teeth if the wheel is not a solid disc,
and may be calculatedas:
where:
The formula is valid for bs / b 0.2 and sR/mn 1. Outside this
range of validity and if the web is not centrallypositioned, CR has
to be specially considered.Note:CR is the ratio between the average
mesh stiffness over the facewidth and the mesh stiffness of a gear
pair ofsolid discs. The local mesh stiffness in way of the web
corresponds to the mesh stiffness with CR = 1. The localmesh
stiffness where there is no web support will be less than
calculated with CR above. Thus, e.g. a centrallypositioned web will
have an effect corresponding to a longitudinal crowning of the
teeth. See also 1.9.3.1regarding KH.B. The non-linear approach.In
the following an example is given on how to consider the
non-linearity.The relation between unit load F/b as a function of
mesh deflection is assumed to be a progressive curve upto 500 N/mm
and from there on a straight line. This straight line when extended
to the baseline is assumed tointersect at 10m.With these
assumptions the unit force F/b as a function of mesh deflection can
be expressed as:
bs = thickness of a central websR = average thickness of rim
(net value from tooth root to inside of rim).
for
for
BRCCqcos0.8c =
( )0.250.75cc +=
( )[ ]n02a01aB 200.0212hh
1.20.51C
++=
12n1n
x0.00635z
0.25791z
0.155510.04723q ++= 122n
22
1n
1 x00529.0z
x0.24188x0.00193z
x0.11654+ + 0.00182 x22
( )( )nR m5/s
sR e5
/bbln1C +=
( )10KbF
= 500bF
>
=
500F/b10K
bF 500
bF
-
Classification Notes - No.41.2, May 2012Sec.1. Basic Principles
and General Influence Factors Page 22
KA K for determination of Kv.KA K Kv for determination of KH.KA
K Kv KH for determination of KH. = mesh deflection (m)K =
applicable stiffness (c' or c) Use of stiffnesses for KV, KH and
KHFor calculation of Kv and KH the stiffness is calculated as
follows:When F/b < 500, the stiffness is determined as
where the increment is chosen as e.g. F/b = 10 and thus
When F / b > 500, the stiffness is c' or c.For calculation of
KH the mesh deflection is used directly,
or an equivalent stiffness determined as .
Bevel gearsIn lack of more detailed relationship between
stiffness and geometry the following may be used.
beff not to be used in excess of 0.85 b in these formulae.Bevel
gears with heightwise and lengthwise crowning have progressive mesh
stiffness. The values mentionedabove are only valid for high loads.
They should not be used for determination of Ceff (see 4.3.2) or KH
(see1.9.3.1).
1.12 Running-in AllowancesThe running-in allowances account for
the influence of running-in wear on the various error elements.y
respectively y are the running-in amounts which reduce the
influence of pitch and profile errors,respectively influence of
localised faceload.Cay is defined as the running-in amount that
compensates for lack of tip relief.The following relations may be
used:For not surface hardened steel
with etc. (N/mm), i.e. unit load incorporating the relevant
factors as:At KKbF
bF
=
F/b------------
50010F/b10
K10F/b +++=
Fb ----------
b0.85b
13c eff=b0.85
b16c eff =
ptHlim
f160y =
xlimH
f320y
=DET NORSKE VERITAS AS
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Classification Notes - No.41.2, May 2012Sec.2. Calculation of
Surface Durability Page 23
with the following upper limits:
For surface hardened steely = 0.075 fpt but not more than 3 for
any speedy = 0.15 Fx but not more than 6 for any speedFor all kinds
of steel
When pinion and wheel material differ, the following
applies:
2. Calculation of Surface Durability2.1 Scope and General
RemarksPart 2 includes the calculations of flank surface durability
as limited by pitting, spalling, case crushing andsubsurface
yielding. Endurance and time limited flank surface fatigue is
calculated by means of 2.2 to 2.12. Ina way also tooth fractures
starting from the flank due to subsurface fatigue is included
through the criteria in2.13.Pitting itself is not considered as a
critical damage for slow speed gears. However, pits can create a
severe notcheffect that may result in tooth breakage. This is
particularly important for surface hardened teeth, but also forhigh
strength through hardened teeth. For high-speed gears, pitting is
not permitted.Spalling and case crushing are considered similar to
pitting, but may have a more severe effect on toothbreakage due to
the larger material breakouts, initiated below the surface.
Subsurface fatigue is considered in2.13.For jacking gears
(self-elevating offshore units) or similar slow speed gears
designed for very limited life, themax. static (or very slow
running) surface load for surface hardened flanks is limited by the
subsurface yieldstrength.For case hardened gears operating with
relatively thin lubrication oil films, grey staining (micropitting)
may bethe limiting criterion for the gear rating. Specific
calculation methods for this purpose are not given here, butare
under consideration for future revisions. Thus depending on
experience with similar gear designs,limitations on surface
durability rating other than those according to 2.2 to 2.13 may be
applied.
2.2 Basic EquationsCalculation of surface durability (pitting)
for spur gears is based on the contact stress at the inner point of
singlepair contact or the contact at the pitch point, whichever is
greater.Calculation of surface durability for helical gears is
based on the contact stress at the pitch point.
V 5 m/s 5-10 m/s > 10 m/sy max none
y max none
Use the larger of fpt1 - y1 and fpt2 - y2 to replace fpt - y in
the calculation of KH see 1.10 and Kv see 1.8.
Use in the calculation of KH see 1.9.
Use in the calculation of Kv see 1.8.
Use in the scuffing calculation see 4 if no design tip relief is
foreseen.
limH
12800 limH
6400
limH
25600 limH
12800
5.145.189718
1C2
limHay +
=
( )21 yy21y +=
( )2ay1aya CC21C +=
( )2ay1ay2a1a CC21CC +==DET NORSKE VERITAS AS
For helical gears with 0 < < 1, a linear interpolation
between the above mentioned applies.
-
Classification Notes - No.41.2, May 2012Sec.2. Calculation of
Surface Durability Page 24
Calculation of surface durability for spiral bevel gears is
based on the contact stress at the midpoint of the zoneof
contact.Alternatively for bevel gears the contact stress may be
calculated with the program BECAL. In that case, KAand Kv are to be
included in the applied tooth force, but not KH and KH. The
calculated (real) Hertzianstresses are to be multiplied with ZK in
order to be comparable with the permissible contact stresses.The
contact stresses calculated with the method in part 2 are based on
the Hertzian theory, but do not alwaysrepresent the real Hertzian
stresses.The corresponding permissible contact stresses HP are to
be calculated for both pinion and wheel.
2.2.1 Contact stress
Cylindrical gears
where:
Ft, KA , K , Kv , KH , KH , see 1.5 to 1.10.d1, b, u, see 1.2 to
1.5.
Bevel gears
where:1.05 is a correlation factor to reach real Hertzian
stresses (when ZK = 1)ZE, KA etc. see above.
ZM = mid-zone factor, see 2.3.3.ZK = bevel gear factor, see
2.7.
Fmt, dv1, uv, see 1.2 1.5.It is assumed that the heightwise
crowning is chosen so as to result in the maximum contact stresses
at or nearthe midpoint of the flanks.
2.2.2 Permissible contact stress
where:
ZB,D = Zone factor for inner point of single pair contact for
pinion resp. wheel (see 2.3.2).ZH = Zone factor for pitch point
(see 2.3.1).ZE = Elasticity factor (see 2.4).Z = Contact ratio
factor (see 2.5).Z = Helix angle factor (see 2.6).
H lim = Endurance limit for contact stresses (see 2.8).ZN = Life
factor for contact stresses (see 2.9).SH = Required safety factor
according to the rules.ZL,Zv,ZR = Oil film influence factors (see
2.10).ZW = Work hardening factor (see 2.11).ZX = Size factor (see
2.12).
( )HHvA
1
tEHDB,H KKKKKbud1uF
ZZZZZ +=
( )HHvA
v1v
vmtKEMH KKKKKbud
1uFZZZ1.05 +=
XWRvLH
NHlimHP ZZZZZS
Z =DET NORSKE VERITAS AS
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Classification Notes - No.41.2, May 2012Sec.2. Calculation of
Surface Durability Page 25
2.3 Zone Factors ZH, ZB,D and ZM2.3.1 Zone factor ZHThe zone
factor, ZH, accounts for the influence on contact stresses of the
tooth flank curvature at the pitch pointand converts the tangential
force at the reference cylinder to the normal force at the pitch
cylinder.
2.3.2 Zone factors ZB,D The zone factors, ZB,D, account for the
influence on contact stresses of the tooth flank curvature at the
innerpoint of single pair contact in relation to ZH. Index B refers
to pinion D to wheel.For 1, ZB,D = 1For internal gears, ZD = 1For =
0 (spur gears)
If ZB < 1, use ZB = 1If ZD < 1, use ZD = 1For 0 < <
1ZB,D = ZB,D (for spur gears) (ZB,D (for spur gears) 1)2.3.3 Zone
factor ZMThe mid-zone factor ZM accounts for the influence of the
contact stress at the mid point of the flank and appliesto spiral
bevel gears.
This factor is the product of ZH and ZM-B in ISO 10300 with the
condition that the heightwise crowning issufficient to move the
peak load towards the midpoint.
2.3.4 Inner contact pointFor cylindrical or bevel gears with
very low number of teeth the inner contact point (A) may be close
to thebase circle. In order to avoid a wear edge near A, it is
required to have suitable tip relief on the wheel.
2.4 Elasticity Factor, ZEThe elasticity factor, ZE, accounts for
the influence of the material properties as modulus of elasticity
andPoissons ratio on the contact stresses.For steel against steel
ZE = 189.8
wtt2
wtbH sincos
coscos2Z =
( )
=
2
2
2b
2a
1
2
1b
1a
wtB
z211
dd
z21
dd
tanZ
( )
=
1
2
1b
1a
2
2
2b
2a
wtD
z211
dd
z21
dd
tanZ
=
btm2
2vb2
2vabtm2
1vb2val
2v1vvtbmM
pddpdd
ddtancos2ZDET NORSKE VERITAS AS
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Classification Notes - No.41.2, May 2012Sec.2. Calculation of
Surface Durability Page 26
2.5 Contact Ratio Factor, ZThe contact ratio factor Z accounts
for the influence of the transverse contact ratio and the overlap
ratio on the contact stresses.
2.6 Helix Angle Factor, ZThe helix angle factor, Z, accounts for
the influence of helix angle (independent of its influence on Z) on
thesurface durability.
2.7 Bevel Gear Factor, ZKThe bevel gear factor accounts for the
difference between the real Hertzian stresses in spiral bevel gears
andthe contact stresses assumed responsible for surface fatigue
(pitting). ZK adjusts the contact stresses in such away that the
same permissible stresses as for cylindrical gears may apply.The
following may be used:ZK = 0.80
2.8 Values of Endurance Limit, Hlim and Static Strength, ,
Hlim is the limit of contact stress that may be sustained for
5107 cycles, without the occurrence of progressive pitting.For most
materials 5107 cycles are considered to be the beginning of the
endurance strength range or lowerknee of the -N curve. (See also
Life Factor ZN). However, for nitrided steels 2106 apply.For this
purpose, pitting is defined by
for not surface hardened gears: pitted area 2% of total active
flank area. for surface hardened gears: pitted area 0.5% of total
active flank area, or 4% of one particular tooth
flank area.
and and are the contact stresses which the given material can
withstand for 105 respectively 103cycles without subsurface
yielding or flank damages as pitting, spalling or case crushing
when adequate casedepth applies.The following listed values for
Hlim, and may only be used for materials subjected to a
qualitycontrol as the one referred to in the rules.Results of
approved fatigue tests may also be used as the basis for
establishing these values.The defined survival probability is
99%.
for 1
for < 1
Hlim
Alloyed case hardened steels (surface hardness 58-63 HRC):- of
specially approved high grade:- of normal grade:
16501500
25002400
31003100
Nitrided steel of approved grade, gas nitrided (surface hardness
700 to 800 HV): 1250 1.3 Hlim 1.3 HlimAlloyed quenched and tempered
steel, bath or gas nitrided(surface hardness 500 to 700 HV): 1000
1.3 Hlim 1.3 HlimAlloyed, flame or induction hardened steel
(surface hardness 500 to 650 HV): 0.75 HV + 750 1.6 Hlim 4.5
HVAlloyed quenched and tempered steel: 1.4 HV + 350 1.6 Hlim 4.5
HVCarbon steel: 1.5 HV + 250 1.6 Hlim 1.6 Hlim
1Z =
( )
1
34
Z +
=
Z1
cos----------------=
H105 H103
H105 H103
H105 H103
H105H103DET NORSKE VERITAS AS
These values refer to forged or hot rolled steel. For cast steel
the values for Hlim are to be reduced by 15%.
-
Classification Notes - No.41.2, May 2012Sec.2. Calculation of
Surface Durability Page 27
2.9 Life Factor, ZNThe life factor, ZN, takes account of a
higher permissible contact stress if only limited life (number of
cycles,NL) is demanded or lower permissible contact stress if very
high number of cycles apply.If this is not documented by approved
fatigue tests, the following method may be used:For all steels
except nitrided:
I.e. ZN = 0.92 for 1010 cycles. The ZN = 1 from 5107 on, may
only be used when the material cleanliness is of approved high
grade (see RulesPt.4 Ch.2) and the lubrication is optimised by a
specially approved filtering process.
(but not less than ZN105)
For nitrided steels:
I.e. ZN = 0.92 for 1010 cycles. The ZN = 1 from 2106 on, may
only be used when the material cleanliness is of approved high
grade (see RulesPt4 Ch2) and the lubrication is optimised by a
specially approved filtering process.
Note that when no index indicating number of cycles is used, the
factors are valid for 5107 (respectively 2106for nitriding)
cycles.
2.10 Influence Factors on Lubrication Film, ZL, ZV and ZRThe
lubricant factor, ZL, accounts for the influence of the type of
lubricant and its viscosity, the speed factor,ZV, accounts for the
influence of the pitch line velocity and the roughness factor, ZR,
accounts for influence ofthe surface roughness on the surface
endurance capacity.
NL 5107: ZN = 1 or
105 < NL < 5107:
NL = 105:
103 < NL < 105:
NL 103
NL 2 106: ZN = 1 or
105 < NL < 2106
NL 105
0157.0
L
7
N N105Z
=
510NlogZ0.37
L
7
N N105Z
=
WXRVLHlim
WstX10H1010NN ZZZZZ
ZZZZ
55
5=
)/Z(Zlog0.5
L
5
10NN
5N103N10
5N10ZZ
==
WXRVLlimH
Wst10X10H10NN ZZZZZ
ZZZZ
33
3
==
0098,0
L
6
N N102Z
=
510NZlog7686.0
L
6
N N102Z
=
XWRVL
10XWst10NN ZZZZZ
ZZ1.3ZZ
55 ==DET NORSKE VERITAS AS
-
Classification Notes - No.41.2, May 2012Sec.2. Calculation of
Surface Durability Page 28
The following methods may be applied in connection with the
endurance limit:
where:
For NL 105: ZL ZV ZR = 1.0
2.11 Work Hardening Factor, ZWThe work hardening factor, ZW,
accounts for the increase of surface durability of a soft steel
gear when meshingthe soft steel gear with a surface hardened or
substantially harder gear with a smooth surface.The following
approximation may be used for the endurance limit:Surface hardened
steel against not surface hardened steel:
where:HB = the Brinell hardness of the soft memberFor HB >
470, use HB = 470For HB < 130, use HB = 130RZeq = equivalent
roughness
If RZeq > 16, then use RZeq = 16If RZeq < 1.5, then use
RZeq = 1.5where:
If values of ZW < 1 are evaluated, ZW = 1 should be used for
flank endurance. However, the low value for ZWmay indicate a
potential wear problem.
Surface hardened steels Not surface hardened steels
ZL
ZV
ZR
40 = Kinematic oil viscosity at 40C (mm2/s).For case hardened
steels the influence of a high bulk temperature (see 4. Scuffing)
should be considered. E.g. bulk temperatures in excess of 120C for
long periods may cause reduced flank surface endurance limits.For
values of 40 > 500, use 40 = 500.
RZrel = The mean roughness between pinion and wheel (after
running in) relative to an equivalent radius of curvature at the
pitch point c = 10mm.
RZrel =
RZ = Mean peak to valley roughness (m) (DIN definition) (roughly
RZ = 6 Ra)
RZH = surface roughness of the hard member before run in. RZS =
surface roughness of the soft member before run in40 = see
2.10.
( )240/1342.136.091.0
++ ( )240/1342.1
68.083.0+
+
( )v/328.014.093.0
++ ( )v/328.0
30.085.0+
+
08.0
ZrelR3
15.0
ZrelR3
( ) 31
cZ2Z1
10RR5.0
+
15.0
ZeqW R
31700
130HB2.1Z
=
ZeqDET NORSKE VERITAS AS
-
Classification Notes - No.41.2, May 2012Sec.2. Calculation of
Surface Durability Page 29
Through hardened pinion against softer wheel:
For u > 20, use u = 20For static strength (< 105
cycles):Surface hardened against not surface hardenedZWst =
1.05Through hardened pinion against softer wheelZWst = 1
2.12 Size Factor, ZXThe size factor accounts for statistics
indicating that the stress levels at which fatigue damage occurs
decreasewith an increase of component size, as a consequence of the
influence on subsurface defects combined withsmall stress
gradients, and of the influence of size on material quality.ZX may
be taken unity provided that subsurface fatigue for surface
hardened pinions and wheels is considered,e.g. as in the following
subsection 2.13.
2.13 Subsurface FatigueThis is only applicable to surface
hardened pinions and wheels. The main objective is to have a
subsurfacesafety against fatigue (endurance limit) or deformation
(static strength) which is at least as high as the safetySH
required for the surface. The following method may be used as an
approximation unless otherwisedocumented.The high cycle fatigue
(>3106 cycles) is assumed to mainly depend on the orthogonal
shear stresses. Staticstrength ( 400, the t400 is to be replaced by
a fictive t400 = 1.6 t550).
In addition the specified surface hardness is not to be less
than the max necessary hardness (at tz = 0.5aH). This
For use ZW = 1
For use
( )
+= 0.00829HBHB0.008981u1Z
2
1W
2.1HBHB
2
1
7.1HBHB
2
1 > 1.7HBHB
2
1=
+
=Z
1
HHR
KHHR Z
1 =DET NORSKE VERITAS AS
applies to all hardening methods.
-
Classification Notes - No.41.2, May 2012Sec.3. Calculation of
Tooth Strength Page 30
For high cycle fatigue (>3 106 cycles) the following
applies:
Where aH is half the hertzian contact width multiplied by an
empirical factor of 1.2 that takes into account thepossible
influence of reduced compressive residual stresses (or even tensile
residual stresses) on the localfatigue strength. If any of the
specified hardness depths including the surface hardness is below
the curve described by HV = f(tz), the actual safety factor against
subsurface fatigue is determined as follows:
reduce SH stepwise in the formula for HV and aH until all
specified hardness depths and surface hardnessbalance with the
corrected curve. The safety factor obtained through this method is
the safety against sub-surface fatigue.
For static strength (
-
Classification Notes - No.41.2, May 2012Sec.3. Calculation of
Tooth Strength Page 31
For bevel gears the calculation is based on force application at
the tooth tip of the virtual cylindrical gear.Subsequently the
stress is converted to load application at the mid point of the
flank due to the heightwisecrowning.Bevel gears may also be
calculated with the program BECAL. In that case, KA and Kv are to
be included in theapplied tooth force, but not KF and KF.In case of
a thin annulus or a thin gear rim etc, radial cracking can occur
rather than tangential cracking (fromroot fillet to root fillet).
Cracking can also start from the compression fillet rather than the
tension fillet. Forrim thickness sR < 3.5mn a special
calculation procedure is given in 3.15 and 3.16, and a simplified
procedurein 3.14.A tooth breakage is often the end of the life of a
gear transmission. Therefore, a high safety SF against breakageis
required.It should be noted that this part 3 does not cover
fractures caused by:
oil holes in the tooth root space wear steps on the flank flank
surface distress such as pits, spalls or grey staining.
Especially the latter is known to cause oblique fractures
starting from the active flank, predominately in spiralbevel gears,
but also sometimes in cylindrical gears.Specific calculation
methods for these purposes are not given here, but are under
consideration for futurerevisions. Thus, depending on experience
with similar gear designs, limitations other than those outlined in
part3 may be applied.
3.2 Tooth Root StressesThe local tooth root stress is defined as
the max. principal stress in the tooth root caused by application
of thetooth force. I.e. the stress ratio R = 0. Other stress ratios
such as for e.g. idler gears (R -1.2), shrunk on gearrims (R >
0), etc. are considered by correcting the permissible stress
level.
3.2.1 Local tooth root stress The local tooth root stress for
pinion and wheel may be assessed by strain gauge measurements or
FEcalculations or similar. For both measurements and calculations
all details are to be agreed in advance.Normally, the stresses for
pinion and wheel are calculated as:Cylindrical gears:
where:
YF = Tooth form factor (see 3.3).YS = Stress correction factor
(see 3.4).Y = Helix angle factor (see 3.6).Ft, KA, K, Kv, KF, KF,
see 1.5 1.10.b, see 1.3.Bevel gears:
where:
YFa = Tooth form factor, see 3.3.YSa = Stress correction factor,
see 3.4.Y = Contact ratio factor, see 3.5.
Fmt, KA, etc., see 1.5 to 1.10.
FFvASFn
tF K K K K K Y Y Ym b
F =
FFvASaFamn
mtF K K K K K Y Y Ym b
F =DET NORSKE VERITAS AS
b, see 1.3.
-
Classification Notes - No.41.2, May 2012Sec.3. Calculation of
Tooth Strength Page 32
3.2.2 Permissible tooth root stressThe permissible local tooth
root stress for pinion respectively wheel for a given number of
cycles, N, is:
Note that all these factors YM etc. are applicable to 3106
cycles when used in this formula for FP. Theinfluence of other
number of cycles on these factors is covered by the calculation of
YN.where:
3.3 Tooth Form Factors YF, YFaThe tooth form factors YF and YFa
take into account the influence of the tooth form on the nominal
bendingstress.YF applies to load application at the outer point of
single tooth pair contact of the virtual spur gear pair and isused
for cylindrical gears.YFa applies to load application at the tooth
tip and is used for bevel gears.Both YF and YFa are based on the
distance between the contact points of the 30-tangents at the root
fillet of thetooth profile for external gears, respectively 60
tangents for internal gears.
Figure 3.1 External tooth in normal section
FE = Local tooth root bending endurance limit of reference test
gear (see 3.7).YM = Mean stress influence factor which accounts for
other loads than constant load direction, e.g. idler gears,
tem-
porary change of load direction, pre-stress due to shrinkage,
etc. (see 3.8).YN = Life factor for tooth root stresses related to
reference test gear dimensions (see 3.9).SF = Required safety
factor according to the rules.YrelT = Relative notch sensitivity
factor of the gear to be determined, related to the reference test
gear (see 3.10).YRrelT = Relative (root fillet) surface condition
factor of the gear to be determined, related to the reference test
gear
(see 3.11).YX = Size factor (see 3.12).YC = Case depth factor
(see 3.13).
CXRrelTrelTF
NMFEFP YYYYS
YY
=DET NORSKE VERITAS AS
Figure 3.2 Internal tooth in normal section
-
Classification Notes - No.41.2, May 2012Sec.3. Calculation of
Tooth Strength Page 33
Definitions:
In the case of helical gears, YF and YFa are determined in the
normal section, i.e. for a virtual number of teeth.YFa differs from
YF by the bending moment arm hFa and Fan and can be determined by
the same procedureas YF with exception of hFe and Fan . For hFa and
Fan all indices e will change to a (tip).The following formulae
apply to cylindrical gears, but may also be used for bevel gears
when replacing:mn with mnmzn with zvnt with vt with m
Fig. 3.3 Dimensions and basic rack profile of the teeth
(finished profile)Tool and basic rack data such as hfP, fp and spr
etc. are referred to mn, i.e. dimensionless.3.3.1 Determination of
parameters
where
z0 = number of teeth of pinion cutterx0 = addendum modification
coefficient of pinion cutterhfP= addendum of pinion cutterfP= tip
radius of pinion cutter.
with undercut without undercut
For external gears
For internal gears
n
2
n
Fn
enFn
Fe
F
cosms
cosmh6
Y
=
n
2
n
Fn
anFn
Fa
Fa
cosms
cosmh6
Y
=
( )n
n
prnfPnfP m cos
ssin 1'tan h4E
=
fPfP ' =( )
0z
1.95fPfP0
fPfP 1.0363.156hx'
++=
xh'G fpfp +=
E2H
=DET NORSKE VERITAS AS
m2z nn
-
Classification Notes - No.41.2, May 2012Sec.3. Calculation of
Tooth Strength Page 34
with
(to be solved iteratively, suitable start value for external
gears and for internal gears).
a) Tooth root chord sFn:For external gears
For bevel gears with a tooth thickness modification:xsm affects
mainly sFn, but also hFe and Fen. The total influence of xsm on YFa
Ysa can be approximatedby only adding 2 xsm to sFn / mn.For
internal gears
b) Root fillet radius F at 30 tangent:
c) Determination of bending moment arm hF:dn = zn mn
dan = dn + 2 hapbn = mn cos n
dbn = dn cos n
Fen = en eFor external gears
For internal gears
for external gears
for internal gears
3 =
6 =
Htan zG2
n=
6---=
3---
+
= '
cosG3
3sinz
ms
fpnn
Fn
+
= '
cosG
6sinz
ms
fPnn
Fn
( )G2coszcos G2'm 2n2
fpn
F
+=
b2n cos
=
( )4
d1pzz
2dd
zz2d
2bn
2
nbn2bn
2an
en +
=
en
bnen d
dcos arc =
ennnn
e inv inv x tan 22
z1
+
+=
( )
=
n
enFenee
n
Fe
mdtan sin cos
21
mh ]'
cosG
3cosz fpn +
( ) = enFeneeFe d tan sin cos1h 'G3cosz fPnDET NORSKE VERITAS
AS
nn m2m cos6
-
Classification Notes - No.41.2, May 2012Sec.3. Calculation of
Tooth Strength Page 35
3.3.2 Gearing with n > 2For deep tooth form gearing produced
with a verified grade of accuracy of 4 or better, and with applied
profile
modification to obtain a trapezoidal load distribution along the
path of contact, the YF may be corrected by thefactor YDT as:
3.4 Stress Correction Factors YS, YSaThe stress correction
factors YS and YSa take into account the conversion of the nominal
bending stress to thelocal tooth root stress. Thereby YS and YSa
cover the stress increasing effect of the notch (fillet) and the
factthat not only bending stresses arise at the root. A part of the
local stress is independent of the bending momentarm. This part
increases the more the decisive point of load application
approaches the critical tooth rootsection.Therefore, in addition to
its dependence on the notch radius, the stress correction is also
dependent on theposition of the load application, i.e. the size of
the bending moment arm.YS applies to the load application at the
outer point of single tooth pair contact, YSa to the load
application attooth tip.YS can be determined as follows:
YSa can be calculated by replacing hFe with hFa in the above
formulae.Note:
a) Range of validity 1 < qs < 8In case of sharper root
radii (i.e. produced with tools having too sharp tip radii), YS
resp. YSa must bespecially considered.
b) b)In case of grinding notches (due to insufficient
protuberance of the hob), YS resp. YSa can riseconsiderably, and
must be multiplied with:
where:
tg = depth of the grinding notchg = radius of the grinding
notch
c) The formulae for YS resp. YSa are only valid for n = 20.
However, the same formulae can be used as asafe approximation for
other pressure angles.
3.5 Contact Ratio Factor YThe contact ratio factor Y covers the
conversion from load application at the tooth tip to the load
applicationat the mid point of the flank (heightwise) for bevel
gears.The following may be used:
( )2.52 n
2.502.05for 0.6662.366Y nnDT =2.05for 1.0Y nDT
-
Classification Notes - No.41.2, May 2012Sec.3. Calculation of
Tooth Strength Page 36
3.6 Helix Angle Factor YThe helix angle factor Y takes into
account the difference between the helical gear and the virtual
spur gearin the normal section on which the calculation is based in
the first step. In this way it is accounted for that theconditions
for tooth root stresses are more favourable because the lines of
contact are sloping over the flank.The following may be used (
input in degrees):Y = 1 /120When > 1, use = 1 and when > 30 ,
use = 30 in the formula.However, the above equation for Y may only
be used for gears with > 25 if adequate tip relief is applied
toboth pinion and wheel (adequate = at least 0.5 Ceff, see
4.3.2).
3.7 Values of Endurance Limit, FEFE is the local tooth root
stress (max. principal) which the material can endure permanently
with 99% survivalprobability. 3106 load cycles is regarded as the
beginning of the endurance limit or the lower knee of the N curve.
FE is defined as the unidirectional pulsating stress with a minimum
stress of zero (disregardingresidual stresses due to heat
treatment). Other stress conditions such as alternating or
pre-stressed etc. arecovered by the conversion factor YM.FE can be
found by pulsating tests or gear running tests for any material in
any condition. If the approval ofthe gear is to be based on the
results of such tests, all details on the testing conditions have
to be approved bythe Society. Further, the tests may have to be
made under the Society's supervision.If no fatigue tests are
available, the following listed values for FE may be used for
materials subjected to aquality control as the one referred to in
the rules.
FEAlloyed case hardened steels 1) (fillet surface hardness 58 to
63 HRC): of specially approved high grade: of normal grade:
- CrNiMo steels with approved process:- CrNi and CrNiMo steels
generally:- MnCr steels generally:
1050
1000920850
Nitriding steel of approved grade, quenched, tempered and gas
nitrided (surface hardness 700 800 HV): 840Alloyed quenched and
tempered steel, bath or gas nitrided (surface hardness 500 to 700
HV):
720Alloyed quenched and tempered steel, flame or induction
hardened 2) (incl. entire root fillet) (fillet surface hardness 500
to 650 HV):
0.7 HV + 300Alloyed quenched and tempered steel, flame or
induction hardened (excl. entire root fillet) (B = u.t.s. of base
material):
0.25 B + 125Alloyed quenched and tempered steel: 0.4 B +
200Carbon steel: 0.25 B + 250Note:All numbers given above are valid
for separate forgings and for blanks cut from bars forged according
to a qualified pro-cedure, see Pt. 4 Ch. 2 Sec. 3. For rolled
steel, the values are to be reduced with 10%. For blanks cut from
forged bars, that are not qualified as mentioned above, the values
are to be reduced with 20%, For cast steel, reduce with 40%.
1) These values are valid for a root radius
being unground. If, however, any grinding is made in the root
fillet area in such a way that the residual stresses may be
affected, FE is to be reduced by 20%. (If the grinding also leaves
a notch, see 3.4).
with fillet surface hardness 58 to 63 HRC. In case of lower
surface hardness than 58 HRC, FE is to be reduced with 20(58 HRC)
where HRC is the detected hardness. (This may lead to a permissible
tooth root stress that varies along the facewidth. If so, the
actual tooth root stresses may also be considered along
facewidth.)
not being shot peened. In case of approved shot peening, FE may
be increased by 200 for gears where FE is reduced by 20% due to
root grinding. Otherwise FE may be increased by 100 for mn 6 and
100 5 (mn - 6) for mn > 6.However, the possible adverse
influence on the flanks regarding grey staining should be
considered, and if necessary the flanks should be masked.DET NORSKE
VERITAS AS
2) The fillet is not to be ground after surface hardening.
Regarding possible root grinding, see 1).
-
Classification Notes - No.41.2, May 2012Sec.3. Calculation of
Tooth Strength Page 37
3.8 Mean stress influence Factor, YMThe mean stress influence
factor, YM, takes into account the influence of other working
stress conditions thanpure pulsations (R = 0), such as e.g. load
reversals, idler gears, planets and shrink-fitted gears.YM (YMst)
are defined as the ratio between the endurance (or static) strength
with a stress ratio R 0, and theendurance (or static) strength with
R = 0.YM and YMst apply only to a calculation method that assesses
the positive (tensile) stresses and is thereforesuitable for
comparison between the calculated (positive) working stress F and
the permissible stress FPcalculated with YM or YMst.For thin rings
(annulus) in epicyclic gears where the compression fillet may be
decisive, specialconsiderations apply, see 3.16.The following
method may be used within a stress ratio1.2 < R < 0.5:
3.8.1 For idlers, planets and PTO with ice class
where:R = stress ratio = min. stress divided by max. stress.For
designs with the same force applied on both forward- and
back-flank, R may be assumed to 1.2.For designs with considerably
different forces on forward- and back-flank, such as e.g. a marine
propulsionwheel with a power take off pinion, R may be assessed
as:
For a power take off (PTO) with ice class, see 1.6.1 c.M
considers the mean stress influence on the endurance (or static)
strength amplitudes.M is defined as the reduction of the endurance
strength amplitude for a certain increase of the mean stressdivided
by that increase of the mean stress.Following M values may be
used:
1) For bevel gears, use Ys = 2 for determination of M.The listed
M values for the endurance limit are independent of the fillet
shape (Ys), except for case hardening.In principle there is a
dependency, but wide variations usually only occur for case
hardening, e.g. smoothsemicircular fillets versus grinding
notches.
3.8.2 For gears with periodical change of rotational
directionFor case hardened gears with full load applied
periodically in both directions, such as side thrusters, the
sameformula for YM as for idlers (with R = 1.2) may be used
together with the M values for endurance limit. Thissimplified
approach is valid when the number of changes of direction exceeds
100 and the total number of loadcycles exceeds 3106.For gears of
other materials, YM will normally be higher than for a pure idler,
provided the number of changesof direction is below 3106. A linear
interpolation in a diagram with logarithmic number of changes of
directionmay be used, i.e. from YM = 0.9 with one change to YM
(idler) for 3106 changes. This is applicable to YM forendurance
limit. For static strength, use YM as for idlers.For gears with
occasional full load in reversed direction, such as the main wheel
in a reversing gear box, YM =
Endurance limit Static strengthCase hardened 0.8 0.15 Ys 1)
0.7If shot peened 0.4 0.6Nitrided 0.3 0.3Induction or flame
hardened 0.4 0.6Not surface hardened steel 0.3 0.5Cast steels 0.4
0.6
M1M1R1
1Yor Y MstM
+
=
branchmain theoffacewidth unit per forcep.t.o. offacewidt unit
per force2.1DET NORSKE VERITAS AS
0.9 may be used.
-
Classification Notes - No.41.2, May 2012Sec.3. Calculation of
Tooth Strength Page 38
3.8.3 For gears with shrinkage stresses and unidirectional
loadFor endurance strength:
FE is the endurance limit for R = 0.For static strength, YMst =
1 and fit accounted for in 3.9.b.fit is the shrinkage stress in the
fillet (30 tangent) and may be found by multiplying the nominal
tangential(hoop) stress with a stress concentration factor:
3.8.4 For shrink-fitted idlers and planetsWhen combined
conditions apply, such as idlers with shrinkage stresses, the
design factor for endurancestrength is:
Symbols as above, but note that the stress ratio R in this
particular connection should disregard the influenceof fit, i.e. R
normally equal 1.2.For static strength:
The effect of fit is accounted for in 3.9 b.
3.8.5 Additional requirements for peak loadsThe total stress
range (max min) in a tooth root fillet is not to exceed:
3.9 Life Factor, YNThe life factor, YN, takes into account that,
in the case of limited life (number of cycles), a higher tooth
rootstress can be permitted and that lower stresses may apply for
very high number of cycles.Decisive for the strength at limited
life is the N curve of the respective material for given
hardening,module, fillet radius, roughness in the tooth root, etc.
I.e. the factors YrelT, YRelT, YX and YM have aninfluence on YN.If
no N curve for the actual material and hardening etc. is available,
the following method may be used.Determination of the N curve:
a) Calculate the permissible stress FP for the beginning of the
endurance limit (3106 cycles), including theinfluence of all
relevant factors as SF, YrelT, YRelT, YX, YM and YC, i.e.FP = FE YM
YrelT YRelT YX YC / SF
b) Calculate the permissible static stress (103 load cycles)
including the influence of all relevant factors as SFst, YrelTst,
YMst and YCst:
for not surface hardened fillets
for surface hardened fillets
FE
fitM
M1
M21Y+
=
n
F.fit m
21.5scf =
( ) ( ) FEfit
M
R1M1M2
M1M1R1
1Y+
+
=
M1M1R1
1YMst
+
=
F
y
S2.25
FSHV5
( )fitCstYrelTstYMstYFstFstS1
FPst =DET NORSKE VERITAS AS
-
Classification Notes - No.41.2, May 2012Sec.3. Calculation of
Tooth Strength Page 39
where Fst is the local tooth root stress which the material can
resist without cracking (surface hardenedmaterials) or unacceptable
deformation (not surface hardened materials) with 99% survival
probability.
c) Calculate YN as:
Guidance on number of load cycles NL for various
applications:
For propulsion purpose, normally NL = 1010 at full load (yachts
etc. may have lower values). For auxiliary gears driving generators
that normally operate with 70 to 90% of rated power, NL = 108
with
rated power may be applied.
3.10 Relative Notch Sensitivity Factor, YrelTThe dynamic
(respectively static) relative notch sensitivity factor, YrelT
(YrelTst) indicate to which extent thetheoretically concentrated
stress lies above the endurance limits (respectively static
strengths) in the case offatigue (respectively overload)
breakage.YrelT is a function of the material and the relative
stress gradient. It differs for static strength and
endurancelimit.The following method may be used:For endurance
limit:for not surface hardened fillets:
for all surface hardened fillets except nitrided:
FstAlloyed case hardened steel 1) 2300Nitriding steel, quenched,
tempered and gas nitrided (surface hardness 700 to 800 HV)
1250Alloyed quenched and tempered steel, bath or gas nitrided
(surface hardness 500 to 700 HV) 1050Alloyed quenched and tempered
steel, flame or induction hardened (fillet surface hardness 500 650
HV) 1.8 HV + 800
Steel with not surface hardened fillets, the smaller value of 2)
1.8 B