DNV Classification Note 41.2: Calculation of Gear Rating for Marine

CLASSIFICATION NOTES

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The electronic

No. 41.2

Calculation of Gear Rating for Marine Transmissions

MAY 2012

DET NORSKE VERITAS AS

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FOREWORD

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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 somecomponents are given as interpretations of the more general rule requirements.

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Classification Notes - No.41.2, May 2012

Changes – 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.



Contents – 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, Kγ .........................................................................................................................111.8 Dynamic Factor, Kv ................................................................................................................................111.9 Face Load Factors, KHβ and KFβ ...........................................................................................................151.10 Transversal Load Distribution Factors, KHα and KFα ...........................................................................201.11 Tooth Stiffness Constants, c´ and cγ .......................................................................................................201.12 Running-in Allowances ..........................................................................................................................22

2. 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, Zε .........................................................................................................................262.6 Helix Angle Factor, Zβ ...........................................................................................................................262.7 Bevel Gear Factor, ZK ............................................................................................................................262.8 Values of Endurance Limit, σHlim 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..................................................................................................................................29

3. 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 Yε..........................................................................................................................353.6 Helix Angle Factor Yβ............................................................................................................................363.7 Values of Endurance Limit, σFE .............................................................................................................363.8 Mean stress influence Factor, YM...........................................................................................................373.9 Life Factor, YN .......................................................................................................................................383.10 Relative Notch Sensitivity Factor, YδrelT ...............................................................................................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..........................................................................................................44

4. Calculation of Scuffing Load Capacity .............................................................................................. 454.1 Introduction.............................................................................................................................................454.2 General Criteria.......................................................................................................................................464.3 Influence Factors.....................................................................................................................................474.4 The Flash Temperature ϑfla ....................................................................................................................49

Appendix A.Fatigue Damage Accumulation ..................................................................................................................... 57

Appendix B.Application Factors for Diesel Driven Gears............................................................................................... 59

Appendix C.Calculation of Pinion-Rack ........................................................................................................................... 61



Sec.1. Basic Principles and General Influence Factors – Page 5

1. Basic Principles and General Influence Factors

1.1 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 < εα < 2. If εα > 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 gears

Bevel gears

The 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 theσ–N 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 σ–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).ha = addendum (mm).




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.

Index m refers to the midsection of the bevel 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 (μm).Ry = peak to valley roughness (μm).Rz = mean peak to valley roughness (μm).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.αn = normal pressure angle at ref. cylinder.αt = transverse pressure angle at ref. cylinder.αa = transverse pressure angle at tip cylinder.αwt = transverse pressure angle at pitch cylinder.β = helix angle at ref. cylinder.βb = helix angle at base cylinder.βa = helix angle at tip cylinder.εα = transverse contact ratio.εβ = overlap ratio.εγ = total contact ratio.ρa0 = tip radius of tool ref. to mn.ρfp = root radius of basic rack ref. to mn ( = ρa0).ρC = effective radius (mm) of curvature at pitch point.ρF = root fillet radius (mm) in the critical section.σB = ultimate tensile strength (N/mm2).σy = yield strength resp. 0.2% proof stress (N/mm2).

Σ = angle between intersection axis.

= angle modification (Klingelnberg)

m0 = tool module (Klingelnberg)

δ = pitch cone angle.

xsm = tooth thickness modification coefficient (midface).

R = pitch cone distance (mm).

Kϑ




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

(for double helix, b is to be taken as the width of one helix).

tan αt = tan αn / cos βtan βb = tan β cos αttan βa = tan β da / dcos αa = db/dad = z mn / cos βmt = mn /cos βdb = d cos αt = dw cos αwta = 0.5 (dw1 + dw2)dw1/dw2 = z1 / z2inv α = tan α - α (radians)inv αwt = inv αt + 2 tan αn (x1 + x2)/(z1 + z2)zn = z / (cos2 βb cos β)

where ξfw1 is to be taken as the smaller of:

—

—

—

and

, where ξfw2 is calculated as ξfw1

substituting the values for the wheel by the values for the pinion and visa versa.

11

2 ≥=z

zu

1

aw1fw1α

T

ξξε

+=

wtfw1 αtanξ =

soi1

b1wtfw1 d

dacostan -tanαξ =

1

2wt

a2

b2fw1

z

ztanα

d

dacostan ξ

−=

2

1fw2aw1

z

zξξ =

11

z

2πT =

( ) +

⋅+−−⋅= −

2sinαρρxhm

2

d2d nfpfp1fpnsoi1

2

1

t

nfpfplfpn2

tanα

)sinαρρx(hm

⋅+−−

nm

sinb

πβ=εβ




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 δ1,2

(δ1 + δ2 = Σ)

Gear ratio:

tan αvt = tan αn/ cos βm

tan βbm = tan βm cos αvt

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 (δa 1,2 – δ1,2)

(when ha is addendum at outer end and δa is the outer cone angle).

εy =

ρC =

v =

pbt =

sat =

san =

βα εε +

( )2b

wt

u1βcos

αsinua

+

311 10dn

60

π −

βcos

αcosmπ tn

−++

at

n

invααinvz

αtanx22

π

d a

aβcossat

1

2

v

vv z

zu =

m

vtnmbtm βcos

αcosmπp =

2,1

2,1m2.1v cos

dd

δ=




Addendum modification coefficients:

Base circle:

dvb 1,2=dv 1,2 cos αvt

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 benecessary.

mn

1,2am2,1am2,1m m2

hhx

−=

btm

vtv2

2vb2

2va2

1vb2

1va

P

αsinadd0.5dd0.5 −−+−=εα

nm

m

mπ

βsinb=εβ

2β

2α εε +=εγ

3m11mt 10dn

60

πv −=

( )2vbm

vtvvvc

u1βcos

αsinua

+=ρ

( )( )( )1εif

ε

ε1ε2ε

βcos

εbl β2

γ

2βα

2γ

bm

αb <

−−−=

1εifβcosε

εbl β

bmγ

αb ≥=

d

T2000Ft =

t

tbt αcos

FF =

mmt d

T2000F =

vt

mtmbt αcos

FF =




For gears with a defined load spectrum the calculation with a KA may be replaced by a fatigue damagecalculation as given in Appendix A.

1.6.1 KA

For gears designed for long or infinite life at nominal rated torque, KA is defined as the ratio between themaximum repetitive cyclic torque applied to the gear set and nominal rated torque.

This definition is suitable for main propulsion gears and most of the auxiliary gears.

KA can be determined by measurements or system analysis, or may be ruled by conventions (ice classes). (Forthe purpose of a preliminary (but not binding) calculation before KA is determined, it is advised to apply eitherthe max. values mentioned below or values known from similar plants.)

a) For main propulsion gears KA can be taken from the (mandatory) torsional vibration analysis, therebyconsidering all permissible driving conditions.*)Unless specially agreed, the rules do not allow KA in excess of 1.35 for diesel propulsion.*) With turbineor electric propulsion KA would normally not exceed 1.2. However, special attention should be given tothrusters that are arranged in such a way that heavy vessel movements and/or manoeuvring can causesevere load fluctuations. This means e.g. thrusters positioned far from the rolling axis of vessels that couldbe susceptible to rolling. If leading to propeller air suction, the conditions may be even worse.The above mentioned movements or manoeuvring will result in increased propeller excitation. If thethruster is driven by a diesel engine, the engine mean torque is limited to 100%. However, thrusters drivenby electric motors can suffer temporary mean torque much above 100% unless a suitable load controlsystem (limiting available e-motor torque) is provided.

b) For main propulsion gears with ice class notation (see Rules Pt.5 Ch.1 Sec.3 J) KA ice has to be taken as thehigher value of the applicable (rule defined) ice shock torque referred to nominal rated torque and the valueunder a).The Baltic ice class notations refer to a few millions ice shock loads. Thus the life factors may be put YN=1 and ZN=1.2 (except for nitrided gears where ZN = 1 applies).Additionally, the calculations with the normal KA (no ice class) are to fulfil the normal requirements.For polar ice class notations, KA ice applies to all criteria and for long or infinite life.

c) For a power take off (PTO) branch from a main propulsion gear with ice class, ice shocks result in negativetorques. It is assumed that the PTO branch is unloaded when the ice shock load occurs. The influence of these reverse shock loads may be taken into account as follows:The negative torque (reversed load), expressed by means of an application factor based on rated forwardload (T or Ft), is KAreverse = KA ice –1 (the minus 1 because no mean torque assumed). KAice to be calculatedas in the ice class rules. This KAreverse should be used for back flank considerations such as pitting and scuffing. The influence on tooth bending strength (forward direction) may be simplified by using the factorYM = 1 - 0.3 · KAreverse /KA.

d) For diesel driven auxiliaries KA can be taken from the torsional vibration analysis, if available. For unitswhere no vibration analysis is required (< 200 kW) or available, it is advised to apply KA as the upperallowable value 1.35.*)

e) For turbine or electro driven auxiliaries the same as for c) applies, however the practical upper value is 1.2.

*) For diesel driven gears, more information on KA for misfiring and normal driving is given in Appendix B.

1.6.2 KAP

The peak overload factor KAP is defined as the ratio between the temporary occasional peak overload torqueand the nominal rated torque.

For plants where high temporary occasional peak torques can occur (i.e. in excess of the above mentioned KA),the gearing (if nitrided) has to be checked with regard to static strength. Unless otherwise specified the samesafety factors as for infinite life apply.

The scuffing safety is to be specially considered, whereby the KA applies in connection with the bulktemperature, and the KAP applies for the flash temperature calculation and should replace KA in the formulaein 4.3.1, 4.3.2 and 4.4.1.KAP can be evaluated from the torsional impact vibration calculation (as required by the rules).

If the overloads have a duration corresponding to several revolutions of the shafts, the scuffing safety has to beconsidered on basis of this overload, both with respect to bulk and flash temperature. This applies to plants withice class notations (Baltic and polar), and to plants with prime movers which have high temporary overloadcapacity such as e.g. electric motors (provided the driven member can have a considerable increase in demandtorque as e.g. azimuth thrusters during manoeuvring).For plants without additional ice class notation, KAP should normally not exceed 1.5.




1.6.3 Frequent overloadsFor plants where high overloads or shock loads occur regularly, the influence of this is to be considered bymeans of cumulative fatigue, (see Appendix A).

1.7 Load Sharing Factor, KγThe load sharing factor Kγ accounts for the maldistribution of load in multiple-path transmissions (dual tandem,epicyclic, double helix etc.). Kγ is defined as the ratio between the max. load through an actual path and theevenly shared load.

1.7.1 General methodKγ mainly depends on accuracy and flexibility of the branches (e.g. quill shaft, planet support, external forcesetc.), and should be considered on basis of measurements or of relevant analysis as e.g.:

For double helical gears:

An external axial force Fext applied from sources outside the actual gearing (e.g. thrust via or from a toothcoupling) will cause a maldistribution of forces between the two helices. Expressed by a load sharing factor the

If the direction of Fext is known, the calculation should be carried out separately for each helix, and with thetangential force corrected with the pertinent Kγ. If the direction of Fext is unknown, both combinations are tobe calculated, and the higher σH or σF to be used.

1.7.2 Simplified methodIf no relevant analysis is available the following may apply:

For epicyclic gears:

where npl = number of planets ( ≥ 3 ).

For multistage gears with locked paths and gear stages separated by quill shafts (see figure below):

Figure 1.0 Locked paths gear

where φ = quill shaft twist (degrees) under full load.

1.8 Dynamic Factor, KvThe dynamic factor Kv accounts for the internally generated dynamic loads.Kv is defined as the ratio between the maximum load that dynamically acts on the tooth flanks and themaximum externally applied load Ft KA Kγ.In the following 2 different methods (1.8.1 and 1.8.2) are described. In case of controversy between themethods, the next following is decisive, i.e. the methods are listed with increasing priority.

δ = total compliance of a branch under full load (assuming even load share) referred to gear mesh.f =

where f1, f2 etc. are the main individual errors that may contribute to a maldistribution between the branches. E.g. tooth pitch errors, planet carrier pitch errors, bearing clearance influences etc. Compensating effects should also be considered.

δ+δ=γ

fK

−−−−+++ 23

22

21 fff

β⋅±=γ tanF

F1K

t

ext

Kγ 1 0.25 npl 3–+=

( )φ+=γ /2.01K




It is important to observe the limitations for the method in 1.8.1. In particular the influence of lateral stiffnessof shafts is often underestimated and resonances occur at considerably lower speed than determined in 1.8.1.1.

However, for low speed gears with v·z1 < 300 calculations may be omitted and the dynamic factor simplifiedto Kv = 1.05.

1.8.1 Single resonance methodFor a single stage gear Kv may be determined on basis of the relative proximity (or resonance ratio) N betweenactual speed n1 and the lowest resonance speed nE1.

Note that for epicyclic gears n is the relative speed, i.e. the speed that multiplied with z gives the meshfrequency.

1.8.1.1 Determination of critical speedIt is not advised to apply this method for multimesh gears for N > 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:

cγ 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 cγ in connection with determination of natural frequencies may need correction. cγis defined as stiffness per unit facewidth, but when used in connection with the total mesh stiffness, it is not assimple as cγ ·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 Kv

Non-dimensional gear accuracy dependent parameters:

1E

1

n

nN =

red

γ

1

3

1E m

c

zπ

1030n

⋅=

21

21red mm

mmm

+=

22,1b

2,12,1

)2/d(b

Im =

( )b/KKF

yf'cB

At

pptp

γ

−=

( )b/KKF

yF'cB

At

ff

γ

α −=




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 2·Ceff (see 4.3.2) the reduction of εα has to beconsidered (see 4.4.3).

Where:

1.8.1.3 Kv in the subcritical range:

Kv = 1 + N K

K = Cv1 Bp + Cv2 Bf + Cv3 Bk

Cv1 accounts for the pitch error influence

Cv1 = 0.32

Cv2 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 Fα = the total profile form deviation (ISO 1328), max. of pinion or wheel (Note: Fα is p.t. not available for bevel

gears, thus use Fα = fpt)yp and yf = the respective running-in allowances and may be calculated similarly to yα in 1.12, i.e. the value of fpt is

replaced by Fα 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 εγ ≤ 2

for εγ > 2

Cv3 = 0.23 for εγ ≤ 2

for εγ > 2

Cylindrical gears: 0.85 < N ≤ 1.15Bevel gears: 0.75 < N ≤ 1.25

Cv4 = 0.90 for εγ ≤ 2

for εγ > 2

/bKKF

c'C1B

γAt

ak ⋅⋅

⋅−=

0.3ε

0.57C

γ2v −

=

1.56ε

0.096C

γ3v −

=

1.44ε

ε0.050.57C

γ

γ4v −

−=




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 = 0.47

Cv6 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 as

Cylindrical 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:

cγ 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 actualpinion-wheel flexible element, need(s) to be considered as critical frequency (ies).

Cylindrical gears: N ≥ 1.5Bevel gears: N ≥ 1.5

Cv6 = 0.47 for εγ ≤ 2

for εγ > 2

Cv7 = 0.75 for εγ ≤ 1.5

for 1.5 < εγ £ 2.5

Cv7 = 1.0 for εγ > 2.5

Cylindrical gears: 1.15 < N < 1.5Bevel gears: 1.25 < N < 1.5

1.74ε

0.12C

γ6v −

=

[ ] 875.0)2(sin 0.125v7C +−επ= β

( ) ( ) ( )[ ]5.1Nv15.1Nv5.1Nvv KK35.0

N5.1KK === −⋅

−+=

( ) ( ) ( )[ ]5.1Nv25.1Nv5.1Nvv KK25.0

N5.1KK === −⋅

−+=




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, KHβ and KFβThe face load factors, KHβ for contact stresses and for scuffing, KFβ for tooth root stresses, account for non-uniform load distribution across the facewidth.

KHβ is defined as the ratio between the maximum load per unit facewidth and the mean load per unit facewidth.

KFβ 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 KHβ and KFβ

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 KFβ(1or2) is to be speciallyconsidered as it may even exceed KHβ.

E.g. in pinion-rack lifting systems for jack up rigs, where b = b2 ≈ mn and b1≈ 3 mn, the typical KHβ ≈ KFβ2 ≈ 1 and KFβ1 ≈ 1.3.

1.9.2 Measurement of face load factorsPrimarily,

KFβ 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 KHβ.

Secondarily,

KHβ 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 KHβ 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 KHβat design load can be found by calculation and extrapolation. See example.

Figure 1.1 Example of experimental determination of KHb

βHF KK =β2(h/b)h/b1

1

++




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 KHβThe 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.

KHβ is to be determined in the plane of contact.

The influence parameters considered in this method are:

— mean mesh stiffness cγ (see 1.11) (if necessary, also variable stiffness over b)— mean unit load Fm/b = Fbt KA Kγ Kv/b (for double helical gears, see 1.7 for use of Kγ)— 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 yβ (see 1.12).

In practice several other parameters such as centrifugal expansion, thermal expansion, housing deflection, etc.contribute to KHβ. 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 KHβ is bymeans of a graphical approach, described in 1.9.3.1.

If cγ can be considered constant over the facewidth, and no helix modifications apply, KHβ 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 δM as a function of Fm /b and cγ, see 1.11. — Draw a base line with length b, and draw up a rectangular with height δM. (The area δM 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 δM 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, butwith one distinct direction. Superimpose these ordinates to the previous load distribution curve.




— 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 δM b.

— If cγ 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 cγ (average over the active facewidth only).Note that the result is to be a curve that covers the same area δM b as before.

— The influence of running in yβ is to be determined as in 1.12 whereby the value for Fβx is to be taken astwice the distance between the peak of the load distribution curve and δM.

— 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 cγ 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,so that the area above and below this zero line are equal, see Fig. 1.2. The max. positive ordinate is ½Δfsh.

M

βH δ

ycurveofpeakK

−=β




— 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 Fβx is not to be taken less than thelargest of each of these elements.

— Calculate the effective mesh misalignment asFβy = Fβx - yβ (yβ see 1.12)

— Determine

or

where cγ as used here is the effective mesh stiffness, see 1.11.

1.9.4 Determination of fsh

fsh 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 KHβ 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 KHβ procedurecan be used iteratively. 2 to 3 iterations will be enough, even for almost triangular load distributions.

1.9.5 Determination of fdefl

fdefl 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 fbe

fbe 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 anangular misalignment in the plane of contact that is superimposed to the working positions determined in 1.9.5.

d + 2 x mn for bending and shear deflectiond + 2 mn (x – ha0 + 0.2) for torsional deflection

deflbemashx ffffF ±±±Δ=β

2KforF2

bFc1K βH

m

βγγH ≤+=β

2KforF

bFc2K Hβ

m

βγγH >=β




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 fma

fma is the mesh misalignment due to manufacturing tolerances (helix slope deviation) of pinion fHβ1, wheelfHβ2 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, KHβ 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 Kγ ¼ 1, this must be taken into account whencalculating the total sun pinion twist (i.e. twist calculated with the force per mesh without Kγ, 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 KHβ for bevel gears

If 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/b–0.6)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 typetest at full load.

........fff 22be

21bebe ++±=

2

2Hβ

2

1Hβma fff +=

testeff

H Kb

b85.185.1K ⋅

−⋅=β




c) Ktest = 1.2if mesh is only checked by toolmaker’s 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, KHα and KFαThe transverse load distribution factors, KHα for contact stresses and for scuffing, KFα 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 εγ ≤ 2

valid for εγ > 2

where:

Limitations of KHα and KFα:

If the calculated values for

KFα = KHα < 1, use KFα = KHα = 1.0

Bevel gears:

For ground or hard metal hobbed gears, KFα = KHα = 1

For lapped gears, KFα = KHα = 1.1

1.11 Tooth Stiffness Constants, c´ and cγThe 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 μm, in the plane of contact.

c´ is the maximum stiffness of a single pair of teeth.

cγ is the mean value of the mesh stiffness in a transverse plane (brief term: mesh stiffness).

Both valid for high unit load. (Unit load = Ft · KA · Kγ/b).

FtH = Ft KA Kγ Kv KHβcγ = See 1.11γα = See 1.12fpt = Maximum single pitch deviation (μm) of pinion or wheel, or maximum total profile form deviation Fα 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 εαn see 3.3.1.c)

( )

−+==

tH

αptγγHαFα F

byfc0.40.9

2

εKK

( ) ( )tH

αptγ

γ

γHαFα F

byfc

ε

1ε20.40.9KK

−−+==

2εα

γH

Zε

εK >α 2

εα

γH

Zε

εK =α

εα

γF Yε

εK >α

εα

γF Yε

εK =α

αnε

0.750.25Y +=ε




Cylindrical gears

The 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 KHβ and KHα. 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 KHβ.

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 δ 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 10μm.

With these assumptions the unit force F/b as a function of mesh deflection δ 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

BRCCq

βcos0.8´c =

( )0.25ε0.75cc αγ +′=

( )[ ]n02a01a

B α200.0212

hh1.20.51C −−

+−+=

12n1n

x0.00635z

0.25791

z

0.155510.04723q −++= 1

2

2n

22

1n

1 x00529.0z

x0.24188x0.00193

z

x0.11654 +−−− + 0.00182 x22

( )( )nR m5/s

sR

e5

/bbln1C +=

( )10δKb

F −= 500b

F >

−=

500

F/b10δK

b

F 500b

F <




KA · Kγ for determination of Kv.

KA · Kγ · Kv for determination of KHβ.

KA · Kγ · Kv · KHβ for determination of KHα.

δ = mesh deflection (μm)

K = applicable stiffness (c' or cγ)

Use of stiffnesses for KV, KHβ and KHαFor calculation of Kv and KHα the stiffness is calculated as follows:

When F/b < 500, the stiffness is determined as

where the increment is chosen as e.g. Δ F/b = 10 and thus

When F / b > 500, the stiffness is c' or cγ.

For calculation of KHβ the mesh deflection δ is used directly,

or an equivalent stiffness determined as .

Bevel gears

In 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 KHβ (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.

yα respectively yβ 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:γAt KK

b

F

b

F ⋅⋅=

ΔF/bΔδ

------------

500

10F/b10

K

10F/bΔδ

+++=

Fb δ⋅----------

b0.85

b13c´ eff=

b0.85

b16c eff

γ =

ptHlim

α fσ

160y =

βxlimH

f320

yσ

=β



Sec.2. Calculation of Surface Durability – Page 23

with the following upper limits:

For surface hardened steel

yα = 0.075 fpt but not more than 3 for any speed

yβ = 0.15 Fβx but not more than 6 for any speed

For all kinds of steel

When pinion and wheel material differ, the following applies:

2. Calculation of Surface Durability

2.1 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.

For helical gears with 0 < εβ < 1, a linear interpolation between the above mentioned applies.

V ≤ 5 m/s 5-10 m/s > 10 m/syα max none

yβ max none

— Use the larger of fpt1 - yα1 and fpt2 - yα2 to replace fpt - yα in the calculation of KHα see 1.10 and Kv see 1.8.

— Use in the calculation of KHβ 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

σ limH

6400

σ

limH

25600

σ limH

12800

σ

5.145.189718

1C

2limH

ay +

−σ

=

( )2β1ββ yy2

1y +=

( )2ay1aya CC2

1C +=

( )2ay1ay2a1a CC2

1CC +==




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 KHβ and KHα. 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 σHP are to be calculated for both pinion and wheel.

2.2.1 Contact stress

Cylindrical gears

where:

Ft, KA , Kγ , Kv , KHβ , KHα , 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).Zε = Contact ratio factor (see 2.5).Zβ = Helix angle factor (see 2.6).

σH 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).

( )HαHβvγA

1

tβεEHDB,H KKKKK

bud

1uFZZZZZσ

+=

( )HαHβvγA

v1v

vmtKEMH KKKKK

bud

1uFZZZ1.05σ

+⋅=

XWRvLH

NHlimHP ZZZZZ

S

Zσσ =




2.3 Zone Factors ZH, ZB,D and ZM

2.3.1 Zone factor ZH

The 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 εβ ≥ 1, ZB,D = 1

For internal gears, ZD = 1

For εβ = 0 (spur gears)

If ZB < 1, use ZB = 1

If ZD < 1, use ZD = 1

For 0 < εβ < 1

ZB,D = ZB,D (for spur gears) – εβ (ZB,D (for spur gears) – 1)

2.3.3 Zone factor ZM

The 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 andPoisson’s ratio on the contact stresses.

For steel against steel ZE = 189.8

wtt2

wtbH

sinααcos

cosαcosβ2Z =

( )

π−ε−−

π−−

α=

α2

2

2b

2a

1

2

1b

1a

wtB

z

211

d

d

z

21

d

d

tanZ

( )

π−ε−−

π−−

α=

α1

2

1b

1a

2

2

2b

2a

wtD

z

211

d

d

z

21

d

d

tanZ

ε−−

ε−−

αβ=αα btm

22vb

22vabtm

21vb

2val

2v1vvtbmM

pddpdd

ddtancos2Z




2.5 Contact Ratio Factor, ZεThe contact ratio factor Zε accounts for the influence of the transverse contact ratio εα and the overlap ratio εβon the contact stresses.

2.6 Helix Angle Factor, ZβThe helix angle factor, Zβ, accounts for the influence of helix angle (independent of its influence on Zε) on thesurface durability.

2.7 Bevel Gear Factor, ZK

The 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, σHlim and Static Strength, ,

σHlim is the limit of contact stress that may be sustained for 5·107 cycles, without the occurrence of progressive pitting.

For most materials 5·107 cycles are considered to be the beginning of the endurance strength range or lowerknee of the σ-N curve. (See also Life Factor ZN). However, for nitrided steels 2·106 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 103

cycles without subsurface yielding or flank damages as pitting, spalling or case crushing when adequate casedepth applies.

The following listed values for σHlim, 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 εβ ≥ 1

for εβ < 1

σHlim

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 σHlim 1.3 σHlimAlloyed quenched and tempered steel, bath or gas nitrided(surface hardness 500 to 700 HV): 1000 1.3 σHlim 1.3 σHlimAlloyed, flame or induction hardened steel (surface hardness 500 to 650 HV): 0.75 HV + 750 1.6 σHlim 4.5 HVAlloyed quenched and tempered steel: 1.4 HV + 350 1.6 σHlim 4.5 HVCarbon steel: 1.5 HV + 250 1.6 σHlim 1.6 σHlimThese values refer to forged or hot rolled steel. For cast steel the values for σHlim are to be reduced by 15%.

αε

1Z =ε

( )α

ββ

αε ε

εε1

3

ε4Z +−

−=

Zβ1

βcos----------------=

σH10

5 σH10

3

σH10

5 σH10

3

σH10

5 σH10

3

σH10

5 σH10

3




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 5·107 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 2·106 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 5·107 (respectively 2·106

for 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 ≥ 5⋅107: ZN = 1 or

105 < NL < 5·107:

NL = 105:

103 < NL < 105:

NL ≤ 103

NL ≥ 2 ⋅106: ZN = 1 or

105 < NL < 2·106

NL ≤ 105

0157.0

L

7

N N

105Z

⋅=

510NlogZ0.37

L

7

N N

105Z

⋅=

WXRVLHlim

WstX10H1010NN ZZZZZσ

ZZσZZ

55

5=

)/Z(Zlog0.5

L

5

10NN

5N103N10

5N

10ZZ

==

WXRVLlimH

Wst10X10H

10NN ZZZZZ

ZZZZ

33

3σ

σ==

0098,0

L

6

N N

102Z

⋅=

510NZlog7686.0

L

6

N N

102Z

⋅=

XWRVL

10XWst10NN ZZZZZ

ZZ1.3ZZ

5

5 ==




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 = 130

RZeq = equivalent roughness

If RZeq > 16, then use RZeq = 16

If RZeq < 1.5, then use RZeq = 1.5

where:

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

ν40 = Kinematic oil viscosity at 40ºC (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 120ºC for long periods may cause reduced flank surface endurance limits.For values of ν40 > 500, use ν40 = 500.

RZrel = The mean roughness between pinion and wheel (after running in) relative to an equivalent radius of curvature at the pitch point ρc = 10mm.

RZrel =

RZ = Mean peak to valley roughness (μm) (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 inν40 = see 2.10.

( )240/1342.1

36.091.0

ν++

( )240/1342.1

68.083.0

ν++

( )v/328.0

14.093.0

++

( )v/328.0

30.085.0

++

08.0

ZrelR

3

15.0

ZrelR

3

( ) 3

1

cZ2Z1 ρ

10RR5.0

+

15.0

ZeqW R

3

1700

130HB2.1Z

−−=

Zeq




Through hardened pinion against softer wheel:

For u > 20, use u = 20

For static strength (< 105 cycles):

Surface hardened against not surface hardened

ZWst = 1.05

Through hardened pinion against softer wheel

ZWst = 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 (>3·106 cycles) is assumed to mainly depend on the orthogonal shear stresses. Staticstrength (<103 cycles) is assumed to depend mainly on equivalent stresses (von Mises). Both are influenced byresidual stresses, but this is only considered roughly and empirically.

The subsurface working stresses at depths inside the peak of the orthogonal shear stresses respectively theequivalent stresses are only dependent on the (real) Hertzian stresses. Surface related conditions as expressedby ZL , ZV and ZR are assumed to have a negligible influence.The real Hertzian stresses σHR are determined as:For helical gears with εβ ≥ 1 :

σHR = σH

For helical gears with εβ < 1 and spur gears:

For bevel gears:

The necessary hardness HV is given as a function of the net depth tz (net = after grinding or hard metal hobbing,and perpendicular to the flank).The coordinates tz and HV are to be compared with the design specification, such as:

— for flame and induction hardening; tHVmin, HVmin — for nitriding; t400min, HV = 400 — for case hardening; t550min, HV = 550; t400min, HV = 400 and t300min, HV = 300 (the latter only if the core

hardness < 300. If the core hardness > 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). Thisapplies to all hardening methods.

For use ZW = 1

For use

( )

−⋅−+= 0.00829

HB

HB0.008981u1Z

2

1W

2.1HB

HB

2

1 ≤

7.1HB

HB

2

1 > 1.7HB

HB

2

1 =

ε

α

ββ ε

ε+ε−

⋅σ=σZ

1

HHR

KHHR Z

1σσ ⋅=



Sec.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 (<103 cycles) the following applies:

In the case of insufficient specified hardness depths, the same procedure for determination of the actual safetyfactor as above applies.

For limited life fatigue (103 < cycles < 3⋅106 ):

For this purpose it is necessary to extend the correction of safety factors to include also higher values thanrequired. I.e. in the case of more than sufficient hardness and depths, the safety factor in the formulae for bothhigh cycle fatigue and static strength are to be increased until necessary and specified values balance.

The actual safety factor for a given number of cycles N between 103 and 3⋅106 is found by linear interpolationin a double logarithmic diagram.

3. Calculation of Tooth Strength

3.1 Scope and General RemarksPart 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurfaceinitiated) and yielding.

For rim thickness sR ≥ 3.5·mn the strength is calculated by means of 3.2 to 3.13. For cylindrical gears thecalculation is based on the assumption that the highest tooth root tensile stress arises by application of the forceat the outer point of single tooth pair contact of the virtual spur gears. The method has, however, a fewlimitations that are mentioned in 3.6.

⋅+

−⋅⋅⋅= 90

0.5a

zt

0.5a

zt

cosSσ0.4HV

H

HHHR

0.5a

zt toapplicable

H≥

for valuethe0.5a

tFor

H

z < applies0.5a

t

H

z =

56300

ρSσ1.2a cHHR

H⋅⋅⋅=

⋅+

−⋅⋅⋅= 90

0.7a

t

0.6a

zt

cosSσ0.19HV

Hst

z

HstHHR

0.6a

ttoapplicable

Hst

z ≥

56300

ρSσa cHHR

Hst⋅⋅=

−⋅−

= ⋅ logN3.477

logSlogSlogS

310H6103HHN 36 10H103H

Slog8628.1Slog8628.0 ⋅+⋅ ⋅




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 KFβ and KFα.

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 < 3.5·mn 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).Yβ = Helix angle factor (see 3.6).

Ft, KA, Kγ, Kv, KFβ, KFα, 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.Yε = Contact ratio factor, see 3.5.

Fmt, KA, etc., see 1.5 to 1.10.

b, see 1.3.

FαFβvγAβSFn

tF K K K K K Y Y Y

m b

Fσ =

FαFβvγASaFamn

mtF K K K K K Y Y Y

m b

Fσ ε=




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 3·106 cycles when used in this formula for σFP. 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

Figure 3.2 Internal tooth in normal section

σFE = 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.YδrelT = 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 YYYY

S

YYδ

σ=σ




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 αFan and can be determined by the same procedureas YF with exception of hFe and αFan . For hFa and αFan 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 mnm

zn with zvn

αt with αvt

β with βm

Fig. 3.3 Dimensions and basic rack profile of the teeth (finished profile)

Tool and basic rack data such as hfP, ρfp 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 cutterρfP= tip radius of pinion cutter.

with undercut without undercut

For external gears

For internal gears

n

2

n

Fn

enFn

Fe

F

α cosm

s

α cosm

h6

Y

=

n

2

n

Fn

anFn

Fa

Fa

α cosm

s

α cosm

h6

Y

=

( )n

n

prnfPnfP m

α cos

sαsin 1'ραtan h

4

πE

−−−−=

fPfP ρ'ρ =

( )0z

1.95fPfP0

fPfP1.0363.156

ρhxρ'ρ

⋅−++=

xh'ρG fpfp +−=

τm

E

2

π

z

2H

nn

−

−=




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 αFen. 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 ρF at 30º tangent:

c) Determination of bending moment arm hF:dn = zn mn

dan = dn + 2 hapbn = π mn cos αn

dbn = dn cos αn

αFen = αen – γeFor external gears

For internal gears

for external gears

for internal gears

3

πτ =

6

πτ =

Htan z

G2

n

−ϑ=ϑ

ϑ π6---=

π3---

−

ϑ+

ϑ−= 'ρ

cos

G3

3

πsinz

m

sfpn

n

Fn

−

ϑ+

ϑ−= 'ρ

cos

G

6

πsinz

m

sfPn

n

Fn

( )G2coszcos

G2'ρ

m

ρ2

n

2

fpn

F

−ϑϑ+=

b2α

αnβcos

εε =

( )4

d1εp

z

z

2

dd

z

z2d

2bn

2

αnbn

2bn

2an

en +

−−

−=

en

bnen d

dcos arcα =

ennnn

e α invα invα x tan 22

π

z

1 −+

+=γ

( )

−=

n

enFenee

n

Fe

m

dαtan sin γ γcos

2

1

m

h ]'ρcos

G

3

πcosz fpn +

ϑ−

ϑ−−

( ) −

⋅⋅−=

n

enFenee

n

Fe

m

dα tanγ sinγ cos

2

1

m

h

−

ϑ−

ϑ−⋅ 'ρ

cos

G3

6

πcosz fPn




3.3.2 Gearing with εαn > 2

For 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, YSa

The 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 notchρg = radius of the grinding notch

c) The formulae for YS resp. YSa are only valid for αn = 20°. However, the same formulae can be used as asafe approximation for other pressure angles.

3.5 Contact Ratio Factor YεThe contact ratio factor Yε 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:

Yε = 0.625

( )2.5ε2 αn ≤≤

2.50ε2.05for 0.666ε2.366Y αnαnDT ≤≤−=

2.05εfor 1.0Y αnDT <=

( )

+

+=/L2.31.21

1

sS qL0.131.2Y

andh

sL:where

Fe

Fn= )3.3see(ρ2

sq

F

Fns =

g

g

ρ

t0.61.3

1.3

−




3.6 Helix Angle Factor YβThe helix angle factor Yβ 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 (β input in degrees):

Yβ = 1 – εβ β/120

When εβ > 1, use εβ = 1 and when β > 30° , use β = 30° in the formula.

However, the above equation for Yβ may only be used for gears with β > 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, σFE

σFE is the local tooth root stress (max. principal) which the material can endure permanently with 99% survivalprobability. 3⋅106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ –N curve. σFE 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.

σFE 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 σFE may be used for materials subjected to aquality control as the one referred to in the rules.

σFEAlloyed 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) (σB = u.t.s. of base material):

0.25 σB + 125Alloyed quenched and tempered steel: 0.4 σB + 200Carbon steel: 0.25 σB + 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, σFE 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, σFE 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, σFE may be increased by 200 for gears where σFE is reduced by 20% due to root grinding. Otherwise σFE 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.

2) The fillet is not to be ground after surface hardening. Regarding possible root grinding, see 1).




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 ¹ 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 σF and the permissible stress σFPcalculated 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 ratio–1.2 < 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 3·106.

For gears of other materials, YM will normally be higher than for a pure idler, provided the number of changesof direction is below 3·106. 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 3·106 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 =0.9 may be used.

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

M1

M1R1

1Yor Y MstM

+−−

=

branchmain theoffacewidth unit per force

p.t.o. offacewidt unit per force2.1−




3.8.3 For gears with shrinkage stresses and unidirectional loadFor endurance strength:

σFE is the endurance limit for R = 0.

For static strength, YMst = 1 and σfit accounted for in 3.9.b.

σfit 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 σfit, i.e. R normally equal – 1.2.

For static strength:

The effect of σfit is accounted for in 3.9 b.

3.8.5 Additional requirements for peak loadsThe total stress range (σmax – σmin) in a tooth root fillet is not to exceed:

3.9 Life Factor, YN

The 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 σ – N – curve of the respective material for given hardening,module, fillet radius, roughness in the tooth root, etc. I.e. the factors YδrelT, YRelT, YX and YM have aninfluence on YN.

If no σ – N – curve for the actual material and hardening etc. is available, the following method may be used.

Determination of the σ – N – curve:

a) Calculate the permissible stress σFP for the beginning of the endurance limit (3·106 cycles), including theinfluence of all relevant factors as SF, YδrelT, YRelT, YX, YM and YC, i.e.σFP = σFE ·YM ·YδrelT ·YRelT ·YX ·YC / SF

b) Calculate the permissible «static» stress (≤103 load cycles) including the influence of all relevant factors as SFst, YδrelTst, YMst and YCst:

for not surface hardened fillets

for surface hardened fillets

FE

fitM σ

σ

M1

M21Y

+−=

n

F.fit m

ρ21.5scf −=

( ) ( ) FE

fitM σ

σ

R1M1

M2

M1

M1R1

1Y

−⋅+−

+−−

=

M1

M1R1

1YMst

+−−

=

F

y

S

σ2.25

FS

HV5

( )fitσCstYδrelTstYMstYFstσFstS

1FPstσ −⋅⋅⋅=




where σFst 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, YδrelT

The dynamic (respectively static) relative notch sensitivity factor, YδrelT (YδrelTst) indicate to which extent thetheoretically concentrated stress lies above the endurance limits (respectively static strengths) in the case offatigue (respectively overload) breakage.

YδrelT 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:

σFstAlloyed 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 σB or 2.25 σy1) This is valid for a fillet surface hardness of 58 to 63 HRC. In case of lower fillet surface hardness than 58 HRC,

σFst is to be reduced with 30·(58 – HRC) where HRC is the actual hardness. Shot peening or grinding notches are not considered to have any significant influence on σFst.

2) Actual stresses exceeding the yield point (σy or σ0.2) will alter the residual stresses locally in the “tension” fillet respectively “compression” fillet. This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the “compression” fillet.

NL > 3·106

YN = 1 or

i.e. Yn = 0.92 for 1010

The YN = 1 from 3·106 on may only be used when special material cleanness applies, see rules Pt.4 Ch.2.

103<NL<3·106

NL < 103

or simply use σFPst as mentioned in b) directly.

0.01

L

6

N N

103Y

⋅=

exp

L

6

N N

103Y

⋅=

cycles6103forσ

cycles310forσlog0.2876exp

FP

FPst

⋅=

cycles6103forσ

cycles310forσNY

FP

FPst

⋅=

( )0.2

4s0,2

4

relTσ1031.33

q21σ101.220.1351Y

⋅⋅−+⋅⋅−+

= −

−

δ

( )1.06

q210.02451Y s

relT

++=δ




for nitrided fillets:


for not surface hardened fillets1):

for surface hardened fillets except nitrided:

YδrelTst = 0.44 YS + 0.12

for nitrided fillets:

YδrelTst = 0.6 + 0.2 YS

1) These values are only valid if the local stresses do not exceed the yield point and thereby alter the residualstress level. See also 3.9b footnote 2.

3.11 Relative Surface Condition Factor, YRrelTThe relative surface condition factor, YRrelT, takes into account the dependence of the tooth root strength onthe surface condition in the tooth root fillet, mainly the dependence on the peak to valley surface roughness.

YRrelT differs for endurance limit and static strength.

The following method may be used:


YRrelT = 1.675 – 0.53 (Ry + 1)0.1

for surface hardened steels and alloyed quenched and tempered steels except nitrided

YRrelT = 5.3 – 4.2 (Ry + 1)0.01

for carbon steels

YδrelT = 4.3 – 3.26 (Ry + 1)0.005

for nitrided steels


YRrelTst = 1 for all Ry and all materials.

For a fillet without any longitudinal machining trace, Ry ≈ Rz.

3.12 Size Factor, YXThe size factor, YX, takes into account the decrease of the strength with increasing size. YX differs forendurance limit and static strength.

The following may be used:



YXst = 1 for all mn and all materials.

3.13 Case Depth Factor, YCThe case depth factor, YC, takes into account the influence of hardening depth on tooth root strength.YC applies only to surface hardened tooth roots, and is different for endurance limit and static strength.In case of insufficient hardening depth, fatigue cracks can develop in the transition zone between the hardenedlayer and the core. For static strength, yielding shall not occur in the transition zone, as this would alter thesurface residual stresses and therewith also the fatigue strength.

YX = 1 for mn ≤ 5 generallyYX = 1.03 – 0.006 mn for 5 < mn < 30 for not surface

hardened steelsYX = 0.85 for mn ≥ 30

YX = 1.05 – 0.01 mn for 5 < mn ≥ 25 for surface hardened steelsYX = 0.8 for mn ≥ 25

( )1.347

q210.1421Y s

relT

++=δ

( )( )( )0.25

0.2

0.250.2s

relTst/σ3000.821

/σ300 1Y0.821Y

+−+=δ




The major parameters are case depth, stress gradient, permissible surface respectively subsurface stresses, andsubsurface residual stresses.

The following simplified method for YC may be used.

YC consists of a ratio between permissible subsurface stress (incl. influence of expected residual stresses) andpermissible surface stress. This ratio is multiplied with a bracket containing the influence of case depth andstress gradient. (The empirical numbers in the bracket are based on a high number of teeth, and are somewhaton the safe side for low number of teeth.)

YC and YCst may be calculated as given below, but calculated values above 1.0 are to be put equal 1.0.



where const. and t are connected as:

For symbols, see 2.13.

In addition to these requirements to minimum case depths for endurance limit, some upper limitations apply tocase hardened gears:

The max. depth to 550 HV should not exceed

1) 1/3 of the top land thickness san unless adequate tip relief is applied (see 1.10).2) 0.25 mn. If this is exceeded, the following applies additionally in connection with endurance limit:

3.14 Thin rim factor YBWhere the rim thickness is not sufficient to provide full support for the tooth root, the location of a bendingfailure may be through the gear rim, rather than from root fillet to root fillet.

YB is not a factor used to convert calculated root stresses at the 30° tangent to actual stresses in a thin rimtension fillet. Actually, the compression fillet can be more susceptible to fatigue.

YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailedcalculation of stresses in both tension and compression fillets are available.

Figure 3.4 Examples on thin rims

YB is applicable in the range 1.75 < sR/mn < 3.5.

YB = 1.15 · ln (8.324 · mn/sR)

(for sR/mn ≥ 3.5, YB = 1)

(for sR/mn ≤ 1.75, use 3.15)

Hardening process t = endurance limit const =

static strength const =

Case hardeningt550 640 1900t400 500 1200t300 380 800

Nitriding t400 500 1200Induction- or flame hardening tHVmin 1.1 HVmin 2.5 HVmin

+

+=nFFE

C m0.2ρ

t31

σ

.constY

+

+=nFFst

Cst m0.2ρ

t31

σ

.constY

−−= 0.25

m

t1Y

n

max550C




σF as calculated in 3.2.1 is to be multiplied with YB when sR/mn < 3.5. Thus YB is used for both high and lowcycle fatigue.

Note: This method is considered to be on the safe side for external gear rims. However, for internal gear rimswithout any flange or web stiffeners the method may not be on the safe side, and it is advised to check with themethod in 3.15/3.16.

3.15 Stresses in Thin RimsFor rim thickness sR < 3.5 mn the safety against rim cracking has to be checked.

The following method may be used.

3.15.1 GeneralThe stresses in the standardised 30º tangent section, tension side, are slightly reduced due to decreasing stresscorrection factor with decreasing relative rim thickness sR/mn. On the other hand, during the complete stresscycle of that fillet, a certain amount of compression stresses are also introduced. The complete stress rangeremains approximately constant. Therefore, the standardised calculation of stresses at the 30º tangent may beretained for thin rims as one of the necessary criteria.

The maximum stress range for thin rims usually occurs at the 60º to 80º tangents, both for «tension» and«compression» side. The following method assumes the 75º tangent to be the decisive. Therefore, in additionto the a.m. criterion applied at the 30º tangent, it is necessary to evaluate the max. and min. stresses at the 75ºtangent for both «tension» (loaded flank) fillet and «compression» (back-flank) fillet. For this purpose thewhole stress cycle of each fillet should be considered, but usually the following simplification is justified:

Figure 3.5 Nomenclature of fillets

Index «T» means «tensile» fillet, «C» means «compression» fillet.

σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor,Y75.

σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the toothbending stresses at 75º tangent.

3.15.2 Stress concentration factors at the 75º tangentsThe nominal rim stress consists of bending stresses due to local bending moments, tangential stresses due tothe tangential force Ft, and radial shear stresses due to Fr.

The major influence is given by the bending stresses. The influence of the tangential stresses is minor, and eventhough its stress concentration factor is slightly higher than for bending, it is considered to be safe enough whenthe sum of these nominal stresses are combined with the stress concentration factor for bending. The influenceof the radial shear stress is neglected.

The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75º tangent may becalculated as:

where ρ75 is the root radius at the 75º tangent ref. to mn. Usually ρ75 is closed to the tool radius ρao, and

ρ75 = ρao

is a safe approximation compensating for the a.m. simplifications to the «unsafe» side.

75n

R75

n

R

75

ρm

sρ1.85

m

s3

Y

+

⋅=




The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness, approximatelywith the empirical factor

3.15.3 Nominal rim stressesThe bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 0.5 sR) and thebending caused by the radial force Fr.

The sectional modulus (first moment of area) which is used for determination of the nominal bending stressesis not necessarily the same for the 2 a.m. bending moments. If flanges, webs, etc. outside the toothed sectioncontribute to stiffening the rim against various deflections, the influence of these stiffeners should beconsidered. E.g. an end flange will have an almost negligible influence on the effective sectional modulus forthe stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not muchinvolve the flange. On the other hand, the radial forces, as for instance from the meshes in an annulus, wouldcause considerable radial deflections that the flange might restrict to a substantial amount. When consideringthe stiffening of such flanges or webs on basis of simplified models, it is advised to use an effective rimthickness sR' = sR + 0.2 mn for the first moment of area of the rim (toothed part) cross section.

For a high number of rim teeth, it may be assumed that the rim bending moments in the fillets adjacent to theloaded tooth are of the same magnitude as right under the applied force. This assumption is reasonable for anannulus, but rather much on the safe side for a hollow pinion.

The influence of Ft on the nominal tangential stress is simplified by half of it for compressive stresses (σ1) andthe other half for tensile stresses (σ2). Applying these assumptions, the nominal rim stresses adjacent to theloaded tooth are:

where σ1, σ2 see Fig. 3.5.

It must be checked if the max. (tensile) stress in the compression fillet really occurs when the fillet is adjacentto the loaded tooth. In principle, the stress variation through a complete rotation should be considered, and themax. value used. The max. value is usually never less than 0. For an annulus, e.g. the tilt moment is zero in themid position between the planet meshes, whilst the bending moment due to the radial forces is half of that atthe mesh but with opposite sign.

If these formulae are applied to idler gears, as e.g. planets, the influence of nominal tangential stresses must becorrected by deleting Ft/(2 A) for σ1, and using Ft/A for σ2. Further, the influence of Fr on the nominal bendingstresses is usually negligible due to the planet bearing support.

3.15.4 Root fillet stressesDetermination of min. and max. stresses in the «tension» fillet:

Minimum stress:

A = minimum area of cross section (usually bR sR).WT = the sectional modulus of rim with respect to tooth tilt moment (usually bRsr

2/6).WR = the sectional modulus of the rim including the influence of stiffeners as flanges, webs etc. (WR ≥ WT).bR = the width of the rim.R = the radius of the neutral axis in the rim, i.e. from wheel centre to midpoint of rim.f(ϑ) = a function for bending moment distribution around the rim.

For a rim (pinion) with one mesh only, the f(ϑ) at the position of load application is 0.24.For an annulus with 3 or more meshes, f(ϑ) at the position of each load application is approx.:3 planets f(ϑ) = 0.194 planets f(ϑ) = 0.145 planets f(ϑ) = 0.116 planets f(ϑ) = 0.09

1.5

R

ncorr s

m

3

11Y

−=

( ) ( )A2

F

W

f RF

W

s0.5hF0.5 t

R

r

T

RFt1 −ϑ−+−=σ

( ) ( )A2

F

W

f RF

W

s0.5hF0.5 t

R

r

T

RFt2 +ϑ−+=σ

751FTmin YσK σ =




Maximum stress:

where:

0.3 is an empirical factor relating the tension stresses (σF) at the 30º tangent to the part of the tension stressesat the 75º tangent which add to the rim related stresses. (0.3 also takes into account that full superposition ofnominal stresses times stress concentration factors from both «sides of the corner fillet» would result in toohigh stresses.)

Determination of min. and max. stresses in the «compression» fillet:

Minimum stress:

Maximum stress:

where:

0.36 is an empirical factor relating the tension stresses (σF) at the 30º tangent to the part of the compressionstresses at the 75º tangent which add to the rim related stresses.

For gears with reversed loads as idler gears and planets there is no distinct «tension» or «compression» fillet.The minimum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive). The maximumstress σFTmax is the maximum of σFTmax and σFCmax (usually the former is decisive).

3.16 Permissible Stresses in Thin Rims

3.16.1 GeneralThe safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solidgears. The «ordinary» criteria at the 30º tangent apply as given in 3.1 through 3.13.

Additionally the following criteria at the 75º tangent may apply.

3.16.2 For >3·106 cyclesThe permissible stresses for the «tension» fillets and for the «compression» fillets are determined by means ofa relevant fatigue diagram.

If the actual tooth root stress (tensile or compressive) exceeds the yield strength to the material, the inducedresidual stresses are to be taken into account.

For determination of permissible stresses the following is defined:

R = stress ratio, i.e respectively

Δσ = stress range, i.e. σFTmax – σFTmin resp. σFCmax – σFCmin

(For idler gears and planets and Δσ = σFmax σFmin).

The permissible stress range Δσρ for the «tension» respectively «compression» fillets can be calculated as:

where:

σFP = see 3.2, determined for unidirectional stresses (YM = 1).

If the yield strength σy is exceeded in either tension or compression, residual stresses are induced. This may beconsidered by correcting the stress ratio R for the respective fillets (tension or compression).

For R > –1

For – ∞ < R < –1

corrF752 Yσ0.3Yσ KσFTmax +=

αβγ ⋅⋅⋅⋅= FFvA KKKKKK

corrF751FCmin Yσ0.36Yσ Kσ −=

752FCmax Yσ Kσ =

σFTmin

σFTmax-------------------

σFCmin

σFCmax-------------------

σFmin

σFmax---------------

FPp σ

R1

R10.31

1.3Δσ

−++

=

FPp σ

R1

R10.151

1.3Δσ

−++

=



Sec.4. Calculation of Scuffing Load Capacity – Page 45

E.g. if |σFCmin| > σy, (i.e. exceeded in compression), the difference Δ = |σFCmin| – σy affects the stress ratio as

Similarly the stress ratio in the tension fillet may require correction.

If the yield strength is exceeded in tension, σFTmax > σy, the difference Δ = σFTmax – σy affects the stress ratio as

Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAPif this exceeds KA.

3.16.3 For ≤ 103 cyclesThe permissible stress range Δσp is not to exceed:

For all values of R, Δσpst is limited by:

Definition of Δσ and R, see 3.16.2, with particular attention to possible correction of R if the yield strength isexceeded.

σFPst see 3.2, determined for unidirectional stresses (YM = 1) and < 103 cycles.

3.16.4 For 103 < cycles < 3·106

Δσp is to be determined by linear interpolation a log-log diagram.

Δσp at NL load cycles is:

4. Calculation of Scuffing Load Capacity

4.1 IntroductionHigh surface temperatures due to high loads and sliding velocities can cause lubricant films to break down. Thisseizure or welding together of areas of tooth surface is termed scuffing.

In contrast to pitting and fatigue breakage which show a distinct incubation period, a single short overloadingcan lead to a scuffing failure. In the ISO-TR13989 two criteria are mentioned. The method used in thisClassification Note is based on the principles of the flash temperature criterion.

Note:Bulk temperature in excess of 120ºC for long periods may have an adverse effect on the surface durability, see2.11.

For R > –1

For – ∞ < R < –1

not surface hardened

surface hardened

Δσ

σ

Δσ

ΔσR

FCmax

y

FCmax

FCmin

+−

=++=

y

FTmin

FTmax

FTmin

σ

Δσ

Δσ

ΔσR

−=−−=

FPstpst σ

R1

R10.51

1.5Δσ

−++

=

FPstpst σ

R1

R10.251

1.5Δσ

−++

=

F

y

S

σ2.25

CF

YS

HV5

6L 103p

exp

L

6

N p ΔσN

103Δσ ⋅

⋅=

6

3

103p

10p

Δσ

Δσlog2876.0exp

⋅

=




4.2 General CriteriaIn no point along the path of contact the local contact temperature may exceed the permissible temperature, i.e.:

where:

The scuffing temperature may be calculated as:

where:

Application of other test methods such as the Ryder, the FZG-Ryder R/46.5/74, and the FZG L-42 Test 141/19.5/110 may be specially considered.

For high speed gears with very short time of contact ϑS, may be increased as follows provided use of EP-oils.

Addition to the calculated scuffing temperature ϑS:

If tc ≥ 18 μs, no addition

If tc < 18 μs, add 18 ⋅ XwrelT ⋅ (18 – tc)

where

tc = contact time (μs) which is the time needed to cross the Hertzian contact width.

ϑB = max. contact temperature along the path of contact.

ϑB = ϑMB + ϑfla max

ϑMB = bulk temperature, see 4.3.4.ϑfla max = max. flash temperature along the path of contact, see 4.4.ϑS = scuffing temperature, see below.ϑoil = oil temperature before it reaches the mesh (max. applicable for the actual load case to be used, i.e. normally

alarm temperature, except for ice classes where a max. expected temperature applies).SS = required safety factor according to the Rules.

XwrelT = relative welding factor.

XwrelTThrough hardened steel 1.0Phosphated steel 1.25Copper-plated steel 1.50Nitrided steel 1.50

Casehard-ened steel

Less than 10% retained austenite 1.1510 – 20% retained austenite 1.020 – 30% retained austenite 0.85

Austenitic steel 0.45

FZG = load stage according to FZG-Test A/8.3/90.(Note: This is the load stage where scuffing occurs. However, due to scatter in test results, calculations are to be made with one load stage less than the specification.)

XL = lubricant factor.= 1.0 for mineral oils.= 0.8 for polyalfaolefins.= 0.7 for non-water-soluble polyglycols.= 0.6 for water-soluble polyglycols.= 1.5 for traction fluids.= 1.3 for phosphate esters.

ν40 = kinematic oil viscosity at 40°C (mm2/s).

oilS

oilSB S

ϑ+ϑ−ϑ≤ϑ

50SB −ϑ≤ϑ

L2

wrelT

0.02

40S XFZGX

ν

1001.120.85780

⋅

⋅++=ϑ




σH as calculated in 2.2.1.

For bevel gears, use uv in stead of u.

4.3 Influence Factors

4.3.1 Coefficient of frictionThe following coefficient of friction may apply:

where:

Cylindrical gears

Bevel gears

4.3.2 Effective tip relief Ceff

Ceff is the effective tip relief; that amount of tip relief, which compensates for the elastic deformation of thegear mesh, i.e. zero load at the tooth tip. It is assumed (simplified) to be equal for both pinion and wheel.

wBt = specific tooth load (N/mm)vΣC = sum of tangential velocities at pitch point.

At pitch line velocities > 50 m/s, the limiting value of vΣC at v = 50 m/s is to be used.ρredC = relative radius of curvature (transversal plane) at the pitch point

(see 1)

(see 1)

ηoil = dynamic viscosity (mPa s) at ϑoil, calculated as

where ρ in kg/m3 approximated as

and νoil is kinematic viscosity at ϑoil and may be calculated by means of the following equation:

Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as

This is defined as the roughness on the new flanks i.e. as manufactured.XL = see 4.2

( ) [ ]μsu1cosβn

uσ340t

b1

Hc +⋅⋅

⋅=

L0.25a

0.05oil

0.2

redC ΣC

Bt X R ηρv

w0.048μ −

=

HαHβvγAbt

Bt KKKKKb

Fw ⋅⋅⋅⋅⋅=

twΣC αsin v2v =

bCredC β cos ρρ =

HαHβvγAtmb

Bt KKKKKb

Fw ⋅⋅⋅⋅⋅=

vtmtΣC αsin v2v =

bmvCredC β cos ρρ =

1000

ρ νη oil

oil =

( ) 7.0 15oil15 −ϑ−ρ=ρ

( ) ( )++=+ 0.8νloglog0.8νloglog 100oil( )

⋅−

ϑ+−313log373log

273log373log oil ( ) ( )( )0.8νloglog0.8νloglog 10040 +−+

( )2a1aa RR 0.5R +=




Cylindrical gears

Alternatively for spur and helical gears the non-linear approach in 1.11 may be used (taking Ceff. = δ).

Bevel gears

where:

4.3.3 Tip relief and extension

Cylindrical gears

The extension of the tip relief is not to result in an effective contact ratio εα < 1 when the gear is unloaded(exceptions to this may only apply for applications where the gear is not to run at light loads). This means thatthe unrelieved part of the path of contact is to be minimum pbt. It is further assumed that this unrelieved part isplaced centrally on the path of contact.

If root relief applies, it has to be calculated as equivalent tip relief. I.e. pinion root relief (at mesh position A)is added to Ca2, and wheel root relief (at mesh position E) is added to Ca1. If no design tip relief or root relief on the mating gear is specified (i.e. if Ca1+ Croot2 = 0 and visa versa), usethe running in amount, see 1.12.

Bevel gears

Bevel gears are to have heightwise crowning, i.e. no distinct relieved/unrelieved area. This may be treated astip and root relief. For calculation purposes the root relief is combined with the tip relief of the mating memberinto an equivalent tip relief. If no resulting tip relieves are specified, the equivalent tip relives may becalculated, as an approximation, based on the tool crowning Ca tool (per mille of tool module m0) as follows:

where:

For helical:

(see 1)

For spur:

(see 1)

(see 1)

γ

γ=cb

KKFC Abt

eff

'cb

KKFC Abt

effγ=

bc

KFC Ambt

effγ

=

γ

α

ε2

ε44c

+⋅=γ

2root1aeq1a CCC +=

1root2aeq2a CCC +=

2

0

n0101atoolal m

)m2(mA1·mCC

−+−⋅=

( )

⋅−−⋅⋅−+= vtvb1

2vb1

21vavt1n1 αtan dddαsin 0.5x1mA

2

0

n020atool22a m

)mm(2A1·mCC

−+−⋅=

( )

⋅−−⋅⋅−+= vtvb2

2vb2

2va2vt2n2 αtan dddαsin 0.5x1mA




If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves, the followingmay be assumed:

Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq.

4.3.4 Bulk temperatureThe bulk temperature may be calculated as:

where:

For high speed gears (v > 50 m/s) it may be necessary to assess the bulk temperature on the basis of a thermalrating of the entire gear transmission.

4.4 The Flash Temperature ϑfla

4.4.1 Basic formulaThe local flash temperature ϑfla may be calculated as

(For bevel gears, replace u with uv)

and is to be calculated stepwise along the path of contact from A to E.

where:

Xs = lubrication factor.= 1.2 for spray lubrication.= 1.0 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed < 5 m/s).= 1.0 for spray lubrication with additional cooling spray (spray on both pinion and wheel, or spray on

pinion and dip lubrication of wheel).= 0.2 for meshes fully submerged in oil.

Xmp = contact factor,

np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher).

ϑflaaverage = average of the integrated flash temperature (see 4.4) along the path of contact.

μ = coefficient of friction, see 4.3.1.Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach

path, due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief.

Ca = tip relief of driven member.Xcorr is only applicable in the approach path and if Ca < Ceff, otherwise 1.0.

2

0

20atool1root1 m

A1mCC

−⋅⋅=

2

0

10atool2root2 m

A1mCC

−⋅⋅=

( )eq2aeq1aeq2aeq1a C andC calculated of sum5.0CC ⋅==

flaaveragempsoilMB X X 5.0 ϑ+ϑ=ϑ

( )pmp nX += 15.0

( )AE

E

Ayyyfla

flaaverage

d

Γ−Γ

ΓΓϑ=ϑ =

( )4/1

redy

y2ly

2/11

4/3Btcorrfla

unXwX325.0

y ρ

ρ−ρ

μ=ϑ Γ

( )3

AD

yaeffcorr 50

CC1X

Γ−ΓεΓ−

+=α




Γy, ρly etc. see 4.4.2.

4.4.2 Geometrical relationsThe various radii of flank curvature (transversal plane) are:

Cylindrical gears

Note that for internal gears, a and u are negative.

Bevel gears

Γ is the parameter on the path of contact, and y is any point between A end E.

At the respective ends, Γ has the following values:

Root pinion/tip wheel

Cylindrical gears

Bevel gears

Tip pinion/root wheel

Cylindrical gears

Bevel gears

At inner point of single pair contact

wBt = unit load, see 4.3.1.= load sharing factor, see 4.4.3.

n1 = pinion r.p.m.

ρ1y = pinion flank radius at mesh point y.ρ2y = wheel flank radius at mesh point y.ρredy = equivalent radius of curvature at mesh point y.

XΓy

y2y1

y2y1redy

ρ+ρρρ

=ρ

twy

1y αsin au1

Γ1ρ

++

=

twy

2y αsin au1

Γuρ

+−

=

vtvv

y1y αsin a

u1

Γ1ρ

++

=

vtvv

yv2y αsin a

u1

Γuρ

+−

=

( )

−

−−= 1

αtan

1/dd

z

zΓ

tw

2b2a2

1

2A

( )

−

−−= 1

αtan

1/dduΓ

vt

2vb2va2

vA

( )1

αtan

1/ddΓ

tw

2b1a1

E −−

=

( )1

αtan

1/ddΓ

vt

2vb1va1

E −−

=




Cylindrical gears

Bevel gears

At outer point of single pair contact

Cylindrical gears

Bevel gears

At pitch point ΓC = 0.

The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintaina minimum contact ratio of unity for unloaded gears) are at

4.4.3 Load sharing factor XΓThe load sharing factor XΓ accounts for the load sharing between the various pairs of teeth in mesh along thepath of contact.

XΓ is to be calculated stepwise from A to E, using the parameter Γy.

4.4.3.1 Cylindrical gears with β = 0 and no tip relief

Figure 4.1

4.4.3.2 Cylindrical gears with β = 0 and tip reliefTip relief on the pinion reduces XΓ in the range G – E and increases correspondingly XΓ in the range F – B.

Tip relief on the wheel reduces XΓ in the range A – F and increases correspondingly XΓ in the range D – G.

Following remains generally:

tw1EB α tan z

π2ΓΓ −=

vtv1EB α tan z

π2ΓΓ −=

tw1AD α tan z

π2ΓΓ +=

vt v1AD αtan z

π2ΓΓ +=

2BA

FΓ+Γ=Γ

2ED

GΓ+Γ=Γ

ByAAB

Ay for3

1

3

1X

yΓ<Γ≤Γ

Γ−ΓΓ−Γ

+=Γ

DyB for1Xy

Γ<Γ≤Γ=Γ

EyDDE

yE for3

1

3

1X

yΓ≤Γ<Γ

Γ−ΓΓ−Γ

+=Γ

DyB for1Xy

Γ≤Γ≤Γ=Γ

2/1XXGF

== ΓΓ




In the following it must be distinguished between Ca < Ceff respectively Ca > Ceff. This is shown by an examplebelow where Ca1 < Ceff and Ca2 > Ceff.

Figure 4.2

Note:When Ca > Ceff the path of contact is shortened by A – A' respectively E' – E. The single pair contact path isextended into B' respectively D'. If this shift is significant, it is necessary to consider the negative effect onsurface durability (B') and bending stresses (B' and D').

Range A - F

Range F - B

effa2 CCFor ≤

−=Γ

eff

2a

C

C1

3

1X

A

FyAeff

2a

AF

Ay for C3

C

6

1XX

AyΓ≤Γ≤Γ

⋅

+Γ−ΓΓ−Γ

+= ΓΓ

effa2 CCFor ≥

'AyAfor0Xy

Γ≤Γ≤Γ=Γ

( )2

1

C

C

1C

C

with

eff

2a

eff

2a

AFA'A−

−Γ−Γ+Γ=Γ

Fy'Aeff

2a

AF

Ay

eff

2a for 2

1

C

C

C

C1X

yΓ≤Γ≤Γ

−

Γ−ΓΓ−Γ

+−=Γ

effa1 CCFor ≤

ByFeff

1a

FB

Fy forC3

C

6

1

2

1X

yΓ≤Γ≤Γ

⋅

+Γ−ΓΓ−Γ

+=Γ

effa1 CCFor ≥

'ByFeff

1a

FB

Fy for2

1

C

C

2

1X

yΓ≤Γ≤Γ

−

Γ−ΓΓ−Γ

+=Γ

By'B for1Xy

Γ≤Γ≤Γ=Γ

1C

C2

with

eff

1a

FBF'B

−

Γ−Γ+Γ=Γ




Range D – G

Range G - E

4.4.3.3 Gears with β > 0, buttressingDue to oblique contact lines over the flanks a certain buttressing may occur near A and E.

This applies to both cylindrical and bevel gears with tip relief < Ceff. The buttressing Xbutt is simplified as alinear function within the ranges A – H respectively I – E.

Figure 4.3

effa2 CCFor ≤

GyDeff

2a

DG

Dy

eff

2a for C 3

C

6

1

C 3

C

3

2X

yΓ≤Γ≤Γ

+

Γ−ΓΓ−Γ

−+=Γ

effa2 CCFor ≥

'DyD for1Xy

Γ≤Γ≤Γ=Γ

( )2

1

C

C

1C

C

with

eff

2a

eff

2a

DGD'D−

−Γ−Γ+Γ=Γ

Gy'Deff

2a

DG

Dy

eff

2a for2

1

C

C

C

CX Γ≤Γ≤Γ

−

Γ−ΓΓ−Γ

−=Γ

effa1 CCFor ≤

−=Γ

eff

1a

C

C1

3

1X

E

EyGeff

1a

GE

Gy forC3

C

6

1

2

1X

yΓ≤Γ≤Γ

⋅

+Γ−ΓΓ−Γ

−=Γ

effa1 CCFor >

'EyGeff

1a

GE

Gy for2

1

C

C

2

1X

yΓ≤Γ≤Γ

−

Γ−ΓΓ−Γ

−=Γ

Ey'E for0Xy

Γ≤Γ≤Γ=Γ

1C

C2

with

eff

1a

GEG'E

−

Γ−Γ+Γ=Γ

1εwhen1.3X βbutt EA,≥=

1εwhenε0.31X ββbutt EA,<+=




Cylindrical gears

Bevel gears

4.4.3.4 Cylindrical gears with εγ ≤ 2 and no tip relief

is obtained by multiplication of in 4.4.3.1 with Xbutt in 4.4.3.3.

4.4.3.5 Gears with εγ > 2 and no tip relief

Applicable to both cylindrical and bevel gears.

Figure 4.4

4.4.3.6 Cylindrical gears with εγ ≤ 2 and tip relief

is obtained by multiplication of in 4.4.3.2 with Xbutt in 4.4.3.3.

4.4.3.7 Cylindrical gears with εγ > 2 and tip relief

Tip relief on the pinion (respectively wheel) reduces in the range G – E (respectively A – F) and increases in the range F – G.

is obtained by multiplication of as described below with Xbutt in 4.4.3.3.

In the XΓ example below the influence of tip relief is shown (without the influence of Xbutt) by means of .

Tip relief > Ceff causes new end points A' respectively E' of the path of contact.

Figure 4.5

Range A – F

bIEAH βsin 0.2ΓΓΓΓ =−=−

bmIEAH βsin 0.2ΓΓΓΓ =−=−

XΓyXΓy

IyH for1

Xy

Γ≤Γ≤Γε

=α

Γ

IyHybutt and forX1

Xy

Γ>ΓΓ<Γε

=α

Γ

XΓyXΓy

XΓyXΓy

XΓyXΓy

eff2aeff1a CC and CC <>

( ) ( )( ) eff

a2αa1α

AF

Ay

eff

2aeff

C 12

C 13εC 1ε

C

CCX

y +εε++−

Γ−ΓΓ−Γ

+ε

−=ααα

Γ

eff2aFyA CC if for ≤Γ≤Γ≤Γ

eff2aFy'A CCiffor and ≥Γ≤Γ≤Γ




Range F – G

Range G – E

4.4.3.8 Bevel gears with εγ more than approx. 1.8 and heightwise crowningFor Ca1 = Ca2 = Ceff the following applies:

Figure 4.6

For tip relief < Ceff, is found by linear interpolation between and as in 4.4.3.5.

The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with theinfluence of Ca1.(For there is a discontinuity at M.)

E.g. with Ca1 = 0.4 Ceff and Ca2 = 0.55 Ceff, then

Range A – M

Range M – E

eff2a'AyA CC iffor0Xy

>Γ≤Γ≤Γ=Γ

( ) ( ) ( )( ) ( ) a2αa1α

αeffa2AFA'A C 1ε 3C 1ε

1ε 2 CC with

++−+−Γ−Γ+Γ=Γ

( )( )( ) GyF

eff

2a1a forC 1 2

CC 11X

yΓ≤Γ≤Γ

+εε+−ε+

ε=

αα

α

αΓ

( ) ( )( ) eff

2a1a

GE

Gy

C 1 2

C 1C 1 3XX

GFy +εε−ε++ε

Γ−ΓΓ−Γ

−=αα

ααΓΓ −

effa1EyG CC if for ≤Γ≤Γ≤Γ

effa1'EyG CC if Γfor and ≥Γ≤Γ≤

eff1aEy'E CC if for0Xy

≥Γ≤Γ≤Γ=Γ

( ) ( ) ( )( ) ( ) 2a1a

eff1aGEE'E C 1C 1 3

1 2 CC with

−ε++ε+ε−Γ−Γ−Γ=Γ

αα

α

( )AEM 5.0 Γ+Γ=Γ

( )( )2

AD3

2My65.1

Xy Γ−Γε

Γ−Γ−

ε=

ααΓ

XΓyXΓy Ca Ceff=( ) XΓy Ca C0=( )

Ca1 Ca2≠

)CC()0C( eff2ay2ayyX 55.0X 45.0X =Γ=ΓΓ +=

)CC()0C( eff1ay1ayyX 4.0X 0.6X =Γ=ΓΓ +=




For tip relief > Ceff the new end points A' and E' are found as

Range A – A'

Range A' – M

Range M – E'

Range E' – E

( )

−Γ−Γε+Γ=Γ α 1

C

C

6 eff

2aADA'A

( )

−Γ−Γε+Γ=Γ α 1

C

C

6 eff

1aADE'E

0Xy

=Γ

( )( )

Γ−Γ

Γ−Γ−

−ε=

αΓ 2

M'A

2My

eff

2a1

C

C4

35.1X

y

( )( )

Γ−Γ

Γ−Γ−

−ε=

αΓ 2

M'E

2My

eff

1a1

C

C4

35.1X

y

0Xy

=Γ



Appendix A Fatigue Damage Accumulation – Page 57

Appendix AFatigue Damage AccumulationThe Palmgren-Miner cumulative damage calculation principle is used. The procedure may be applied asfollows:

A.1 Stress SpectrumFrom the individual torque classes, the torques (Ti) at the peak values of class intervals and the associatednumber of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque.

(In case of a cyclic torque variation within the torque classes, it is advised to use the peak torque. If the cyclicvariation is such that the same teeth will repeatedly suffer the peak torque, this is a must.)

The stress spectra for tooth roots and flanks (σFi, σHi) with all relevant factors (except KA) are to be calculatedon the basis of the torque spectrum. The load dependent K-factors are to be determined for each torque class.

A.2 σ–N–curveThe stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on thebasis of permissible stresses (i.e. including the demanded minimum safety factors) as determined in 2respectively 3. If different safety levels for high cycle fatigue and low cycle fatigue are desired, this may beexpressed by different demand safety factors applied at the endurance limit respectively at static strength.

A.3 Damage accumulationThe individual damage ratio Di at ith stress level is defined as

where:

Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) donot contribute to the damage sum. However, calculating the actual safety factor Sact as described below all theσi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute tothe damage sum and thus to the determination of Sact. The final value of S is decisive.

(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi, thereby findingthe actual life factor. This life factor can be solved with regard to load cycles, i.e. NFi.)

The damage sum ΣDi is not to exceed unity.

If ΣDi ≠ 1, the safety against cumulative fatigue damage is different from the applied demand safety factor. Fordetermination of this theoretical safety factor an iteration procedure is required as described in the followingflowchart:

NLi = The number of applied cycles at ith stress.NFi = The number of cycles to failure at ith stress.

Fi

Lii N

ND =



Appendix A Fatigue Damage Accumulation – Page 58

S is correction factor with which the actual safety factor Sact can be found.

Sact is the demand safety factor (used in determination of the permissible stresses in the σ–N–curve) times thecorrection factor S.

The full procedure is to be applied for pinion and wheel, tooth roots and flanks.

Note:If alternating stresses occur in a spectrum of mainly pulsating stresses, the alternating stresses may be replacedby equivalent pulsating stresses, i.e. by means of division with the actual mean stress influence factor YM.



Appendix B Application Factors for Diesel Driven Gears – Page 59

Appendix BApplication Factors for Diesel Driven Gears

For diesel driven gears the application factor KA depends on torsional vibrations. Both normal operation andmisfiring conditions have to be considered.

Normally these two running conditions can be covered by only one calculation.

B.1 Definitions

where:

where:

The normal operation is assumed to last for a very high number of cycles, such as 1010.

The misfiring operation is assumed to last for a limited duration, such as 107 cycles.

B.2 Determination of decisive loadAssuming life factor at 1010 cycles as YN = ZN = 0.92 which usually is relevant, the calculation may beperformed only once with the combination having the highest value of application factor/life factor.

For bending stresses and scuffing, the higher value of

For contact stresses, the higher value of

B.3 Simplified procedureNote that this is only a guidance, and is not a binding convention.

where TV ideal = vibratory torque with all cylinders perfectly equal. See also rules Pt.4 Ch.3 Sec.1 G300.

When using trends from torsional vibration analysis and measurements, the following may be used:

TV ideal / To is close to zero for engines with few cylinders and using a suitable elastic coupling, and increaseswith relative coupling stiffness and number of cylinders.

TV misf /To may be high for engines with few cylinders and decreases with number of cylinders.

T0 = rated nominal torque

Tv norm = vibratory torque amplitude for normal operation (see rules Pt.4 Ch.3 Sec.1 G301 for definition of “normal” irregularity)

T = remaining nominal torque when one cylinder out of action

Tv misf = vibratory torque amplitude in misfiring condition. This refers to a permissible misfiring condition, i.e. a con-dition that does not require automatic or immediate corrective actions as speed or pitch reduction.

and

and (but for nitrided gears)

0

normv0Anorm T

TTKoperationNormal

+=

0

misfvmisfA T

TTKoperationMisfiring

+=

0.92

K normA

0.98

K misfA

0.92

K normA

1.13

K misfA

0.97

K misfA

cylinders ofnumber ZwhereTZ

1ZT o =⋅−=

( ) ideal Videal Vmisf Vnorm V TTT24

ZT +−=



Appendix B Application Factors for Diesel Driven Gears – Page 60

This can be indicated as:

Inserting this into the formulae for the two application factors, the following guidance can be given:

KA misf ≈ 1.12 to 1.18

KA norm ≈ 1.10 to 1.15

Since KA norm is to be combined with the lower life factors, the decisive load condition will be the normal one,and a KA of 1.15 will cover most relevant cases, when a suitable elastic coupling is chosen.

and

200

Z

T

T

ο

idealV ≈

80

Z0.4

T

T

o

misfV −≈



Appendix C Calculation of Pinion-Rack – Page 61

Appendix CCalculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear. Withnormal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stressesfor static strength will be decisive for the lay out, however, with exception of surface hardened pinions wherecase crushing has to be considered.

In the following the use of part 1 and 3 for pinion-racks is shown, including relevant simplifications.

C.1 Pinion tooth root stressesSince the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000),the static strength is decisive.

The actual stress is calculated as:

b1 is limited to b2 + 2·mn.

YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurategears.

Pinions often use a non-involute profile in the dedendum part of the flank, e.g. a constant radius equal the radiusof curvature at reference circle. For such pinions sFn and hFa are to be measured directly on a sectional drawingof the pinion tooth.

Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 1.0. However, when b1 > b2,then KFβ1 >1.0.

If no detailed documentation of KFβ1 is available, the following may be used:

KFβ1 = 1 + 0.15·(b1/b2 – 1)

The permissible stress (not surface hardened) is calculated as:

The mean stress influence due to leg lifting may be disregarded.

The actual and permissible stresses should be calculated for the relevant loads as given in the rules.

C.2 Rack tooth root stressesThe actual stress is calculated as:

See C.1 for details.

The permissible stress is calculated as:

For alloyed steels (Ni, Cr, Mo) with high toughness and ductility the value of YδrelTst may be put equal to YSa.

C.3 Surface hardened pinionsFor surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank.

In principle the calculation described in 2.13 may be used, but when the theoretical Hertzian stress exceeds theapproximately 1.8 times the yield strength of the rack material, plastic deformation will occur. This will limitthe peak Hertzian stress but increases the contact width, and thus the penetration of stresses into the depth.

An approximation may be based on an assessment of contact width determined by means of equal areas underthe theoretical (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 1.8 · σy) with theunknown width.

β1FSaFan1

tF1 KYY

mb

Fσ ⋅⋅⋅

⋅=

δrelTstF

Fst1FP1 Y

S

σσ ⋅=

SaFan2

tF2 YY

mb

Fσ ⋅⋅

⋅=

δrelTstF

Fst2FP2 Y

S

σσ ⋅=


Documents

DNV Classification Note 41.2: Calculation of Gear Rating for Marine