Calculation of Gear Rating for Marine Transmissions

CLASSIFICATION NOTES

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No. 41.2

Calculation of Gear Rating for Marine Transmissions

MAY 2012DET NORSKE VERITAS AS

FOREWORDDET NORSKE VERITAS (DNV) is an autonomous and independent foundation with the objectives of safeguarding life,property and the environment, at sea and onshore. DNV undertakes classification, certification, and other verification andconsultancy services relating to quality of ships, offshore units and installations, and onshore industries worldwide, andcarries out research in relation to these functions.Classification NotesClassification Notes are publications that give practical information on classification of ships and other objects. Examplesof design solutions, calculation methods, specifications of test procedures, as well as acceptable repair methods for some

components are given as interpretations of the more general rule requirements. Det Norske Veritas AS May 2012

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Classification Notes - No.41.2, May 2012Changes Page 3

CHANGES

GeneralThis document supersedes CN 41.2, May 2003.

Text affected by the main changes in this edition is highlighted in red colour. However, if the changes involvea whole chapter, section or sub-section, normally only the title will be in red colour.

Main Changes 2.6 Minor changes in one formula.DET NORSKE VERITAS AS

Classification Notes - No.41.2, May 2012Contents Page 4

CONTENTS

1. Basic Principles and General Influence Factors ................................................................................. 51.1 Scope and Basic Principles .......................................................................................................................51.2 Symbols, Nomenclature and Units ...........................................................................................................51.3 Geometrical Definitions............................................................................................................................71.4 Bevel Gear Conversion Formulae and Specific Formulae .......................................................................81.5 Nominal Tangential Load, Ft, Fbt, Fmt and Fmbt.......................................................................................91.6 Application Factors, KA and KAP.............................................................................................................91.7 Load Sharing Factor, K .........................................................................................................................111.8 Dynamic Factor, Kv ................................................................................................................................111.9 Face Load Factors, KH and KF ...........................................................................................................151.10 Transversal Load Distribution Factors, KH and KF ...........................................................................201.11 Tooth Stiffness Constants, c and c .......................................................................................................201.12 Running-in Allowances ..........................................................................................................................222. Calculation of Surface Durability....................................................................................................... 232.1 Scope and General Remarks ...................................................................................................................232.2 Basic Equations.......................................................................................................................................232.3 Zone Factors ZH, ZB,D and ZM ...............................................................................................................252.4 Elasticity Factor, ZE................................................................................................................................252.5 Contact Ratio Factor, Z .........................................................................................................................262.6 Helix Angle Factor, Z ...........................................................................................................................262.7 Bevel Gear Factor, ZK ............................................................................................................................262.8 Values of Endurance Limit, Hlim and Static Strength, , .......................................................................262.9 Life Factor, ZN........................................................................................................................................272.10 Influence Factors on Lubrication Film, ZL, ZV and ZR..........................................................................272.11 Work Hardening Factor, ZW...................................................................................................................282.12 Size Factor, ZX........................................................................................................................................292.13 Subsurface Fatigue..................................................................................................................................293. Calculation of Tooth Strength ............................................................................................................ 303.1 Scope and General Remarks ...................................................................................................................303.2 Tooth Root Stresses ................................................................................................................................313.3 Tooth Form Factors YF, YFa...................................................................................................................323.4 Stress Correction Factors YS, YSa ..........................................................................................................353.5 Contact Ratio Factor Y..........................................................................................................................353.6 Helix Angle Factor Y............................................................................................................................363.7 Values of Endurance Limit, FE .............................................................................................................363.8 Mean stress influence Factor, YM...........................................................................................................373.9 Life Factor, YN .......................................................................................................................................383.10 Relative Notch Sensitivity Factor, YrelT ...............................................................................................393.11 Relative Surface Condition Factor, YRrelT .............................................................................................403.12 Size Factor, YX .......................................................................................................................................403.13 Case Depth Factor, YC............................................................................................................................403.14 Thin rim factor YB ..................................................................................................................................413.15 Stresses in Thin Rims..............................................................................................................................423.16 Permissible Stresses in Thin Rims..........................................................................................................444. Calculation of Scuffing Load Capacity .............................................................................................. 454.1 Introduction.............................................................................................................................................454.2 General Criteria.......................................................................................................................................464.3 Influence Factors.....................................................................................................................................474.4 The Flash Temperature fla ....................................................................................................................49Appendix A.Fatigue Damage Accumulation ..................................................................................................................... 57Appendix B.Application Factors for Diesel Driven Gears............................................................................................... 59Appendix C.Calculation of Pinion-Rack ........................................................................................................................... 61DET NORSKE VERITAS AS

Classification Notes - No.41.2, May 2012Sec.1. Basic Principles and General Influence Factors Page 5

1. Basic Principles and General Influence Factors1.1 Scope and Basic PrinciplesThe gear rating procedures given in this Classification Note are mainly based on the ISO6336 Part 1 to 5(cylindrical gears), and partly on ISO 10300 Part 1 to 3 (bevel gears) and ISO Technical Reports on Scuffingand Fatigue Damage Accumulation, but especially applied for marine purposes, such as marine propulsion andimportant auxiliaries onboard ships and mobile offshore units.The calculation procedures cover gear rating as limited by contact stresses (pitting, spalling or case crushing),tooth root stresses (fatigue breakage or overload breakage), and scuffing resistance. Even though no calculationprocedures for other damages such as wear, grey staining (micropitting), etc. are given, such damages may limitthe gear rating. The Classification Note applies to enclosed parallel shaft gears, epicyclic gears and bevel gears (withintersecting axis). However, open gear trains may be considered with regard to tooth strength, i.e. part 1 and 3may apply. Even pinion-rack tooth strength may be considered, but since such gear trains often are designedwith non-involute pinions, the calculation procedure of pinion-racks is described in Appendix C.Steel is the only material considered.The methods applied throughout this document are only valid for a transverse contact ratio 1 < < 2. If > 2,either special considerations are to be made, or suggested simplification may be used.All influence factors are defined regarding their physical interpretation. Some of the influence factors aredetermined by the gear geometry or have been established by conventions. These factors are to be calculatedin accordance with the equations provided. Other factors are approximations, which are clearly stated in thetext by terms as may be calculated as. These approximations are substitutes for exact evaluations where suchare lacking or too extensive for practical purposes, or factors based on experience. In principle, any suitablemethod may replace these approximations.Bevel gears are calculated on basis of virtual (equivalent) cylindrical gears using the geometry of themidsection. The virtual (helical) cylindrical gear is to be calculated by using all the factors as a real cylindricalgear with some exceptions. These exceptions are mentioned in connection with the applicable factors.Wherever a factor or calculation procedure has no reference to either cylindrical gears or bevel gears, it isgenerally valid, i.e. combined for both cylindrical and bevel. In order to minimise the volume of this Classification Note such combinations are widely used, and everywhereit is necessary to distinguish, it is clearly pointed out by local headings such as:Cylindrical gearsBevel gearsThe permissible contact stresses, tooth root stresses and scuffing load capacity depend on the safety factors asrequired in the respective Rule sections.Terms as endurance limit and static strength are used throughout this Classification Note.Endurance limit is to be understood as the fatigue strength in the range of cycles beyond the lower knee of theN curves, regardless if it is constant or drops with higher number of cycles.Static strength is to be understood as the fatigue strength in the range of cycles less than at the upper knee ofthe N curves.For gears that are subjected to a limited number of cycles at different load levels, a cumulative fatiguecalculation applies. Information on this is given in Appendix A.When the term infinite life is used, it means number of cycles in the range 108 to 1010.

1.2 Symbols, Nomenclature and UnitsThe main symbols as influence factors (K, Z, Y and X with indeces) etc. are presented in their respectiveheadings. Symbols which are not explained in their respective sections are as follows:

a = centre distance (mm).b = facewidth (mm).d = reference diameter (mm).da = tip diameter (mm).db = base diameter (mm).dw = working pitch diameter (mm).DET NORSKE VERITAS AS

ha = addendum (mm).


Index 1 refers to the pinion, 2 to the wheel.Index n refers to normal section or virtual spur gear of a helical gear.Index w refers to pitch point.Special additional symbols for bevel gears are as follows:

Index v refers to the virtual (equivalent) helical cylindrical gear.

ha0 = addendum of tool ref. to mn.hfp = dedendum of basic rack ref. to mn (= ha0).hFe = bending moment arm (mm) for tooth root stresses for application of load at the outer point of single tooth pair

contact.hFa = bending moment arm (mm) for tooth root stresses for application of load at tooth tip.HB = Brinell hardness.HV = Vickers hardness.HRC = Rockwell C hardnessmn = normal module.n = rev. per minute.NL = number of load cycles.qs = notch parameter.Ra = average roughness value (m).Ry = peak to valley roughness (m).Rz = mean peak to valley roughness (m).san = tooth top land thickness (mm).sat = transverse top land thickness (mm).sFn = tooth root chord (mm) in the critical section.spr = protuberance value of tool minus grinding stock, equal residual undercut of basic rack, ref. to mn.T = torque (Nm).u = gear ratio (per stage).v = linear speed (m/s) at reference diameter.x = addendum modification coefficient.z = number of teeth.zn = virtual number of spur teeth.n = normal pressure angle at ref. cylinder.t = transverse pressure angle at ref. cylinder.a = transverse pressure angle at tip cylinder.wt = transverse pressure angle at pitch cylinder. = helix angle at ref. cylinder.b = helix angle at base cylinder.a = helix angle at tip cylinder. = transverse contact ratio. = overlap ratio. = total contact ratio.a0 = tip radius of tool ref. to mn.fp = root radius of basic rack ref. to mn ( = a0).C = effective radius (mm) of curvature at pitch point.F = root fillet radius (mm) in the critical section.B = ultimate tensile strength (N/mm2).y = yield strength resp. 0.2% proof stress (N/mm2).

= angle between intersection axis.= angle modification (Klingelnberg)

m0 = tool module (Klingelnberg) = pitch cone angle.xsm = tooth thickness modification coefficient (midface).R = pitch cone distance (mm).

KDET NORSKE VERITAS AS

Index m refers to the midsection of the bevel gear.


1.3 Geometrical DefinitionsFor internal gearing z2, a; da2, dw2, d2 and db2 are negative, x2 is positive if da2 is increased, i.e. the numericvalue is decreased.The pinion has the smaller number of teeth, i.e.

For calculation of surface durability b is the common facewidth on pitch diameter.For tooth strength calculations b1 or b2 are facewidths at the respective tooth roots. If b1 or b2 differ much fromb above, they are not to be taken more than 1 module on either side of b.Cylindrical gears

tan t = tan n / cos tan b = tan cos ttan a = tan da / dcos a = db/dad = z mn / cos mt = mn /cos db = d cos t = dw cos wta = 0.5 (dw1 + dw2)dw1/dw2 = z1 / z2inv = tan - (radians)inv wt = inv t + 2 tan n (x1 + x2)/(z1 + z2)zn = z / (cos2 b cos )

where fw1 is to be taken as the smaller of:

and

, where fw2 is calculated as fw1substituting the values for the wheel by the values for the pinion and visa versa.

11

2 =zz

u

1

aw1fw1 T

+=

wtfw1 tan =

soi1

b1wtfw1 d

dacostan -tan =

1

2wt

a2

b2fw1 z

ztan

d

dacostan

=

2

1fw2aw1 z

z =

11 z

2T =

( ) +

+=

2sinxhm

2d2d nfpfp1fpnsoi1

21

t

nfpfplfpn2

tan)sinx(hm

+

nmsinb

=DET NORSKE VERITAS AS

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


1.4 Bevel Gear Conversion Formulae and Specific FormulaeConversion of bevel gears to virtual equivalent helical cylindrical gears is based on the bevel gear midsection.The conversion formulae are:Number of teeth:

zv1.2 = z1,2/ cos 1,2(1 + 2 = )

Gear ratio:

tan vt = tan n/ cos mtan bm = tan m cos vt

Base pitch:

Reference, pitch, diameters:

Centre distance:av = 0.5 (dv1 + dv2)

Tip diameters:dva 1.2 = dv 1,2 + 2 ham 1,2

Addenda:for gears with constant addenda (Klingelnberg):

ham 1,2 = mmn (1 + xm 1,2)for gears with variable addenda (Gleason):

ham 1,2 = ha 1,2 b/2 tan (a 1,2 1,2)

y =

C =

v =

pbt =

sat =

san =

+

( )2bwt

u1cossinua+

311 10dn60

coscosm tn

+

+

at

ninvinv

z

tanx22

d a

acossat

1

2

v

vv z

zu =

m

vtnmbtm cos

cosmp =

2,1

2,1m2.1v cos

dd


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


Addendum modification coefficients:

Base circle:dvb 1,2=dv 1,2 cos vt

Transverse contact ratio:*)

Overlap ratio*) (theoretical value for bevel gears with no crowning, but used as approximations in thecalculation procedures):

Total contact ratio:*)

(* Note that index v is left out in order to combine formulae for cylindrical and bevel gears.)Tangential speed at midsection:

Effective radius of curvature (normal section):

Length of line of contact:

1.5 Nominal Tangential Load, Ft, Fbt, Fmt and FmbtThe nominal tangential load (tangential to the reference cylinder with diameter d and perpendicular to an axialplane) is calculated from the nominal (rated) torque T transmitted by the gear set.Cylindrical gears

Bevel gears

1.6 Application Factors, KA and KAPThe application factor KA accounts for dynamic overloads from sources external to the gearing.It is distinguished between the influence of repetitive cyclic torques KA (1.6.1) and the influence of temporaryoccasional peak torques KAP (1.6.2).Calculations are always to be made with KA. In certain cases additional calculations with KAP may be

mn

1,2am2,1am2,1m m2

hhx

=

btm

vtv2

2vb2

2va2

1vb2

1va

P

sinadd0.5dd0.5 +=

nm

mmsinb

=

22 +=

3m11mt 10dn60

v =

( )2vbmvtvv

vcu1cossinua

+=

( )( )( )1if

12cosb

l 2

22bm

b 0.85, as the influence of higher modes hasto be considered, see 1.8.2. In case of significant lateral shaft flexibility (e.g. overhung mounted bevel gears),the influence of coupled bending and torsional vibrations between pinion and wheel should be considered if N 0.75 , see 1.8.2.

where:c is the actual mesh stiffness per unit facewidth, see 1.11.For gears with inactive ends of the facewidth, as e.g. due to high crowning or end relief such as often appliedfor bevel gears, the use of c in connection with determination of natural frequencies may need correction. cis defined as stiffness per unit facewidth, but when used in connection with the total mesh stiffness, it is not assimple as c b, as only a part of the facewidth is active. Such corrections are given in 1.11.mred is the reduced mass of the gear pair, per unit facewidth and referred to the plane of contact.For a single gear stage where no significant inertias are closely connected to neither pinion nor wheel, mred iscalculated as:

The individual masses per unit facewidth are calculated as

where I is the polar moment of inertia (kgmm2).The inertia of bevel gears may be approximated as discs with diameter equal the midface pitch diameter andwidth equal to b. However, if the shape of the pinion or wheel body differs much from this idealised cylinder,the inertia should be corrected accordingly.For all kind of gears, if a significant inertia (e.g. a clutch) is very rigidly connected to the pinion or wheel, itshould be added to that particular inertia (pinion or wheel). If there is a shaft piece between these inertias, thetorsional shaft stiffness alters the system into a 3-mass (or more) system. This can be calculated as in 1.8.2, butalso simplified as a 2-mass system calculated with only pinion and wheel masses.

1.8.1.2 Factors used for determination of KvNon-dimensional gear accuracy dependent parameters:

1E

1nnN =

red

1

3

1E mc

z1030n =

21

21red mm

mmm+

=

22,1b

2,12,1

)2/d(b

Im =

( )b/KKF

yf'cB

At

pptp

=

( )yF'cB ff =DET NORSKE VERITAS AS

b/KKF At


Non-dimensional tip relief parameter:

For gears of quality grade (ISO 1328) Q = 7 or coarser, Bk = 1.For gears with Q 6 and excessive tip relief, Bk is limited to max. 1.For gears (all quality grades) with tip relief of more than 2Ceff (see 4.3.2) the reduction of has to beconsidered (see 4.4.3).Where:

1.8.1.3 Kv in the subcritical range:

Kv = 1 + N KK = Cv1 Bp + Cv2 Bf + Cv3 Bk

Cv1 accounts for the pitch error influenceCv1 = 0.32Cv2 accounts for profile error influence

Cv3 accounts for the cyclic mesh stiffness variation

1.8.1.4 Kv in the main resonance range:

Running in this range should preferably be avoided, and is only allowed for high precision gears.Kv = 1 + Cv1 Bp + Cv2 Bf + Cv4 Bk

Cv4 accounts for the resonance condition with the cyclic mesh stiffness variation.

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

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

replaced by F for yf.c = the single tooth stiffness, see 1.11Ca = the amount of tip relief, see 4.3.3. In case of different tip relief on pinion and wheel, the value that results

in the greater value of Bk is to be used. If Ca is zero by design, the value of running-in tip relief Cay (see 1.12) may be used in the above formula.

Cylindrical gears: N 0.85Bevel gears: N 0.75

Cv2 = 0.34 for 2

for > 2

Cv3 = 0.23 for 2

for > 2

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

Cv4 = 0.90 for 2

for > 2

/bKKFc'C

1BAt

ak

=

0.30.57C2v

=

1.560.096C3v

=

1.440.050.57

C

4v



1.8.1.5 Kv in the supercritical range:

Special care should be taken as to influence of higher vibration modes, and/or influence of coupled bending(i.e. lateral shaft vibrations) and torsional vibrations between pinion and wheel. These influences are notcovered by the following approach.

Kv = Cv5 Bp + Cv6 Bf + Cv7 Cv5 accounts for the pitch error influence.Cv5 = 0.47Cv6 accounts for the profile error influence.

Cv7 relates the maximum externally applied tooth loading to the maximum tooth loading of ideal, accurategears operating in the supercritical speed sector, when the circumferential vibration becomes very soft.

1.8.1.6 Kv in the intermediate range:

Comments raised in 1.8.1.4 and 1.8.1.5 should be observed.Kv is determined by linear interpolation between Kv for N = 1.15 respectively 1.25 and N = 1.5 asCylindrical gears

Bevel gears

1.8.2 Multi-resonance methodFor high speed gear (v > 40 m/s), for multimesh medium speed gears, for gears with significant lateral shaftflexibility etc. it is advised to determine Kv on basis of relevant dynamic analysis.Incorporating lateral shaft compliance requires transformation of even a simple pinion-wheel system into alumped multi-mass system. It is advised to incorporate all relevant inertias and torsional shaft stiffnesses intoan equivalent (to pinion speed) system. Thereby the mesh stiffness appears as an equivalent torsional stiffness:

c b (db1/2)2 (Nm/rad)

The natural frequencies are found by solving the set of differential equations (one equation per inertia). Notethat for a gear put on a laterally flexible shaft, the coupling bending-torsionals is arranged by introducing thegear mass and the lateral stiffness with its relation to the torsional displacement and torque in that shaft.Only the natural frequency (ies) having high relative displacement and relative torque through the actual

Cylindrical gears: N 1.5Bevel gears: N 1.5

Cv6 = 0.47 for 2

for > 2

Cv7 = 0.75 for 1.5

for 1.5 < 2.5

Cv7 = 1.0 for > 2.5

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

1.740.12C6v

=

[ ] 875.0)2(sin 0.125v7C +=

( ) ( ) ( )[ ]5.1Nv15.1Nv5.1Nvv KK35.0N5.1KK

===

+=

( ) ( ) ( )[ ]5.1Nv25.1Nv5.1Nvv KK25.0N5.1KK

===

+=DET NORSKE VERITAS AS

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


Kv may be determined by means of the method mentioned in 1.8.1 thereby using N as the least favourable ratio(in case of more than one pinion-wheel dominated natural frequency). I.e. the N-ratio that results in the highestKv has to be considered.The level of the dynamic factor may also be determined on basis of simulation technique using numeric timeintegration with relevant tooth stiffness variation and pitch/profile errors.

1.9 Face Load Factors, KH and KFThe face load factors, KH for contact stresses and for scuffing, KF for tooth root stresses, account for non-uniform load distribution across the facewidth.KH is defined as the ratio between the maximum load per unit facewidth and the mean load per unit facewidth.KF is defined as the ratio between the maximum tooth root stress per unit facewidth and the mean tooth rootstress per unit facewidth. The mean tooth root stress relates to the considered facewidth b1 respectively b2.Note that facewidth in this context is the design facewidth b, even if the ends are unloaded as often applies toe.g. bevel gears.The plane of contact is considered.

1.9.1 Relations between KH and KF

where h/b is the ratio tooth height/facewidth. The maximum of h1/b1, and h2/b2 is to be used, but not higherthan 1/3. For double helical gears, use only the facewidth of one helix.If the tooth root facewidth (b1 or b2) is considerably wider than b, the value of KF(1or2) is to be speciallyconsidered as it may even exceed KH.E.g. in pinion-rack lifting systems for jack up rigs, where b = b2 mn and b1 3 mn, the typical KH KF2 1 and KF1 1.3.1.9.2 Measurement of face load factorsPrimarily,KF may be determined by a number of strain gauges distributed over the facewidth. Such strain gauges mustbe put in exactly the same position relative to the root fillet. Relations in 1.9.1 apply for conversion to KH.Secondarily,KH may be evaluated by observed contact patterns on various defined load levels. It is imperative that thevarious test loads are well defined. Usually, it is also necessary to evaluate the elastic deflections. Some teethat each 90 degrees are to be painted with a suitable lacquer. Always consider the poorest of the contact patterns.After having run the gear for a suitable time at test load 1 (the lowest), observe the contact pattern with respectto extension over the facewidth. Evaluate that KH by means of the methods mentioned in this section. Proceedin the same way for the next higher test load etc., until there is a full face contact pattern. From these data, theinitial mesh misalignment (i.e. without elastic deflections) can be found by extrapolation, and then also the KHat design load can be found by calculation and extrapolation. See example.

HF KK =2(h/b)h/b1

1++DET NORSKE VERITAS AS

Figure 1.1 Example of experimental determination of KHb


It must be considered that inaccurate gears may accumulate a larger observed contact pattern than the actualsingle mesh to mesh contact patterns. This is particularly important for lapped bevel gears. Ground or hardmetal hobbed bevel gears are assumed to present an accumulated contact pattern that is practically equal theactual single mesh to mesh contact patterns. As a rough guidance the (observed) accumulated contact patternof lapped bevel gears may be reduced by 10% in order to assess the single mesh to mesh contact pattern whichis used in 1.9.9.

1.9.3 Theoretical determination of KHThe methods described in 1.9.3 to 1.9.8 may be used for cylindrical gears. The principles may to some extentalso be used for bevel gears, but a more practical approach is given in 1.9.9.General: For gears where the tooth contact pattern cannot be verified during assembly or under load, allassumptions are to be well on the safe side.KH is to be determined in the plane of contact.The influence parameters considered in this method are:

mean mesh stiffness c (see 1.11) (if necessary, also variable stiffness over b) mean unit load Fm/b = Fbt KA K Kv/b (for double helical gears, see 1.7 for use of K) misalignment fsh due to elastic deflections of shafts and gear bodies (both pinion and wheel) misalignment fdefl due to elastic deflections of and working positions in bearings misalignment fbe due to bearing clearance tolerances misalignment fma due to manufacturing tolerances helix modifications as crowning, end relief, helix correction running in amount y (see 1.12).In practice several other parameters such as centrifugal expansion, thermal expansion, housing deflection, etc.contribute to KH. However, these parameters are not taken into account unless in special cases when beingconsidered as particularly important.When all or most of the a.m. parameters are to be considered, the most practical way to determine KH is bymeans of a graphical approach, described in 1.9.3.1.If c can be considered constant over the facewidth, and no helix modifications apply, KH can be determinedanalytically as described in 1.9.3.2.

1.9.3.1 Graphical methodThe graphical method utilises the superposition principle, and is as follows:

Calculate the mean mesh deflection M as a function of Fm /b and c, see 1.11. Draw a base line with length b, and draw up a rectangular with height M. (The area M b is proportional

to the transmitted force). Calculate the elastic deflection fsh in the plane of contact. Balance this deflection curve around a zero line,

so that the areas above and below this zero line are equal.

Figure 1.2 fsh balanced around zero line

Superimpose these ordinates of the fsh curve to the previous load distribution curve. (The area under thisnew load distribution curve is still M b.).

Calculate the bearing deflections and/or working positions in the bearings and evaluate the influence fdeflin the plane of contact. This is a straight line and is balanced around a zero line as indicated in Fig. 1.4, butDET NORSKE VERITAS AS

with one distinct direction. Superimpose these ordinates to the previous load distribution curve.


The amount of crowning, end relief or helix correction (defined in the plane of contact) is to be balancedaround a zero line similarly to fsh.

Figure 1.3 Crowning Cc balanced around zero line

Superimpose these ordinates to the previous load distribution curve. In case of high crowning etc. as e.g.often applied to bevel gears, the new load distribution curve may cross the base line (the real zero line). Theresult is areas with negative load that is not real, as the load in those areas should be zero. Thus correctiveactions must be made, but for practical reasons it may be postponed to after next operation.

The amount of initial mesh misalignment, fma + fbe (defined in the plane of contact), is to be balancedaround a zero line. If the direction of fma + fbe is known (due to initial contact check), or if the direction offbe is known due to design (e.g. overhang bevel pinion), this should be taken into account. If directionunknown, the influence of fma + fbe in both directions as well as equal zero, should be considered.

Figure 1.4 fma+fbe in both directions, balanced around zero line.Superimpose these ordinates to the previous load distribution curve. This results in up to 3 different curves, ofwhich the one with the highest peak is to be chosen for further evaluation.

If the chosen load distribution curve crosses the base line (i.e. mathematically negative load), the curve isto be corrected by adding the negative areas and dividing this with the active facewidth. The (constant)ordinates of this rectangular correction area are to be subtracted from the positive part of the loaddistribution curve.It is advisable to check that the area covered under this new load distribution curve is still equal M b.

If c cannot be considered as constant over b, then correct the ordinates of the load distribution curve withthe local (on various positions over the facewidth) ratio between local mesh stiffness and average meshstiffness c (average over the active facewidth only).Note that the result is to be a curve that covers the same area M b as before.

The influence of running in y is to be determined as in 1.12 whereby the value for Fx is to be taken astwice the distance between the peak of the load distribution curve and M.

Determine

1.9.3.2 Simplified analytical method for cylindrical gearsThe analytical approach is similar to 1.9.3.1 but has a more limited application as c is assumed constant overthe facewidth and no helix modification applies.

Calculate the elastic deflection fsh in the plane of contact. Balance this deflection curve around a zero line,

M

H

ycurveofpeakK


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


Calculate the initial mesh alignment as

The negative signs may only be used if this is justified and/or verified by a contact pattern test. Otherwise,always use positive signs. If a negative sign is justified, the value of Fx is not to be taken less than thelargest of each of these elements.

Calculate the effective mesh misalignment asFy = Fx - y (y see 1.12)

Determine

or

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

1.9.4 Determination of fshfsh is the mesh misalignment due to elastic deflections. Usually it is sufficient to consider the combined meshdeflection of the pinion body and shaft and the wheel shaft. The calculation is to be made in the plane of contact(of the considered gear mesh), and to consider all forces (incl. axial) acting on the shafts. Forces from othermeshes can be parted into components parallel respectively vertical to the considered plane of contact. Forcesvertical to this plane of contact have no influence on fsh.It is advised to use following diameters for toothed elements:

Usually, fsh is calculated on basis of an evenly distributed load. If the analysis of KH shows a considerablemaldistribution in term of hard end contact, or if it is known by other reasons that there exists a hard endcontact, the load should be correspondingly distributed when calculating fsh. In fact, the whole KH procedurecan be used iteratively. 2 to 3 iterations will be enough, even for almost triangular load distributions.

1.9.5 Determination of fdeflfdefl is the mesh misalignment in the plane of contact due to bearing deflections and working positions (housingdeflection may be included if determined). First the journal working positions in the bearings are to be determined. The influence of external moments andforces must be considered. This is of special importance for twin pinion single output gears with all 3 shafts inone plane.For rolling bearings fdefl is further determined on basis of the elastic deflection of the bearings. An elasticbearing deflection depends on the bearing load and size and number of rolling elements. Note that the bearingclearance tolerances are not included here. For fluid film bearings fdefl is further determined on basis of the lift and angular shift of the shafts due tolubrication oil film thickness. Note that fbe takes into account the influence of the bearing clearance tolerance. When working positions, bearing deflections and oil film lift are combined for all bearings, the angularmisalignment as projected into the plane of the contact is to be determined. fdefl is this angular misalignment(radians) times the facewidth.

1.9.6 Determination of fbefbe is the mesh misalignment in the plane of contact due to tolerances in bearing clearances. In principle fbe andfdefl could be combined. But as fdefl can be determined by analysis and has a distinct direction, and fbe isdependent on tolerances and in most cases has no distinct direction (i.e. tolerance), it is practicable to separatethese two influences.Due to different bearing clearance tolerances in both pinion and wheel shafts the two shaft axis will have an

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

deflbemashx ffffF =

2KforF2

bFc1K H

m

H +=

2KforF

bFc2K H

m

H >=DET NORSKE VERITAS AS

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


fbe is the facewidth times this angular misalignment. Note that fbe may have a distinct direction or be given asa tolerance, or a combination of both. For combination of tolerance it is adviced to use

fbe is particularly important for overhang designs, for gears with widely different kinds of bearings on eachside, and when the bearings have wide tolerances on clearances. In general it shall be possible to replacestandard bearings without causing the real load distribution to exceed the design premises. For slow speed gearswith journal bearings, the expected wear should also be considered.

1.9.7 Determination of fmafma is the mesh misalignment due to manufacturing tolerances (helix slope deviation) of pinion fH1, wheelfH2 and housing bore.For gear without specifically approved requirements to assembly control, the value of fma is to be determined as

For gears with specially approved assembly control, the value of fma will depend on those specificrequirements.

1.9.8 Comments to various gear typesFor double helical gears, KH is to be determined for both helices. Usually an even load share between thehelices can be assumed. If not, the calculation is to be made as described in 1.7.1.For planetary gears the free floating sun pinion suffers only twist, no bending. It must be noted that the totaltwist is the sum of the twist due to each mesh. If the value of K 1, this must be taken into account whencalculating the total sun pinion twist (i.e. twist calculated with the force per mesh without K, and multipliedwith the number of planets).When planets are mounted on spherical bearings, the mesh misalignments sun-planet respectively planet-annulus will be balanced. I.e. the misalignment will be the average between the two theoretical individualmisalignments. The faceload distribution on the flanks of the planets can take full advantage of this. However,as the sun and annulus mesh with several planets with possibly different lead errors, the sun and annulus cannotobtain the above mentioned advantage to the full extent.

1.9.9 Determination of KH for bevel gearsIf a theoretical approach similar to 1.9.3 to 1.9.8 is not documented, the following may be used.

beff / b represents the relative active facewidth (regarding lapped gears, see 1.9.2 last part).Higher values than beff / b = 0.90 are normally not to be used in the formula.For dual directional gears it may be difficult to obtain a high beff / b in both directions. In that case the smallerbeff / b is to be used.Ktest represents the influence of the bearing arrangement, shaft stiffness, bearing stiffness, housing stiffness etc.on the faceload distribution and the verification thereof. Expected variations in length- and height-wise toothprofile is also accounted for to some extent.

a) Ktest = 1For ground or hard metal hobbed gears with the specified contact pattern verified at full rating or at fulltorque slow turning at a condition representative for the thermal expansion at normal operation.It also applies when the bearing arrangement/support has insignificant elastic deflections and thermal axialexpansion. However, each initial mesh contact must be verified to be within acceptance criteria that arecalibrated against a type test at full load. Reproduction of the gear tooth length- and height-wise profilemust also be verified. This can be made through 3D measurements or by initial contact movements causedby defined axial offsets of the pinion (tolerances to be agreed upon).

b) Ktest = 1 + 0.4(beff/b0.6)For designs with possible influence of thermal expansion in the axial direction of the pinion. The initialmesh contact verified with low load or spin test where the acceptance criteria are calibrated against a type

........fff 22be21bebe ++=

22H

21Hma fff +=

testeff

H Kbb85.185.1K


test at full load.


c) Ktest = 1.2if mesh is only checked by toolmakers blue or by spin test contact. For gears in this category beff./b > 0.85is not to be used in the calculation.

1.10 Transversal Load Distribution Factors, KH and KFThe transverse load distribution factors, KH for contact stresses and for scuffing, KF for tooth root stressesaccount for the effects of pitch and profile errors on the transversal load distribution between 2 or more pairsof teeth in mesh.The following relations may be used:Cylindrical gears:

valid for 2

valid for > 2 where:

Limitations of KH and KF:If the calculated values for

KF = KH < 1, use KF = KH = 1.0

Bevel gears:For ground or hard metal hobbed gears, KF = KH = 1For lapped gears, KF = KH = 1.1

1.11 Tooth Stiffness Constants, c and cThe tooth stiffness is defined as the load which is necessary to deform one or several meshing gear teeth having1 mm facewidth by an amount of 1 m, in the plane of contact.c is the maximum stiffness of a single pair of teeth.c is the mean value of the mesh stiffness in a transverse plane (brief term: mesh stiffness).

FtH = Ft KA K Kv KHc = See 1.11 = See 1.12fpt = Maximum single pitch deviation (m) of pinion or wheel, or maximum total profile form deviation F of

pinion or wheel if this is larger than the maximum single pitch deviation.Note: In case of adequate equivalent tip relief adapted to the load, half of the above mentioned fpt can be introduced.

A tip relief is considered adequate when the average of Ca1 and Ca2 is within 40% of the value of Ceff in 4.3.2:

If the calculated value of use

If the calculated value of use

where

(for n see 3.3.1.c)

( )

+==tH

ptHF F

byfc0.40.9

2

KK

( ) ( )tH

pt

HF F

byfc

120.40.9KK

+==

2

HZ

K > 2

HZ

K =

F Y

K >

F Y

K =

n0.750.25Y +=DET NORSKE VERITAS AS

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


Cylindrical gearsThe real stiffness is a combination of the progressive Hertzian contact stiffness and the linear tooth bendingstiffnesses. For high unit loads the Hertzian stiffness has little importance and can be disregarded. Thisapproach is on the safe side for determination of KH and KH. However, for moderate or low loads Kv maybe underestimated due to determination of a too high resonance speed.The linear approach is described in A.An optional approach for inclusion of the non-linear stiffness is described in B.A. The linear approach.

and

where:

(for internal gears, use zn2 equal infinite and x2 = 0).ha0 = hfp for all practical purposes.CR considers the increased flexibility of the wheel teeth if the wheel is not a solid disc, and may be calculatedas:

where:

The formula is valid for bs / b 0.2 and sR/mn 1. Outside this range of validity and if the web is not centrallypositioned, CR has to be specially considered.Note:CR is the ratio between the average mesh stiffness over the facewidth and the mesh stiffness of a gear pair ofsolid discs. The local mesh stiffness in way of the web corresponds to the mesh stiffness with CR = 1. The localmesh stiffness where there is no web support will be less than calculated with CR above. Thus, e.g. a centrallypositioned web will have an effect corresponding to a longitudinal crowning of the teeth. See also 1.9.3.1regarding KH.B. The non-linear approach.In the following an example is given on how to consider the non-linearity.The relation between unit load F/b as a function of mesh deflection is assumed to be a progressive curve upto 500 N/mm and from there on a straight line. This straight line when extended to the baseline is assumed tointersect at 10m.With these assumptions the unit force F/b as a function of mesh deflection can be expressed as:

bs = thickness of a central websR = average thickness of rim (net value from tooth root to inside of rim).

for

for

BRCCqcos0.8c =

( )0.250.75cc +=

( )[ ]n02a01aB 200.0212hh

1.20.51C

++=

12n1n

x0.00635z

0.25791z

0.155510.04723q ++= 122n

22

1n

1 x00529.0z

x0.24188x0.00193z

x0.11654+ + 0.00182 x22

( )( )nR m5/s

sR e5

/bbln1C +=

( )10KbF

= 500bF

>

=

500F/b10K

bF 500

bF


KA K for determination of Kv.KA K Kv for determination of KH.KA K Kv KH for determination of KH. = mesh deflection (m)K = applicable stiffness (c' or c) Use of stiffnesses for KV, KH and KHFor calculation of Kv and KH the stiffness is calculated as follows:When F/b < 500, the stiffness is determined as

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

When F / b > 500, the stiffness is c' or c.For calculation of KH the mesh deflection is used directly,

or an equivalent stiffness determined as .

Bevel gearsIn lack of more detailed relationship between stiffness and geometry the following may be used.

beff not to be used in excess of 0.85 b in these formulae.Bevel gears with heightwise and lengthwise crowning have progressive mesh stiffness. The values mentionedabove are only valid for high loads. They should not be used for determination of Ceff (see 4.3.2) or KH (see1.9.3.1).

1.12 Running-in AllowancesThe running-in allowances account for the influence of running-in wear on the various error elements.y respectively y are the running-in amounts which reduce the influence of pitch and profile errors,respectively influence of localised faceload.Cay is defined as the running-in amount that compensates for lack of tip relief.The following relations may be used:For not surface hardened steel

with etc. (N/mm), i.e. unit load incorporating the relevant factors as:At KKbF

bF

=

F/b------------

50010F/b10

K10F/b +++=

Fb ----------

b0.85b

13c eff=b0.85

b16c eff =

ptHlim

f160y =

xlimH

f320y


Classification Notes - No.41.2, May 2012Sec.2. Calculation of Surface Durability Page 23

with the following upper limits:

For surface hardened steely = 0.075 fpt but not more than 3 for any speedy = 0.15 Fx but not more than 6 for any speedFor all kinds of steel

When pinion and wheel material differ, the following applies:

2. Calculation of Surface Durability2.1 Scope and General RemarksPart 2 includes the calculations of flank surface durability as limited by pitting, spalling, case crushing andsubsurface yielding. Endurance and time limited flank surface fatigue is calculated by means of 2.2 to 2.12. Ina way also tooth fractures starting from the flank due to subsurface fatigue is included through the criteria in2.13.Pitting itself is not considered as a critical damage for slow speed gears. However, pits can create a severe notcheffect that may result in tooth breakage. This is particularly important for surface hardened teeth, but also forhigh strength through hardened teeth. For high-speed gears, pitting is not permitted.Spalling and case crushing are considered similar to pitting, but may have a more severe effect on toothbreakage due to the larger material breakouts, initiated below the surface. Subsurface fatigue is considered in2.13.For jacking gears (self-elevating offshore units) or similar slow speed gears designed for very limited life, themax. static (or very slow running) surface load for surface hardened flanks is limited by the subsurface yieldstrength.For case hardened gears operating with relatively thin lubrication oil films, grey staining (micropitting) may bethe limiting criterion for the gear rating. Specific calculation methods for this purpose are not given here, butare under consideration for future revisions. Thus depending on experience with similar gear designs,limitations on surface durability rating other than those according to 2.2 to 2.13 may be applied.

2.2 Basic EquationsCalculation of surface durability (pitting) for spur gears is based on the contact stress at the inner point of singlepair contact or the contact at the pitch point, whichever is greater.Calculation of surface durability for helical gears is based on the contact stress at the pitch point.

V 5 m/s 5-10 m/s > 10 m/sy max none

y max none

Use the larger of fpt1 - y1 and fpt2 - y2 to replace fpt - y in the calculation of KH see 1.10 and Kv see 1.8.

Use in the calculation of KH see 1.9.

Use in the calculation of Kv see 1.8.

Use in the scuffing calculation see 4 if no design tip relief is foreseen.

limH

12800 limH

6400

limH

25600 limH

12800

5.145.189718

1C2

limHay +

=

( )21 yy21y +=

( )2ay1aya CC21C +=

( )2ay1ay2a1a CC21CC +==DET NORSKE VERITAS AS

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


Calculation of surface durability for spiral bevel gears is based on the contact stress at the midpoint of the zoneof contact.Alternatively for bevel gears the contact stress may be calculated with the program BECAL. In that case, KAand Kv are to be included in the applied tooth force, but not KH and KH. The calculated (real) Hertzianstresses are to be multiplied with ZK in order to be comparable with the permissible contact stresses.The contact stresses calculated with the method in part 2 are based on the Hertzian theory, but do not alwaysrepresent the real Hertzian stresses.The corresponding permissible contact stresses HP are to be calculated for both pinion and wheel.

2.2.1 Contact stress

Cylindrical gears

where:

Ft, KA , K , Kv , KH , KH , see 1.5 to 1.10.d1, b, u, see 1.2 to 1.5.

Bevel gears

where:1.05 is a correlation factor to reach real Hertzian stresses (when ZK = 1)ZE, KA etc. see above.

ZM = mid-zone factor, see 2.3.3.ZK = bevel gear factor, see 2.7.

Fmt, dv1, uv, see 1.2 1.5.It is assumed that the heightwise crowning is chosen so as to result in the maximum contact stresses at or nearthe midpoint of the flanks.

2.2.2 Permissible contact stress

where:

ZB,D = Zone factor for inner point of single pair contact for pinion resp. wheel (see 2.3.2).ZH = Zone factor for pitch point (see 2.3.1).ZE = Elasticity factor (see 2.4).Z = Contact ratio factor (see 2.5).Z = Helix angle factor (see 2.6).

H lim = Endurance limit for contact stresses (see 2.8).ZN = Life factor for contact stresses (see 2.9).SH = Required safety factor according to the rules.ZL,Zv,ZR = Oil film influence factors (see 2.10).ZW = Work hardening factor (see 2.11).ZX = Size factor (see 2.12).

( )HHvA

1

tEHDB,H KKKKKbud1uF

ZZZZZ +=

( )HHvA

v1v

vmtKEMH KKKKKbud

1uFZZZ1.05 +=

XWRvLH

NHlimHP ZZZZZS

Z =DET NORSKE VERITAS AS


2.3 Zone Factors ZH, ZB,D and ZM2.3.1 Zone factor ZHThe zone factor, ZH, accounts for the influence on contact stresses of the tooth flank curvature at the pitch pointand converts the tangential force at the reference cylinder to the normal force at the pitch cylinder.

2.3.2 Zone factors ZB,D The zone factors, ZB,D, account for the influence on contact stresses of the tooth flank curvature at the innerpoint of single pair contact in relation to ZH. Index B refers to pinion D to wheel.For 1, ZB,D = 1For internal gears, ZD = 1For = 0 (spur gears)

If ZB < 1, use ZB = 1If ZD < 1, use ZD = 1For 0 < < 1ZB,D = ZB,D (for spur gears) (ZB,D (for spur gears) 1)2.3.3 Zone factor ZMThe mid-zone factor ZM accounts for the influence of the contact stress at the mid point of the flank and appliesto spiral bevel gears.

This factor is the product of ZH and ZM-B in ISO 10300 with the condition that the heightwise crowning issufficient to move the peak load towards the midpoint.

2.3.4 Inner contact pointFor cylindrical or bevel gears with very low number of teeth the inner contact point (A) may be close to thebase circle. In order to avoid a wear edge near A, it is required to have suitable tip relief on the wheel.

2.4 Elasticity Factor, ZEThe elasticity factor, ZE, accounts for the influence of the material properties as modulus of elasticity andPoissons ratio on the contact stresses.For steel against steel ZE = 189.8

wtt2

wtbH sincos

coscos2Z =

( )

=

2

2

2b

2a

1

2

1b

1a

wtB

z211

dd

z21

dd

tanZ

( )

=

1

2

1b

1a

2

2

2b

2a

wtD

z211

dd

z21

dd

tanZ

=

btm2

2vb2

2vabtm2

1vb2val

2v1vvtbmM

pddpdd

ddtancos2ZDET NORSKE VERITAS AS


2.5 Contact Ratio Factor, ZThe contact ratio factor Z accounts for the influence of the transverse contact ratio and the overlap ratio on the contact stresses.

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

2.7 Bevel Gear Factor, ZKThe bevel gear factor accounts for the difference between the real Hertzian stresses in spiral bevel gears andthe contact stresses assumed responsible for surface fatigue (pitting). ZK adjusts the contact stresses in such away that the same permissible stresses as for cylindrical gears may apply.The following may be used:ZK = 0.80

2.8 Values of Endurance Limit, Hlim and Static Strength, ,

Hlim is the limit of contact stress that may be sustained for 5107 cycles, without the occurrence of progressive pitting.For most materials 5107 cycles are considered to be the beginning of the endurance strength range or lowerknee of the -N curve. (See also Life Factor ZN). However, for nitrided steels 2106 apply.For this purpose, pitting is defined by

for not surface hardened gears: pitted area 2% of total active flank area. for surface hardened gears: pitted area 0.5% of total active flank area, or 4% of one particular tooth

flank area.

and and are the contact stresses which the given material can withstand for 105 respectively 103cycles without subsurface yielding or flank damages as pitting, spalling or case crushing when adequate casedepth applies.The following listed values for Hlim, and may only be used for materials subjected to a qualitycontrol as the one referred to in the rules.Results of approved fatigue tests may also be used as the basis for establishing these values.The defined survival probability is 99%.

for 1

for < 1

Hlim

Alloyed case hardened steels (surface hardness 58-63 HRC):- of specially approved high grade:- of normal grade:

16501500

25002400

31003100

Nitrided steel of approved grade, gas nitrided (surface hardness 700 to 800 HV): 1250 1.3 Hlim 1.3 HlimAlloyed quenched and tempered steel, bath or gas nitrided(surface hardness 500 to 700 HV): 1000 1.3 Hlim 1.3 HlimAlloyed, flame or induction hardened steel (surface hardness 500 to 650 HV): 0.75 HV + 750 1.6 Hlim 4.5 HVAlloyed quenched and tempered steel: 1.4 HV + 350 1.6 Hlim 4.5 HVCarbon steel: 1.5 HV + 250 1.6 Hlim 1.6 Hlim

1Z =

( )

1

34

Z +

=

Z1

cos----------------=

H105 H103

H105 H103

H105 H103

H105H103DET NORSKE VERITAS AS

These values refer to forged or hot rolled steel. For cast steel the values for Hlim are to be reduced by 15%.


2.9 Life Factor, ZNThe life factor, ZN, takes account of a higher permissible contact stress if only limited life (number of cycles,NL) is demanded or lower permissible contact stress if very high number of cycles apply.If this is not documented by approved fatigue tests, the following method may be used:For all steels except nitrided:

I.e. ZN = 0.92 for 1010 cycles. The ZN = 1 from 5107 on, may only be used when the material cleanliness is of approved high grade (see RulesPt.4 Ch.2) and the lubrication is optimised by a specially approved filtering process.

(but not less than ZN105)

For nitrided steels:

I.e. ZN = 0.92 for 1010 cycles. The ZN = 1 from 2106 on, may only be used when the material cleanliness is of approved high grade (see RulesPt4 Ch2) and the lubrication is optimised by a specially approved filtering process.

Note that when no index indicating number of cycles is used, the factors are valid for 5107 (respectively 2106for nitriding) cycles.

2.10 Influence Factors on Lubrication Film, ZL, ZV and ZRThe lubricant factor, ZL, accounts for the influence of the type of lubricant and its viscosity, the speed factor,ZV, accounts for the influence of the pitch line velocity and the roughness factor, ZR, accounts for influence ofthe surface roughness on the surface endurance capacity.

NL 5107: ZN = 1 or

105 < NL < 5107:

NL = 105:

103 < NL < 105:

NL 103

NL 2 106: ZN = 1 or

105 < NL < 2106

NL 105

0157.0

L

7

N N105Z

=

510NlogZ0.37

L

7

N N105Z

=

WXRVLHlim

WstX10H1010NN ZZZZZ

ZZZZ

55

5=

)/Z(Zlog0.5

L

5

10NN

5N103N10

5N10ZZ

==

WXRVLlimH

Wst10X10H10NN ZZZZZ

ZZZZ

33

3

==

0098,0

L

6

N N102Z

=

510NZlog7686.0

L

6

N N102Z

=

XWRVL

10XWst10NN ZZZZZ

ZZ1.3ZZ

55 ==DET NORSKE VERITAS AS


The following methods may be applied in connection with the endurance limit:

where:

For NL 105: ZL ZV ZR = 1.0

2.11 Work Hardening Factor, ZWThe work hardening factor, ZW, accounts for the increase of surface durability of a soft steel gear when meshingthe soft steel gear with a surface hardened or substantially harder gear with a smooth surface.The following approximation may be used for the endurance limit:Surface hardened steel against not surface hardened steel:

where:HB = the Brinell hardness of the soft memberFor HB > 470, use HB = 470For HB < 130, use HB = 130RZeq = equivalent roughness

If RZeq > 16, then use RZeq = 16If RZeq < 1.5, then use RZeq = 1.5where:

If values of ZW < 1 are evaluated, ZW = 1 should be used for flank endurance. However, the low value for ZWmay indicate a potential wear problem.

Surface hardened steels Not surface hardened steels

ZL

ZV

ZR

40 = Kinematic oil viscosity at 40C (mm2/s).For case hardened steels the influence of a high bulk temperature (see 4. Scuffing) should be considered. E.g. bulk temperatures in excess of 120C for long periods may cause reduced flank surface endurance limits.For values of 40 > 500, use 40 = 500.

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

RZrel =

RZ = Mean peak to valley roughness (m) (DIN definition) (roughly RZ = 6 Ra)

RZH = surface roughness of the hard member before run in. RZS = surface roughness of the soft member before run in40 = see 2.10.

( )240/1342.136.091.0

++ ( )240/1342.1

68.083.0+

+

( )v/328.014.093.0

++ ( )v/328.0

30.085.0+

+

08.0

ZrelR3

15.0

ZrelR3

( ) 31

cZ2Z1

10RR5.0

+

15.0

ZeqW R

31700

130HB2.1Z

=

ZeqDET NORSKE VERITAS AS


Through hardened pinion against softer wheel:

For u > 20, use u = 20For static strength (< 105 cycles):Surface hardened against not surface hardenedZWst = 1.05Through hardened pinion against softer wheelZWst = 1

2.12 Size Factor, ZXThe size factor accounts for statistics indicating that the stress levels at which fatigue damage occurs decreasewith an increase of component size, as a consequence of the influence on subsurface defects combined withsmall stress gradients, and of the influence of size on material quality.ZX may be taken unity provided that subsurface fatigue for surface hardened pinions and wheels is considered,e.g. as in the following subsection 2.13.

2.13 Subsurface FatigueThis is only applicable to surface hardened pinions and wheels. The main objective is to have a subsurfacesafety against fatigue (endurance limit) or deformation (static strength) which is at least as high as the safetySH required for the surface. The following method may be used as an approximation unless otherwisedocumented.The high cycle fatigue (>3106 cycles) is assumed to mainly depend on the orthogonal shear stresses. Staticstrength ( 400, the t400 is to be replaced by a fictive t400 = 1.6 t550).

In addition the specified surface hardness is not to be less than the max necessary hardness (at tz = 0.5aH). This

For use ZW = 1

For use

( )

+= 0.00829HBHB0.008981u1Z

2

1W

2.1HBHB

2

1

7.1HBHB

2

1 > 1.7HBHB

2

1=

+

=Z

1

HHR

KHHR Z

1 =DET NORSKE VERITAS AS

applies to all hardening methods.

Classification Notes - No.41.2, May 2012Sec.3. Calculation of Tooth Strength Page 30

For high cycle fatigue (>3 106 cycles) the following applies:

Where aH is half the hertzian contact width multiplied by an empirical factor of 1.2 that takes into account thepossible influence of reduced compressive residual stresses (or even tensile residual stresses) on the localfatigue strength. If any of the specified hardness depths including the surface hardness is below the curve described by HV = f(tz), the actual safety factor against subsurface fatigue is determined as follows:

reduce SH stepwise in the formula for HV and aH until all specified hardness depths and surface hardnessbalance with the corrected curve. The safety factor obtained through this method is the safety against sub-surface fatigue.

For static strength (


For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear.Subsequently the stress is converted to load application at the mid point of the flank due to the heightwisecrowning.Bevel gears may also be calculated with the program BECAL. In that case, KA and Kv are to be included in theapplied tooth force, but not KF and KF.In case of a thin annulus or a thin gear rim etc, radial cracking can occur rather than tangential cracking (fromroot fillet to root fillet). Cracking can also start from the compression fillet rather than the tension fillet. Forrim thickness sR < 3.5mn a special calculation procedure is given in 3.15 and 3.16, and a simplified procedurein 3.14.A tooth breakage is often the end of the life of a gear transmission. Therefore, a high safety SF against breakageis required.It should be noted that this part 3 does not cover fractures caused by:

oil holes in the tooth root space wear steps on the flank flank surface distress such as pits, spalls or grey staining.

Especially the latter is known to cause oblique fractures starting from the active flank, predominately in spiralbevel gears, but also sometimes in cylindrical gears.Specific calculation methods for these purposes are not given here, but are under consideration for futurerevisions. Thus, depending on experience with similar gear designs, limitations other than those outlined in part3 may be applied.

3.2 Tooth Root StressesThe local tooth root stress is defined as the max. principal stress in the tooth root caused by application of thetooth force. I.e. the stress ratio R = 0. Other stress ratios such as for e.g. idler gears (R -1.2), shrunk on gearrims (R > 0), etc. are considered by correcting the permissible stress level.

3.2.1 Local tooth root stress The local tooth root stress for pinion and wheel may be assessed by strain gauge measurements or FEcalculations or similar. For both measurements and calculations all details are to be agreed in advance.Normally, the stresses for pinion and wheel are calculated as:Cylindrical gears:

where:

YF = Tooth form factor (see 3.3).YS = Stress correction factor (see 3.4).Y = Helix angle factor (see 3.6).Ft, KA, K, Kv, KF, KF, see 1.5 1.10.b, see 1.3.Bevel gears:

where:

YFa = Tooth form factor, see 3.3.YSa = Stress correction factor, see 3.4.Y = Contact ratio factor, see 3.5.

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

FFvASFn

tF K K K K K Y Y Ym b

F =

FFvASaFamn

mtF K K K K K Y Y Ym b

F =DET NORSKE VERITAS AS

b, see 1.3.


3.2.2 Permissible tooth root stressThe permissible local tooth root stress for pinion respectively wheel for a given number of cycles, N, is:

Note that all these factors YM etc. are applicable to 3106 cycles when used in this formula for FP. Theinfluence of other number of cycles on these factors is covered by the calculation of YN.where:

3.3 Tooth Form Factors YF, YFaThe tooth form factors YF and YFa take into account the influence of the tooth form on the nominal bendingstress.YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and isused for cylindrical gears.YFa applies to load application at the tooth tip and is used for bevel gears.Both YF and YFa are based on the distance between the contact points of the 30-tangents at the root fillet of thetooth profile for external gears, respectively 60 tangents for internal gears.

Figure 3.1 External tooth in normal section

FE = Local tooth root bending endurance limit of reference test gear (see 3.7).YM = Mean stress influence factor which accounts for other loads than constant load direction, e.g. idler gears, tem-

porary change of load direction, pre-stress due to shrinkage, etc. (see 3.8).YN = Life factor for tooth root stresses related to reference test gear dimensions (see 3.9).SF = Required safety factor according to the rules.YrelT = Relative notch sensitivity factor of the gear to be determined, related to the reference test gear (see 3.10).YRrelT = Relative (root fillet) surface condition factor of the gear to be determined, related to the reference test gear

(see 3.11).YX = Size factor (see 3.12).YC = Case depth factor (see 3.13).

CXRrelTrelTF

NMFEFP YYYYS

YY


Figure 3.2 Internal tooth in normal section


Definitions:

In the case of helical gears, YF and YFa are determined in the normal section, i.e. for a virtual number of teeth.YFa differs from YF by the bending moment arm hFa and Fan and can be determined by the same procedureas YF with exception of hFe and Fan . For hFa and Fan all indices e will change to a (tip).The following formulae apply to cylindrical gears, but may also be used for bevel gears when replacing:mn with mnmzn with zvnt with vt with m

Fig. 3.3 Dimensions and basic rack profile of the teeth (finished profile)Tool and basic rack data such as hfP, fp and spr etc. are referred to mn, i.e. dimensionless.3.3.1 Determination of parameters

where

z0 = number of teeth of pinion cutterx0 = addendum modification coefficient of pinion cutterhfP= addendum of pinion cutterfP= tip radius of pinion cutter.

with undercut without undercut

For external gears

For internal gears

n

2

n

Fn

enFn

Fe

F

cosms

cosmh6

Y

=

n

2

n

Fn

anFn

Fa

Fa

cosms

cosmh6

Y

=

( )n

n

prnfPnfP m cos

ssin 1'tan h4E

=

fPfP ' =( )

0z

1.95fPfP0

fPfP 1.0363.156hx'

++=

xh'G fpfp +=

E2H


m2z 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 Fen. The total influence of xsm on YFa Ysa can be approximatedby only adding 2 xsm to sFn / mn.For internal gears

b) Root fillet radius F at 30 tangent:

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

dan = dn + 2 hapbn = mn cos n

dbn = dn cos n

Fen = en eFor external gears

For internal gears

for external gears

for internal gears

3 =

6 =

Htan zG2

n=

6---=

3---

+

= '

cosG3

3sinz

ms

fpnn

Fn

+

= '

cosG

6sinz

ms

fPnn

Fn

( )G2coszcos G2'm 2n2

fpn

F

+=

b2n cos

=

( )4

d1pzz

2dd

zz2d

2bn

2

nbn2bn

2an

en +

=

en

bnen d

dcos arc =

ennnn

e inv inv x tan 22

z1

+

+=

( )

=

n

enFenee

n

Fe

mdtan sin cos

21

mh ]'

cosG

3cosz fpn +

( ) = enFeneeFe d tan sin cos1h 'G3cosz fPnDET NORSKE VERITAS AS

nn m2m cos6


3.3.2 Gearing with n > 2For deep tooth form gearing produced with a verified grade of accuracy of 4 or better, and with applied profile

modification to obtain a trapezoidal load distribution along the path of contact, the YF may be corrected by thefactor YDT as:

3.4 Stress Correction Factors YS, YSaThe stress correction factors YS and YSa take into account the conversion of the nominal bending stress to thelocal tooth root stress. Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the factthat not only bending stresses arise at the root. A part of the local stress is independent of the bending momentarm. This part increases the more the decisive point of load application approaches the critical tooth rootsection.Therefore, in addition to its dependence on the notch radius, the stress correction is also dependent on theposition of the load application, i.e. the size of the bending moment arm.YS applies to the load application at the outer point of single tooth pair contact, YSa to the load application attooth tip.YS can be determined as follows:

YSa can be calculated by replacing hFe with hFa in the above formulae.Note:

a) Range of validity 1 < qs < 8In case of sharper root radii (i.e. produced with tools having too sharp tip radii), YS resp. YSa must bespecially considered.

b) b)In case of grinding notches (due to insufficient protuberance of the hob), YS resp. YSa can riseconsiderably, and must be multiplied with:

where:

tg = depth of the grinding notchg = radius of the grinding notch

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

3.5 Contact Ratio Factor YThe contact ratio factor Y covers the conversion from load application at the tooth tip to the load applicationat the mid point of the flank (heightwise) for bevel gears.The following may be used:

( )2.52 n

2.502.05for 0.6662.366Y nnDT =2.05for 1.0Y nDT


3.6 Helix Angle Factor YThe helix angle factor Y takes into account the difference between the helical gear and the virtual spur gearin the normal section on which the calculation is based in the first step. In this way it is accounted for that theconditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank.The following may be used ( input in degrees):Y = 1 /120When > 1, use = 1 and when > 30 , use = 30 in the formula.However, the above equation for Y may only be used for gears with > 25 if adequate tip relief is applied toboth pinion and wheel (adequate = at least 0.5 Ceff, see 4.3.2).

3.7 Values of Endurance Limit, FEFE is the local tooth root stress (max. principal) which the material can endure permanently with 99% survivalprobability. 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the N curve. FE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregardingresidual stresses due to heat treatment). Other stress conditions such as alternating or pre-stressed etc. arecovered by the conversion factor YM.FE can be found by pulsating tests or gear running tests for any material in any condition. If the approval ofthe gear is to be based on the results of such tests, all details on the testing conditions have to be approved bythe Society. Further, the tests may have to be made under the Society's supervision.If no fatigue tests are available, the following listed values for FE may be used for materials subjected to aquality control as the one referred to in the rules.

FEAlloyed case hardened steels 1) (fillet surface hardness 58 to 63 HRC): of specially approved high grade: of normal grade:

- CrNiMo steels with approved process:- CrNi and CrNiMo steels generally:- MnCr steels generally:

1050

1000920850

Nitriding steel of approved grade, quenched, tempered and gas nitrided (surface hardness 700 800 HV): 840Alloyed quenched and tempered steel, bath or gas nitrided (surface hardness 500 to 700 HV):

720Alloyed quenched and tempered steel, flame or induction hardened 2) (incl. entire root fillet) (fillet surface hardness 500 to 650 HV):

0.7 HV + 300Alloyed quenched and tempered steel, flame or induction hardened (excl. entire root fillet) (B = u.t.s. of base material):

0.25 B + 125Alloyed quenched and tempered steel: 0.4 B + 200Carbon steel: 0.25 B + 250Note:All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified pro-cedure, see Pt. 4 Ch. 2 Sec. 3. For rolled steel, the values are to be reduced with 10%. For blanks cut from forged bars, that are not qualified as mentioned above, the values are to be reduced with 20%, For cast steel, reduce with 40%.

1) These values are valid for a root radius

being unground. If, however, any grinding is made in the root fillet area in such a way that the residual stresses may be affected, FE is to be reduced by 20%. (If the grinding also leaves a notch, see 3.4).

with fillet surface hardness 58 to 63 HRC. In case of lower surface hardness than 58 HRC, FE is to be reduced with 20(58 HRC) where HRC is the detected hardness. (This may lead to a permissible tooth root stress that varies along the facewidth. If so, the actual tooth root stresses may also be considered along facewidth.)

not being shot peened. In case of approved shot peening, FE may be increased by 200 for gears where FE is reduced by 20% due to root grinding. Otherwise FE may be increased by 100 for mn 6 and 100 5 (mn - 6) for mn > 6.However, the possible adverse influence on the flanks regarding grey staining should be considered, and if necessary the flanks should be masked.DET NORSKE VERITAS AS

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


3.8 Mean stress influence Factor, YMThe mean stress influence factor, YM, takes into account the influence of other working stress conditions thanpure pulsations (R = 0), such as e.g. load reversals, idler gears, planets and shrink-fitted gears.YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R 0, and theendurance (or static) strength with R = 0.YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is thereforesuitable for comparison between the calculated (positive) working stress F and the permissible stress FPcalculated with YM or YMst.For thin rings (annulus) in epicyclic gears where the compression fillet may be decisive, specialconsiderations apply, see 3.16.The following method may be used within a stress ratio1.2 < R < 0.5:

3.8.1 For idlers, planets and PTO with ice class

where:R = stress ratio = min. stress divided by max. stress.For designs with the same force applied on both forward- and back-flank, R may be assumed to 1.2.For designs with considerably different forces on forward- and back-flank, such as e.g. a marine propulsionwheel with a power take off pinion, R may be assessed as:

For a power take off (PTO) with ice class, see 1.6.1 c.M considers the mean stress influence on the endurance (or static) strength amplitudes.M is defined as the reduction of the endurance strength amplitude for a certain increase of the mean stressdivided by that increase of the mean stress.Following M values may be used:

1) For bevel gears, use Ys = 2 for determination of M.The listed M values for the endurance limit are independent of the fillet shape (Ys), except for case hardening.In principle there is a dependency, but wide variations usually only occur for case hardening, e.g. smoothsemicircular fillets versus grinding notches.

3.8.2 For gears with periodical change of rotational directionFor case hardened gears with full load applied periodically in both directions, such as side thrusters, the sameformula for YM as for idlers (with R = 1.2) may be used together with the M values for endurance limit. Thissimplified approach is valid when the number of changes of direction exceeds 100 and the total number of loadcycles exceeds 3106.For gears of other materials, YM will normally be higher than for a pure idler, provided the number of changesof direction is below 3106. A linear interpolation in a diagram with logarithmic number of changes of directionmay be used, i.e. from YM = 0.9 with one change to YM (idler) for 3106 changes. This is applicable to YM forendurance limit. For static strength, use YM as for idlers.For gears with occasional full load in reversed direction, such as the main wheel in a reversing gear box, YM =

Endurance limit Static strengthCase hardened 0.8 0.15 Ys 1) 0.7If shot peened 0.4 0.6Nitrided 0.3 0.3Induction or flame hardened 0.4 0.6Not surface hardened steel 0.3 0.5Cast steels 0.4 0.6

M1M1R1

1Yor Y MstM

+

=

branchmain theoffacewidth unit per forcep.t.o. offacewidt unit per force2.1DET NORSKE VERITAS AS

0.9 may be used.


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

FE is the endurance limit for R = 0.For static strength, YMst = 1 and fit accounted for in 3.9.b.fit is the shrinkage stress in the fillet (30 tangent) and may be found by multiplying the nominal tangential(hoop) stress with a stress concentration factor:

3.8.4 For shrink-fitted idlers and planetsWhen combined conditions apply, such as idlers with shrinkage stresses, the design factor for endurancestrength is:

Symbols as above, but note that the stress ratio R in this particular connection should disregard the influenceof fit, i.e. R normally equal 1.2.For static strength:

The effect of fit is accounted for in 3.9 b.

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

3.9 Life Factor, YNThe life factor, YN, takes into account that, in the case of limited life (number of cycles), a higher tooth rootstress can be permitted and that lower stresses may apply for very high number of cycles.Decisive for the strength at limited life is the N curve of the respective material for given hardening,module, fillet radius, roughness in the tooth root, etc. I.e. the factors YrelT, YRelT, YX and YM have aninfluence on YN.If no N curve for the actual material and hardening etc. is available, the following method may be used.Determination of the N curve:

a) Calculate the permissible stress FP for the beginning of the endurance limit (3106 cycles), including theinfluence of all relevant factors as SF, YrelT, YRelT, YX, YM and YC, i.e.FP = FE YM YrelT YRelT YX YC / SF

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

for not surface hardened fillets

for surface hardened fillets

FE

fitM

M1

M21Y+

=

n

F.fit m

21.5scf =

( ) ( ) FEfit

M

R1M1M2

M1M1R1

1Y+

+

=

M1M1R1

1YMst

+

=

F

y

S2.25

FSHV5

( )fitCstYrelTstYMstYFstFstS1

FPst =DET NORSKE VERITAS AS


where Fst is the local tooth root stress which the material can resist without cracking (surface hardenedmaterials) or unacceptable deformation (not surface hardened materials) with 99% survival probability.

c) Calculate YN as:

Guidance on number of load cycles NL for various applications:

For propulsion purpose, normally NL = 1010 at full load (yachts etc. may have lower values). For auxiliary gears driving generators that normally operate with 70 to 90% of rated power, NL = 108 with

rated power may be applied.

3.10 Relative Notch Sensitivity Factor, YrelTThe dynamic (respectively static) relative notch sensitivity factor, YrelT (YrelTst) indicate to which extent thetheoretically concentrated stress lies above the endurance limits (respectively static strengths) in the case offatigue (respectively overload) breakage.YrelT is a function of the material and the relative stress gradient. It differs for static strength and endurancelimit.The following method may be used:For endurance limit:for not surface hardened fillets:

for all surface hardened fillets except nitrided:

FstAlloyed case hardened steel 1) 2300Nitriding steel, quenched, tempered and gas nitrided (surface hardness 700 to 800 HV) 1250Alloyed quenched and tempered steel, bath or gas nitrided (surface hardness 500 to 700 HV) 1050Alloyed quenched and tempered steel, flame or induction hardened (fillet surface hardness 500 650 HV) 1.8 HV + 800

Steel with not surface hardened fillets, the smaller value of 2) 1.8 B

Calculation of Gear Rating for Marine Transmissions

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