CALCULATION OF GEAR RATING FOR MARINE TRANSMISSIONS · PDF fileCALCULATION OF GEAR RATING FOR...

CLASSIFICATION NOTES

NO. 41.2

CALCULATION OF GEAR RATING FOR MARINE TRANSMISSIONS

MAY2003

DET NORSKE VERITAS Veritasveien l, N-1322 H0vik, Norway Tel.: +47 67 57 99 00 Fax: +47 67 57 99 l l

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Provisions

1t is assumed that the execution of the provisions of this Classification Note is entrusted to appropriately qualified and experienced people, fo r whose use it has been prepared.

© Det Norske Veritas 2003

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CONTENTS

1. 1.1 1.2 l.3 1.4

1.5 1.6 1.6. l 1.6.2 1.6.3 1.7 1.7.1 1.7.2 1.8 1.8. l l.8.2 l.9 l.9.1 1.9.2 1.9.3 1.9 .4 1.9.5 1.9.6 1.9.7 1.9.8 1.9.9 1.10 1.11 1.12 2. 2.1 2.2 2.2.l 2.2.2 2.3 2.3. l 2.3.2 2.3.3 2.3.4 2.4 2.5 2.6 2.7 2.8

2.9 2.10

2.11 2.12 2.13 3. 3.1 3.2 3.2.1 3.2.2

Basic Principles and General Influence Fact ors ... 4 Scope and Basic Principles ....... ................................ .4 Symbols, Nomenclature and Units .... ........................ 4 Geometrical Definitions ........ .... ........... .. .......... ....... ... 5 Bevel Gear Conversion Formulae and Specific Formulae ................................. ................................... 6 Nominal Tangential Load, F1, Fb<> Fmt and Fmhl .......... 6 Application Factors, KA and KAP ...... ......................... 7 KA ..................... ......................................................... 7 KAP ............................................................................. 7 Frequent overloads ...... ................ .. .... ... .... ...... .. ..... ..... 8 Load Sharing Factor, Ky ............................................ 8 General method ....... ... .. .. .... .. ......................... .......... ... 8 Simplified method ................ ..................................... 8 Dyna1nic Factor, Kv ...... .................................. ........... 8 Single resonance method ...... .. ........... ......... ........ ....... 8 Multi-resonance method .. ..................... ................ ... I 0 Face Load Factors, KHr1 and KFp ......................... ..... 10 Relations between KHri and Kt.-p ........................... ..... 10 Measurement of face load factors ................ ............ l 0 Theoretical determination of Ku11 ..... .... .................... l l Determination off.., .......................... ............ ........... 12 Determination offdefl··· ............ ...................... ........... 13 Dctcm1ination of fbe ................................................. 13 Determination of f.na ........ .... .... ................................ 13 Comments to various gear types ........................ .... .. 13 Determination ofKHp for bevel gears ...................... 13 Transversal Load Distribution Factors, K Hc. and Kpal4 Tooth Stiffness Constants, c' and Cy ............ .......... . . 14 Running-in Allowances ........................................... 15 Calculation of Surface Durability ......................... 17 Scope and General Remarks ... .. ........ ..... .. ..... ........... 17 Basic Equations ...................................... ................. 17 Contact stress ........................................................... 17 Permissible contact stress ......... ............. .................. 17 Zone Factors ZH, Zli,D and ZM .............................. .... 18 Zone factor ZH ............ ............ ..... ....... ......... ............ 18 Zone factors Za,l) ......................... ............................ I 8 Zone factor ZM ......................................................... 18 Inner contact point ................................................... 18 Elasticity Factor, Z£ ................................................. I 8 Contact Ratio Factor, Zc ........................................... 18 Helix Angle Factor, Zp ...................... ................ .. ..... 18 Bevel Gear Factor, ZK .... ......... ...... .. .... ........ ............. I 8 Values of Endurance Limit, crHiim and Static Strength, crH

10s ,aH lO; ... ... .. ..... .. .. .. ........ . .. . . .. . .. ....... 18

Life Factor, ZN .......................... .................. .......... ... 19 Influence Factors on Lubrication Film, ZL, Zv and ZR ........................................................ ... ..... ............. 19 Work Hardening Factor, Zw .................................... 20 Size Factor, Zx ............. ............................................ 20 Subsurface Fatigue ................................................... 20 Calculation of Tooth Strength .............................. 22 Scope and General Remarks .................................... 22 Tooth Root Stresses ............................. .................... 22 Local tooth root stress .. .... ................ ............. ....... .... 22 Permissible tooth root stress ......... .... ........ .... ....... .... 22

3.3 3.3 .1 3.3 .2 3.4 3.5 3.6 3.7 3.8 3.8.1 3.8.2

3.8.3

3.8.4 3.8.5 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.15.1

Tooth Form Factors Yr, Y Fa ........ .. ....... ... ......... .. ..... 23 Dctennination of parameters ................................... 23 Gearing with eun > 2 ................................................ 24 Stress Correction Factors Ys, Ys0 .. ............ .............. 24 Contact Ratio Factor Ye ........................................... 25 Helix Angle Factor Yp ............................................. 25 Values of Endurance Limit, <>PE ...... ........................ 25 Mean stress influence Factor, YM ............................ 26 for idlers, planets and PTO with ice class .............. 26 For gears with periodical change of rotational direction .................................................................. 26 For gears with shrinkage stresses and unidirectional load .. ............. ..... .. ... .. ..... ... ... .... ........... .......... .. ... ... ... 26 For shrink-fitted idlers and planets ........................ .. 26 Additional requirements for peak loads .................. 27 Life .factor, Y N ................................................. ..... .. 27 Relative Notch Sensitivity Factor, Yore\T ................. 28 Relative Surface Condition .factor, Y RreJT ............... 28 Size Factor, Y x ....................................... ................. 28 Case Depth Factor, Y c ............................................ 28 Thin rim factor Yu ................................................... 29 Slresses in Thin Rims .............................................. 29 General .................................................................... 29

3 .15 .2 Stress concentration factors at the 7 5° tangents .. .... . 30 3.1 5.3 Nominal rim stresses ....................... ........................ 30 3. 15 .4 Root fillet stresses ........................................... ........ 30 3.16 Permissible Stresses in Thin Rims .......................... 31 3.16.1 General .................................................................... 31 3.16.2 For >3·106 cycles ..................................................... 31 3.16.3 For S: 103 cycles ....................................................... 31 3.16.4 For l03 <cycles < 3·106

.......................................... 32 4. Calculation of Scuffing L<>ad Capacity ............... 33 4.1 Introduction ... .. .... .... ................................................ 33 4.2 General Criteria ....................................................... 33 4.3 Influence Factors ..................................................... 34 4.3. 1 Coefficient of friction ........ .. ................................. ... 34 4.3.2 Effective tip relief Cefl' ................................... .... ...... 34 4.3.3 Tip relief and extension ........................................... 34 4.3.4 Bulk temperature ..................................................... 35 4.4 The Flash Temperature Sna ................................... 35

4.4.1 Basic formula .......................................................... 35 4.4.2 Geometrical relations .............................................. 35 4.4.3 Load sharing factorXr ............................................ 36 Appendix A. Fatigue Damage Accumulation ................. 40 A.1 Stress Spectrum ....................................................... 40 A.2 cr-N-curve ........... ......... ........................ ................... 40 A.3 Damage accumulation .......... .................... ............... 40 Appendix B. Application Factors for Diesel Driven

Gears ........•...•...•...•...•..•...•....•...................•...•....•...• 41 B.1 Definitions ............................................................... 41 B .2 Determination of decisive load ............................... 41 B.3 Simplified procedure ............................................... 41 Appendix C. Calculation of Pinion-Rack ............. .......... 42 C.1 Pinion tooth root stresses .. ... ... ....... ....... ..... ...... .. ..... . 42 C.2 Rack tooth root stresses ........................................... 42 C.3 Surface hardened pinions ................ ~ ................... .... 42

DET NORSK£ VERrT AS

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1. Basic Principles and General Intluence Factors

t.l Scope and Basic Principles

The gear rating procedures given in this Classi!ication Note arc mainly based on the IS0-6336 Part 1-5 (cylindrical gears), and partly on ISO 10300 Part 1-3 (bevel gears) and £SO Technical Reports on Scuffing and Fatigue Damage Accumulation, but especially applied for marine purposes, such as marine propulsion and important auxiliaries onboard ships and mobile offshore units.

The calculation procedures cover gear rating as limited by conlact stresses (pitting, spalling or case crushing), looth root stresses (fatigue breakage or overload breakage), and scuffing resistance. Even though no calculation procedures for other damages such as wear, grey staining (micropitting), etc. are given, such damages may limit the gear rating.

The Classification Note applies to enclosed parallel shaft gears, epicyclic g~rs and bevel gears (with intersecting axis). However, open gear trains may be considered with regard to tooth strength, i.e. part l and 3 may apply. Even pinion-rack tooth strength may be considered, but since such gear trains often are designed with non-involute pinions, the calculation procedure of pinion-racks is described in Appendix C.

Steel is the only malerial considered.

The methods applied throughout this document are only valid for a transverse contact ratio 1 < lia< 2. If En> 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 inlluence factors are detennined by the gear geometry or have been established by conventions. These factors are to be calculated in accordance with the equations provided. Other factors are approximations, which are clearly stated in the text by tc1ms as «may be calculated as». These approximations are substitutes for exact evaluations where such arc lacking or too extensive for practical purposes, or factors based on experience. In principle, any suitable method may replace these approximations.

Bevel gears are calculated on basis of virtual (equivalent) cylindrical gears using the geometry of the midsection. The virtual (helical) cylindrical gear is to be calculated by using all the factors as a real cylindrical gear with some exceptions. These exceptions are mentioned in connection with the applicable factors. Wherever a faclor or calculation procedure has no reference to either cylindrical gears or bevel gears, it is generally valid, i.e. combined for both cylindrical and bevel.

Jn order to minimise the volume of this Classification Note such combinations are widely used, and everywhere it is necessary to distinguish, it is clearly pointed out by local head ings such as:

Cylindrical gears

Bevel gears

Classification Notes- No. 41.2

May2003

The permissible contact stresses, tooth root stresses and scuffing load capacity depend on the safety factors as required 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 the o-N curves, regardless if it is constant or drops wilh higher number of cycles.

Static strength is to be understood as the fatigue strength in the range of cycles less than at lhe upper knee of the o- N curves.

For gears that are subjected to a limited number of cycles at different load levels, a cumulative fatigue calculation applies. Information on this is given in Appendix A.

When the term infinite life is used, it means number of cycles in the range 108-1010.

1.2 Symbols, Nomenclature and Units

The symbols are generally from TSO 701, 1SO/R31 and TSO 1328, with a few additional symbols. Only SI units arc used.

The ma.in symbols as influence factors (K, Z, Y and X with indeces) etc. are presented in their respective headings. Symbols which are not explained in their respeclivc Secs. are as follows:

HB nv HRC

s •• s.1 Sf n

Spr

T u v

centre distance (nun). faccwidth (mm). reference diameter (mm). tip diameter (mm). base diameter (mm). working pitch diameter (mm). addendum (mm). addendum of tool ref. to m0 •

dedendum of basic rack ref. to m0 (= hao). bending moment arm (mm) for tooth root stresses for application of load at the outer point of single tooth pair contact. bending moment arm (mm) for tooth root stresses for application of load at tooth tip. Brinell hardness. Vickers hardness. Rockwell C hardness nonnal module. rev. per minute. number of load cycles. notch parameter. average roughness value (µm). peak to valley roughness (µm). mean peak to valley roughness (µm). tooth top land thickness (mm). transverse top land thickness (mm). tooth root chord (mm) in the critical section. protuberance value of tool minus grinding stock, equal residual undercut of basic rack, ref lo mn. torque (Nm). gear ratio (per stage). linear speed (mis) at reference diameter.

D ET NORSKE VERITAS


May 2003

addendum modification coefficient. number of teeth. virtual number of spur teeth. normal pressure angle at ref. cylinder. transverse pressure angle at ref. cylinder. transverse pressure angle at tip cylinder. transverse pressure angle at pitch cylinder. helix angle at ref. cylinder. helix angle at base cylinder. helix angle at tip cylinder. transverse contact ratio. overlap ratio. total contact ratio. tip radius of tool ref. to mn. root radius of basic rack ref. to mn ( = p.o). effective radius (mm) of curvature at pitch point. root fillet radius (mm) in the crilical section. ultimate tensile strenglh (N/mm2). yield strength resp. 0.2% proof stress (N/mm2

).

lndex I 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:

1: angle between intersection axis . .9 K angle modification (Klingelnberg)

111-0 tool module (Klingelnberg) S pitch cone angle. x.rn tooth thickness modification coefficient (mid-

facc). R pitch cone distance (mm).

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

Index m refers to the midsection of the bevel gear.

1.3 Geometrical Definitions

For internal gearing z2, a; d.2, dw2, d2 and db2 are negati ve, Xz

is positive if da2 is increased, i.e. the numeric value is decreased.

The pinion has the smaller number of teeth, i.e.

For caJculation 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 bi differ much from b above, they are not to be taken more than 1 module on either side of b.

Cylindrical gears

tan a,, I cos ~ tan~ cos Ot

tan p. cos a.,.

d 'f. m0 I COS p ma/cos~ d cos Ut = dw cos awl

a = 0.5 (dw1 + dw2)

Zt I Z2

tan a - a. (radians)

inv «w1 = inv a.1 + 2 tan a,, (x, + x2)/(z1 + Z2)

Zn z I (cos2 Ph cos~)

S1W1 +~awl ii =

a T I

where ~f\vl is Lo be taken as the smaller of :

• Stwt =tan <lwt

•

( db2 l Z2 • ~twt = Lanacos- -tano."'1 -

d~2 z,

and

Z 1 . Saw l = l;rw2 - , where l;fw2 is calculated as Siv,1

Z2

5

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

d~il 02{(%-m, (hr, - x, -p0 +Pr, ·sina,)J2 + I

bsin~ eli=-

mnn

(mn (hq, - Xi -Pti> +pfp ·Sina" ))2]2

tano.1

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

ey = en +i>p

DET NORSK.E VERIT AS

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1t rn n cos n1 P ht =

cos~

[~+2xtana l s.1 = da 2 z n + inva, - invaa

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

Number of teeth:

Zv1.2 = Z1 ,2/ COS 01,2

(01 + ()2 ~ ~)

Gear ratio:

Zv2 Uv = --

Zvt

tan avt = tan r:J.nl cos ~,

tan i~hm = tan ~m cos a,1

Base pitch:

Reference, pitch, diameters:

dm l,2

COSOt,2

Centre distance:

Tip diameters:

dv• 1.2 = dv 1,2 + 2 ham 1,2

Addenda:

for gears with constant addenda (Klingelnberg):

ham 1,2 = IDmn (1 + Xm 1,2)

for gears with variable addenda (Gleason):

h.m 1,2 = ha l,2 - b/2 tan (o. 1.2 -- 01,2)

(when h. is addendum at outer end and 00 is the outer cone angle).

Addendum modification coefficient'>:


Xm 1,2

Base circle:

ham 1,2 - h an12,l

2mmn

dvb l,2 = dv 1,2 COS °'vi

Transverse contact ratio:*)

May 2003

o.sJd~aJ -d~·hl + o.sJd~a2 -d~b2 - av sin n.,l e.,=

Pbtm

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

Total contact ratio: ·i

(* Note lhat index «V» is left out in order to combine fonnulae for cylindrical and bevel gears.)

Tangential speed at midsection:

1C d 0- 3 = -n1 11 60 m

Effective radius of curvature (normal section):

Pvc av uvsinavt

cos ~bm (1 + u J2

Length of line of contact:

J.5 Nominal Tangential Load, F0 Fbt' F mt and Fmbt

The nominal tangential load (tangential lo the reference cylinder with diameter d and perpendicular to an axial plane) is calculated from the nominal (rated) torque T transmitted by the gear set.

Cylindrical gears

Bevel gee1rs

F = 2000T 111\ d

m

F _ Fmt mix ----

cosa.,1

DETNORSKE VERITAS


May 2003

1.6 Application Factors, KA and KAP The application factor KA accounts for dynamic overloads from sources extemal to the gearing.

It is distinguished between the influence of repetitive cyclic torques KA (1.6.1) and the inl1uence of temporary occasional peak torques KAP (1.6.2).

Calculations are always to be made with KA. Tn certain cases additional calculations with KAT' may be necessary.

For gears with a defined load spectrum the calculation with a KA may be replaced by a fatigue damage calculation 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 the maximum repetitive cyclic torque applied lO 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). (For the purpose or a preliminary (but not binding) calculation before KA is detennined, it is advised to apply either the max. values mentioned below or values known from similar plants.)

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

b) For main propulsion gears with ice class notation (sec Rules Pt.5 Ch. I Sec JSOO) KA ice has to be taken as the higher value of the applicable (rule defined) ice shock torque referred to nominal rated torque and the value under a). The Baltic ice class notations refer to a few millions ice shock loads. Thus the life factors may be put Y N= l and ZN= 1.2 (except for nitrided gears where ZN= l 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 oQ"(PTO) branch from a main propulsion gear with ice class, ice shocks result in negative torques. Tl is assumed that the PTO branch is unloaded when the ice shock load occurs.

7

The influence of these reyerse shock loads may be taken into account as follows: The negative torque (reversed load), expressed by means of an application factor based on rated forward load (T or FJ, is K An:vcne =KA ke -1 (the minus I because no mean torque assumed). KAicc to be calculated as in !he ice class rules. This KAruverse should be used for back flank considerations such as pitling and scuHing. The influence on tooth bending strength (forward direction) may be simplified by using the factor Y M = 1 - 0.3 · KArcversefKA·

d) for diesel driven auxiliaries KA can be taken from the torsional vibralion analysis, if available. For units where oo vibration analysis is required ( < 200 kW) or available, it is advised to apply KA as the upper allowable value 1.35.*)

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

•i For diesel driven gears, more information on KA for misfiring and nonnal driving is given in Appendix B.

1.6.2 KAP

The peak overloa.d factor KAP is defined as the ratio between the temporary occasional peak overload torque and 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 (ifnitrided) has to be checked with regard to static strength. Unless otherwise specified the same safety factors as for infinite lite apply.

The scuffing safety is to be specially considered, whereby the KA applies in connection with the bulk lcmperature, and the KAP applies for the flash temperature calculation and should replace KA in the fonnulae in 4.3.1, 4.3.2 and 4.4. l.

KAP can be evaluated from the torsional impact vibration calculation (as required by the rules).

lfthe overloads have a duration corresponding to several revolutions of the shafts, the scuffing safety has to be considered on basis of this overload, both wi lh respect to bulk and flash temperature. This applies to plants with ice class notations (Baltic and polar), and to plants with prime movers which have high temporary overload capacity such as e.g. electric motors (provided the driven member can have a considerable increase in demand torque as e.g. azimuth thrusters during manoeuvring).

For plants without additional ice class notation, KArshould normally not exceed 1.5.

DET NORSK.E VERITAS

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1.6.3 Frequent overloads

For plants where high overloads or shock loads occur regularly, the influence of this is to be considered by means of cumulative fatigue, (see Appendix A).

1.7 Load Sharing Factor, K, The load sharing faclor Ky accounts for the maldistributi0n of load in multiple-path transmissions (dual tandem, epicycl ic, double helix etc.). K.1 is defined as the ratio between the max. load through an actual path and the evenly shared load.

1.7.l General method

Kv mainly depends on accuracy and flexibility of the b~anches (e.g. quill shaft, planet support, external forces etc.), and should be considered on basis of measurements or of relevant analysis as e.g.:

K,,. o+f 'Y 0

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

f = ~f12 + fi + r; +- - - - where fr. f2 etc. are the

main individual en-ors that may contribute to a maldistribution between the branches. E.g. tooth pitch errors, planet carrier pitch errors, bearing clearance influences etc. Compensating elTects should also be considered.

For double helical gears:

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

K =I ± Fcxt Y F1 • tanp

[fthe direction of Fm is known, the calculation should be carried out separately for each helix, and with the tangential force corrected with the pertinent Ky. lfthe direction ofFexi is unknown, both combinations are to be calculated, and the higher <>H or op to be used.

l.7.2 Simplified method

lf no relevant analysis is available the following may apply:

For epicyclic gears:

Ky = t+ 0.25~np1 -3

where np1 = 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

~ = 1+(0.21¢)

May2003

where ¢ = quill shaft twist (deb'Tees) under full load.

1.8 Dynamic Factor, Kv The 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 the maximum externally applied load F1 KA K.1.

In the following 2 difterent methods ( l .8.1 and 1.8.2) are described. Tn case of controversy between the methods, the next following is decisive, i.e. the methods are listed with increasing priority.

lt is important to observe the limitations for the method in 1.8. l. In particular the in11uence oflateral stiffness of shafts is often underestimated and resonances occur at considerably lower speed than detennined in 1.8.1.1.

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

1.8.l Single resonance method

For a single stage gear Kv may be detennined on basis of the relative proximity (or resonance ratio) N between actual speed n 1 and the lowest resonance speed nEt·

N= ~ nn1

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

1.8. I. I Determination of critical speed

Tt is not advised to apply this method for multimesh gears for N > 0.85, as the influence of higher modes has to 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 ifN 2: 0.75 , see 1.8.2

30 .103

where:

ey is the actual mesh stiffness per unit facewidth, see 1.11.

DET NORSKE VERTTAS


May 2003

For gears with inactive ends of the facewidth, as e.g. due to high crowning or end relief such as o ften applied for bevel gears, the use of Cy in connection with determination of natura l frequencies may need correction. Cy is defined as stiffness per unit facewidth, but when used in connection with the total mesh stiffness, it is not as simple as ey·b, as only a part o f the facewidth is active. Such corrections are given in 1.1 l.

mre<t 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, mn:d is calculated as:

IDn;u =

The individual masses per unit facewidth arc calculated as

11 2 m, .2 = '

2 b(dbl,2 / 2)

where I is the polar moment of inertia (kgrrun2).

The inertia of bevel gears may be approximated as discs with d iameter equal the midface pitch diameter and width 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, it should be added to that particular inertia (pinion or wheel). Tf there is a shaft piece between these inertias, the torsional shaft stiffness alters the system into a 3-mass (or more) system. This can be calculated as in 1.8.2, but also simplified as a 2-mass system calculated with only pinion and wheel masses.

1.8.1.2 Factors used for determination ofKv

Non-dimensional gear accuracy dependent parameters:

B _ c'(fpi -yPJ

p -r~KAKy / b

_ c'(F.., - yf) Br ----'~-...........

F1KAK.y lb

Non-dimensional tip relief parameter:

Bk = 1- Ca·c' f 1 ·KA·K/ b

For gears of quality grade (ISO l328) Q = 7 or coarser, Bk"" I.

For gears with Q :::: 6 and excessive tip relief, Bk is limited to max. I.

For gears (all quality grades) with lip reliefofmore than 2·Ceff (see 4.3.2) the reduction of r. i:x has to be considered

(see 4.4.3).

where:

l~t

Y11 and Yr

c'

the sing le pitch deviation (TSO 1328), max. of pinion or wheel

the total profile form deviation (ISO 1328), max. of pinion or wheel (Note: F" is p.t. not available for bevel gears, thus use Fa= fll(.)

the respective running-in allowances and may be calculated similarly to Yu in 1.12, i.e. the value of fpt is replaced by Fa for Yf·

the single tooth stiffness, see 1.11

the amount o f tip relief, see 4.3.3. Jn case of different tip relief on pinion and wheel, the value that results in the greater value of Bk is to be used. lfC. is zero by design, the value of ruruiing-in tip relief C ay (see 1.12) may be used in the above fonnula.

1.8.J.3 Kv in the subcritical range:

Cylindrical gears: N ~ 0.85

Bevel gears: N S 0.75

Kv = l + N K

K = C.1 Br + C.2 Br+ Cv3 Bk

C,.1 accounts for the pitch error influence

Cv1 = 0.32

Cv2 accounts for profile error influence

C v2 = 0.34

C - 0.57

v2 -r.y -0.3

for er > 2

c.J accounts for the cyclic mesh stitlitcss variation

Cvl = 0.23

C - 0.096 vJ -

&y -1.56 for i::1 > 2

1.8. J.4 Kv in the main resonance range:

Cylindrical gear.-;: 0.85 < N :5 1.15

Bevel gears: 0.75 < N :5 1.25

Rwming in this range should preferably be avoided, and is only allowed for high precision gears.

K" = 1 + Cv1 Bp + Cv2 Br+ C,4 Bk

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

Cv4 = 0.90

D ETNORSKE VERITAS

9

lO

1.8.J.5 Kv in the supercritical range:

Cylimlrical gears:

Bevel f(ear.;;:

N 2:. 1.5

N 2:. 1.5

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 vihrations between pinion and wheel. These influences are not covered by the following approach.

Kv = Cvs BP + Cv(. Br+ C,1

Cv.s accounts for the pitch error influence.

Cvs = 0.47

Cv6 account..;; for the profile error intluence.

Cv6 ""' 0.47 forr.y :::: 2

0.12 fon:y > 2 c -v6 -

i;., -1.74

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

Cv1 '- 0.75 for i;'I :S 1.5

Cv1 = 0.125sin(n-(6) - 2)]+ 0.875 for 1.5 < f,1 ~2.5 C.1 = 1.0 for i;., > 2.5

1.8.1.6 Kv in the intermediate range:

Cylindrical gears; 1.15 < N < 1.5

Bevel gears; 1.25 < N < 1.5

Comments raised in 1.8.1.4 and 1.8. l.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 method

For high speed gear (v>40 m/s), for multimesh medium speed gears, for gears with significant lateral shaft flexibility etc. it is advised to determine Kv on basis of relevant dynamic analysis.

lncorporating lateral shaft compliance requires transfonnation of even a simple pinion-whee l system into a lumped multi-mass system. It is advised to incorporate all relevant inertias and torsional shaft stiffnesses into an equivalent (to pinion speed) system. Thereby the me.sh stiffness appears as an equivalent torsional stillness:


May2003

(Nm/rad)

The natural frequencies are found by solving the .set of difforcntial equations (one equation per inertia). Note that for a gear put on a laterally flexible shaft, tJ1e coupling bendingtor.sionals is arranged by introducing the gear mass and the lateral stiffness with its relation to the torsional displacement and torque in that shaft.

Only che natural frequency (ies) having high relative displacement and relative torque through the actual pinionwheel flexible element, nccd(s) to be considered as critical frequency (ies).

Kv may he dctcnnined by means of the method mentioned in 1.8.l thereby using N as the least favourable ratio (in case of more than one pinion-wheel dominated natural frequency). Le. the N-ratio that resul ts in the highest Kv has to be considered.

The level of tJ1c dynamic factor may also be detennined on basis of simulation technique using numeric time integration with relevant tooth stiffness variation and pitch/profile errors.

·1.9 Face Load Factors, Knp and KFf\

The face load factors, KHri for contact stresses and for scuffing, KF~ for tooth root stresses, account for non-uniform load distribution across the Caccwidth.

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

K~.p is defined as the ratio between the maximum tooth root stress per unit facewidth and the mean tooth root stress per unit facewidth. The mean tooth root stress relates to the considered facewidth b1 respectively b2.

Note that facewidth in chis context is the design facewidth b, even ifthe ends are unloaded as often applies to e.g. bevel gears.

The plane of contact is considered.

1.9.1 Relations between K"11 and Ki;p

l exp= -----~

1 + h/b + (h/b)2

where h/b is the ratio tooth height/facewidth. The maximum of' h1/b1, and h2/b2 is to be used, but not higher than 1/3 . For double helical gears, use only the facewidth of one helix.

If the tooth root facewidch (b1 or b2) is considerably wider than b, the value of K Ffl<lor21 is to be specially considered as it may even exceed KHP·

E.g. in pinion-rack lifting systems for j ack up rigs, where b = b2 ::::: m0 and b1 ::::: 3 m., the typical Kn~::::: Kr~2 ::::: l and KF111 ::::: 1.3.

'1.9.2 Meusurement of face load facto1·s

Primarily,

Krp may be determined by a number of strain gauges distributed over the faccwidth. Such strain gauges must be put in

DET NORSKE Y ERITAS


May 2003

exactly the same position relative to the root fillet. Relations in I. 9.1 apply for conversion to KHP·

Secondarily,

KHJJ may he evaluated by observed contact patterns on various defined load levels. It is imperative that the various test loads arc well defined. Usually, it is also necessary to evaluate the elastic deflections. Some teeth at each 90 degrees arc to he painLCd with a suitable lacquer. Always consider the poorest of the contact patterns.

After having run the gear for a suitable time at test load I (the lowest), ubscrvc the contact pattem with respect to extension over the raccwidth. Evaluate that KHp by means of the methods mentioned in this section. Proceed in the same way for the next higher test load etc., until there is a full fat.:c contact pattern. From these data, the initial mesh misalignment (i.e. without elastic deflections) can be found by extrapolation, and then also the K11p at design load can be found by calculation and extrapolation. See example.

'- - ... ........ -. ----...

Test 1 Test 2 Test 3 Norn. load

Figure 1.1 Example of experimental determination of KHp

lt must be considered that inaccurate gears may accumulate a larger observed contact pattern than the actual single mesh to mesh contact patterns. This is pa11icularly important for lapped bevel gears. Ground or hard metal hobbed bevel gears are asswned to present an accumulated contact pattern that is practically equal the actual single mesh to mesh contact patterns. As a rough guidance the (observed) accumulated contat.:t pattern or lapped bevel gears may be reduced by I 0% in order to assess the single mesh to mesh contact pattern which is used in 1.9.9.

1.9.3 Theoretical determination of KH11 The methods described in I. 9.3 to 1.9.8 may be used for cylindrical gears. The principles may to some extent also 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, all assumptions are to be well on the sate side.

KHp is to be detennined in the plane of contact.

The influence parameters considered in this method arc:

• mean mesh stilfoess c., (see l.l 1) (if necessary, also variable stiffness over b)

• mean unit load Y mlb = F bt KA Ky Kv/h (for double helical gears, see 1.7 for use of Ky)

11

• misalignment t~h due to elastic deilections of shafts and gear bodies (both pinion and wheel)

• misalignment fuefl due to elastic deflections of and work-ing positions in bearings

• misalignment fhc due to bearing clearance tolerances • misalignment fma due to manufacturing tolerances • helix modifications as crowning, end relief; helix correc-

tion • running in amount y11 (sec 1.12).

In practice several other parameters such as centrifugal expansion, thermal expansion, housing deflection, etc. contribute to KHfl· However, these parameters are not taken into account unless in special cases when being considered as particularly important.

When all or most of the a.m. parameters arc to be considered, the most practical way to determine KHft is by means of a graphical approach, described in 1.9.3.1.

If c.1 can be considered constant over the facewidth, and no helix modifications apply, KH~ can be determined analytically as described in 1.9.3.2.

1.9.3.J Graphical method

The graphical method utilises the superposition principle, and is as follows:

• Calculate the mean mesh deflection oM as a function of

Fn,/h and Cy , see 1.11.

•

•

Draw a base line with length b, and draw up a rectangular with height 6M. (The area OM b is proportional to the transmitted force). Calculate the elastic deflection fsh in the plane of contact. Balance this deflection curve aroun<l a zern line, so that the areas above and below this zero line are equal.

Zero line +

Figure 1.2 t~,, balanced around zero line

• Superimpose these ordinates of the fs11 curve to the previous load distribution curve. (The area under this new load distribution curve is still OM b.)

• Calculate the bearing dcllcctions and/or working positions in the bearings and evaluate the influence fdefl in the plane of contact. This is a straight line and is balanced around a zero line as indit.:atcd in Fig. 1.4, but with one distinct direction. Superimpose these ordinates to the previous load distribution curve.

DET NORSKE VERITAS

12

• The amount of crowning, end relief or helix correction (defined in the plane of contact) is lo be balanced around a zero hne similarly to fsh

+

Zero line

Figure 1.3 Crowning Cc balanced aound zero line

• Superimpose these ordinates to the previous load distribution curve. In case of high crowning etc. as e.g. oJkn applie<l to bevel gears, the new load distribution curve may cross the base line (the real ~ro line). The result is areas with negative load that is not real, a-; the load in those areas should be 7.ero. Thus corrective aetions must be made, but {or practical reasons it may be postponed to after next operation.

• The amount of initial mesh misalignment, fm(\ + f hc (defined in the plane of contact), is to be bal-

anced arnu11d a zero line. lfthe direction of fma + fbe is

known (due to initial contact check), or ifthe direction or li,0 is known due to design (e.g. overhang bevel pinion), this should be taken into account. If direction unknown, the influence of fma + fbc in both directions as well as equal zero, should he considered.

+ o l!ci.e---··-----· _," ..

,,, ..

+

--- -~-~ ---- -

Figure 1.4 fm.+fbc in both directions, balanced around zero line.

Superimpose these ordinates to Lhc previous load distribution curve. This results in up to 3 diflerent curves, of which the one with the highest peak is lo be chosen for further evaluation.

• l fthe chosen load distribution curve crosses the base line (i.e. mathematically negative load), the curve is to 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 load distribution curve. It is advisable to check that the area covered under this new load distribution curve is still equal oM b.

Classification Notes~ No. 41.2

May2003

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

• The influence of running in y~ is to be determined as in l.12 whereby the value for f11x is to be taken as twice the distance between the peak of the load distribution curve and oM.

• Determine peak of curve - y~

KHll = oM 1.9.3.2 Simplified analytical method for cylindrical gears

The analytical approach is simi Jar to 1 .9. 3. I but has a more limited application as Cy is assumed constant over the facewidLh and no helix modification applies.

• Calculate the elastic deflection fsh in the plane or contact. Balance this dcllcction curve around a zero line, so that the area above and helow this zero line are equal, sec Fig. 1.2. The max. positive ordinate is Y2t.f,h.

• Calculate the initial mesh alignment as

f11x= 1~ fsh ± rma ± fbc ± f<J..,n l The negative signs may only be used if this is justified and/or verified by a contact paltcrn test. Otherwise, always use positive signs. lf a negative sign is justified, the value ofFiix is not to be taken less than the largest of each of these elements.

• Calculate the effective mesh misalignment as 'F~y = l'px - Yp (Y11 sec 1.12)

• Determine Cy Fll''I h

KH~ =1 + for Kn~ ~2 2 I'm

or

2 c Fl}r b KHp = ~ for KHI\ > 2

ID

where Cr as used here is the effective mesh stiffness, sec 1.11.

1.9.4 Determination of fsh

f.i, is the mesh misalignment due to elastic det1ections. Usually it is sufficient to consider the combined mesh delkction 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 other meshes can be parted into components parallel respectively vertical to the considered plane of contact. Forces vertical lo I.his plane of contact have no influence on fsh·

It is advised to use following diameters for toothed clements:

d+2x m. for bending and shear deflection

<l + 2 mn (x - hau + 0.2) for torsional deflection

Usually, 1~ll is calculated on basis of an evenly distributed load. If the analysis ofKHp shows a considerable maldis-

DET NORSKE VERITAS


May 2003

tribution in term of hard end contact, or if it is known hy other reasons that there exists a hard end contact, the load should be correspondingly distributed when calculating f"". In fact, the whole Kup procedure can be used iteratively. 2-3 iterations will be enough, even for almost triangular load distributions.

1.9.5 Determination of fd.n

fdefl is the mesh misalignment in the plane or contact due to bearing deflections and working positions (housing deflecti on may be included ir determined).

First the journal working positions in the bearings are to be determined. The influence or external moments and forces must be considered. This is of special importance for twin pinion single output gears with all 3 shafts in one plane.

For rolling bearings f:1on is fu1t her determined on basis of the elastic deflection of the bearings. An elastic hearing deflection depends on the bearing load and size and number of rolling elements. Note that the bearing clearance tolerances are not included here.

For fluid film bearings fden is further determined on basis of !he lift and angular shift of the shafts due to lubrication oil film thickness. Note that fbe takes into account !he influence of the bearing clearance tolerance.

When working positions, bearing deflections and oil lilm lift are combined for all bearings, the angular misalignment as projected into the plane of the contact is to be determined. fdetl is this angular misalignment (radians) times the facewidth.

1.9.6 Determination of fh•

fbc is the mesh misalignment in the plane of contact due to tolerances in bearing clearances. In principle fhc and fdefl could be combined. But as fdc1l can be detcnnined by analysis and has a distinct direction, and fb~ is dependent on tolerances and in most cases has no distinct direction (i.e. ± tolerance), it is practicable to separate these two influences.

Due to different bearing clearance tolerances in both pinion and wheel shafts the two shaft axis will have an 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 as a :!: tolerance, or a combination of both. for combination of :t tolerance it is adv iced to use

fbe= ± )rnc: + fbe~ + ........

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

1.9.7 Determinntion off,na

frna is the mesh misalignment due to manufacturing tolerances (helix slope deviation) of pinion fH~1, wheel fH112 and housing bore .

For gear without specifically approved requirements to assembly control, the value oft;,,a is to be detcnnined as

f;na=- fA~I + fr)p2

For gears with specially approved assembly control, the value off.n" will depend on those specific requirements.

1.9.8 Comments to various gear types

13

For double helical gears, KHr1 is to be detcnnined for both helices. Usually an even load share between the helices can be assumed. If not, the calculation is to be made as described in 1.7. I.

For planetary gears the free floating sun pinion suffers only twist, no bending. lt must be noted that the total twist is the sum of the twist due to each mesh. ff the value of Ky -t: l,

this must be taken into account when calculating the total sun pinion twist (i.e. twist calculated with the force per mesh without Ky, and multiplied with 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 individual misalignments. The faceload distribution on the flanks of the planets can take full advantage of this. However, as the sun and annulus mesh with seveml planets with possibly different lead errors, the sun and annulus cannot obtain the above mentioned advantage to the foll extent.

1.9.9 Determination of KHJl for bevel gears

If a theoretical approach similar to 1.9.3-1.9.8 is not documented, the following may be used.

KH~ = 1.85 { 1.85 - b~} Kiest

b.1rf b represents the relative active facewidth (regarding lapped gears, see 1.9.2 last parl).

Higher values than befl·/ b = 0.90 arc normally nol to be used in the fomm la.

For dual directional gears it may be difficult to obtain a high bcn I b in both directions. T n that case the smaller berr I h is to be used.

K1esi represents the influence of the bcaru1g arrangement, shaft stiffness, bearing stiffness, housing stiffness etc. on the faceload distribution and the verification thereof. Expected variations in length- and height-wise tooth profile is also accounted for to some cxlcnt.

a) Krest = 1 For ground or hard metal hobbed gears with the spe~i fi cd contact pattern verified at full rating or at full torque slow turning at a condition representative for the thermal expansion at normal operution.

DETNORSKE VERlTAS

14

It also applies when the bearing arrangement/support has insignificant elastic deflections and thermal axial expansion. However, each initial mesh contact must be verified to be within acceptance criteria that are calibrated against a type test at full load. ReproducLion of the gear tooth length- and height-wise profile must also be verified. This can be made through 3D measurements or by initial contact movements caused by defined axial offsets of the pinion (tolerances to be agreed upon).

b) Kiest = l + 0.4·(bcnlb- 0.6) For designs with possible influence ofthennal expansion in the axial direction or the pinion. The initial mesh contact verified with low load or spin test where the acceptance criteria are calibrated against a type test at full load.

c) Ktesr =- l.2 if mesh is only checked by toolmaker's blue or by spin test contact. For gears in this category befl.'/b > 0.85 is not to be used in the calculation.

I. I 0 Transversal Load Distribution Factors, Kna and KFR The transverse load distribution factors, Kuo. for contact stresses and for scuffing, Kfo for tooth root stresses account for the effects of pitch and profile errors on the transversal load distribution hetween 2 or more pairs of teeth in mesh.

The following relations may be used:

Cylindrical gears:

Kfo = KHu = .:r_(o.9+ 0.4 cy(fp• _- Ya)b ) 2 ftH

valid for e7

:::; 2

valid for Ey > 2

where:

Yu

Note:

r. KA Ky Kv KIL~

See 1.11

= Seel .12

Maximum single pitch deviation (µm) of pinion or wheel, or maximum total profile form deviation FCl of pinion or wheel if this is larger than the maximum single pitch deviation.

In case of adequate equivalent tip relief adapted to the load, half of the above mentioned fflt can be int.roduced. A tip relief is considered adequate when the average of C,1 and Ca2 is within ±40% of the value of C cff in 4.3 .2:


May 2003

Limitations of KHn and K1,0 :

If the calculated values for

use KF'n = ~u = 1.0

c g If the calculated value ofKHn > __ 'Y -

2 , use Ku11. = _ _ 'I -

2 Ea Zr. ~: a 2£

fl E Jr the calculated value of KFN > _ _ 'Y - , use Kra = _ _ 'Y -

Ctt Yt i;a Yr.

0.75 where Yr. = 0.25 + - -

Eun

Bevel geu1·s:

(for l::o.n see 3.3. I .c)

For ground or hard metal hobbed gears K r-u-= K.ia = 1

For lapped gears KFa = KHu = 1.1

l.11 Tooth Stiffness Constants, c' and Cy

The tooth stiffnc.'>s is defined as the load which is necessary to deform one or several meshing gear teeth having I mm facewidth by an amount of I µm, in the plane of contact. c· is the maximum sLiffoess of a single pair of teeth . c., is the mean value or the mesh stiITness in a transverse plane (brief term: mesh stiffness).

Both valid for high unit load. (Unit load ,..., F1 • K" · Ky!b).

Cyli1Ulrical gears The real stiffness is a combination of the progressive Hertzian contact stillness and the linear tooth bc11ding stiffnesses. For high unit loads the Hertzian stiffness has little importance and can be disregarded. This approach is on the sate side for determination ofKn13 and KHn.· However, for moderate or low loads Kv may be 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.

·'- 0.8cos~ C C C- R B

q

and

Cy = c'(0.75ea+ 0.25)

where:

Cn =[l+0.5(1.2 - hao1~h ao2 )Jri-0.02(20-an)J

q = 0.04?2)+ 0. 15551 + 0.25791 _ 0.00635 x, 7•ni :G112

+ 0.00182 x/

DET NORSKE YERlT AS


May 2003

(for internal gears, use z02 equal infinite and x2 = 0). hao = hrp for all practical purposes. CR considers the increased flexibility of the wheel teeth if the wheel is not a solid disc, and may he calculated as:

C _ 1

ln{b/ b) R - + Se(sR/ 5m0 )

where:

b, thickness of a central web

average thickness of rim (net value from tooth root to inside of rim).

The formula is valid for bs I b ~ 0.2 and sR/0111 ~ 1. Outside this range of validity and if the web is not centrally positioned, CR has to be specially considered.

Note: CR is the ratio between the average mesh stiffness over the facewidth and the mesh stirlness of a gear pair of solid discs. The local mesh stiffness in way of the web corresponds to the mesh stiffness with CR = l. The local mesh stiffness where there is no web support will be less than calculated with CR above. Thus, e.g. a centrally positioned web will have an effect corresponding to a longitudinal crowning of the teeth. See also 1.9.3.1 regarding KHJI·

B. The non-linear approach.

Jn the following an example is given on how to consider the non-linearity.

The relation between unit load Fib as a function of mesh deflection 8 is assumed to be a progressive curve up to 500 N/mm and from there on a straight line. This straight line when extended to the baseline is assumed to intersect al

IOµm.

With these assumptions the unit force Fib as a function of mesh dcllection o can be expressed as:

£.=K(o - 10) b

F (- [f./b) b=K o-10V5oO

F for - > 500

b

. F 00 lor - < 5 b

with .!::. =.!!_· KA· K. etc. (N/mm), i.e. unit load ineorporat-b b 1

ing the relevant factors as:

KA· K7 for detennination ofKv.

KA· K.1 · K. for determination of KIIB·

K" · Ky· Kv · Kttr1 for determination of KHu·

o == mesh deflection (~un)

K = applicable stiffness ( c' or cy)

Use qf sti.ffnessesfor Kv. Kup and KHa

For calculation of Kv and KHa Lhc stiffness is calculated as follows:

When Fib < 500,

h ·.a;: • d . d 6.F/b t e sh.uness ts etermme as - -Ao where the increment is chosen as e.g. 6. Fib= 10 and thus

60 = F/b+ 10+ 10 Fib+ 10 K 500

When f1 I b > 500, the stiffness is c' or c1

15

For calculation of Ku~ the mesh deflection o is used directly,

or an equivalent stiffness determined as ~ . b·o

Bevel gears

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

c'= 13 beff 0.85b

C ~ 16 boll' 'Y 0.85b

bcrr not to he used in excess of0.85 bin these formulae.

Bevel gears with heightwise and lengthwise crowning have progressive mesh stiffness. The values mentioned above are only valid for high loads. They should not be used for determination ofCetr (see 4.3.2) or Kno (see 1.9.3. J ).

1.12 Running-in Allowances

The running-in allowances account for the influence of running-in wear on the various error elements.

y11 respectively y~ are the running-in amounts which reduce the influence of pitch and profile errors, respectively inlluence of localised faceload.

Cay is defined as the running-in amount that compensates for lack or tip relief.

The following relations may be used:

For not surface hardened steel

160 Yrx =-- fpl

0'1mm

320 f' y~ :..: -- ~x

CTntim

with the following upper limits:

v :'S 5 mis 5-10 mis

YQmnx none 12800 - -O'Htim

Y~max none 25600 - -O"H Jim

Por surface hardened steel

> 10 mis

6400 - -O'Htim

12800 - -O'H tim

y .. '"" 0.075 1~1

y~ = 0.15 F~x

but not more than 3 for any speed

but not more than 6 for any speed

DET NORSKE VERlTAS

16

For all kinds of steel

C = _!_( crHlim - 18.45)2

+1.5 ay 18 97

When pinion and wheel material differ, the following applies:

•

•

•

•


May2003

Use the larger of fp11 -Ya.1 and fi,12 - Yo.2 to replace frt ~Ya in the calculation of KHu and Kv.

Use y ll = ..!_ (y ~ 1 + y llJ in the calculation of KHP . 2

Use Ca = _!._ (cayl + Cay2) in the calculation of .K.,. 2

Use Ca1 = Ca2 = ..!.(cayi + Cay2 ) in the scuffmg 2

calculation if no design tip relief is foreseen.

DET NORSKE VERIT AS


May 2003

2. Calculation of Surface Durability

2. 1 Scope and General Remarks Part 2 includes the calculations of flank surface durability as limited by pitting, spalling, case crushing and subsurface yielding. Endurance and time limitod llank surface fatigue is calculated by means of2.2 - 2. 12. In a way also tooth fractures starting from the flank due to subsurface fatigue is included through the criteria in 2.13.

Pitting itself is not considered as a critical damage for slow speed gears. However, pits can creati:; a severe notch effect that may result in tooth breakage. This is particularly important for surface hardened teeth, but also for high strength through hardened teeth. f'or high-speed gears, pitting is not permitted.

Spalling and case crushing are considered similar to pitting, but may have a more severe eflcct on tooth breakage due to the larger material breakouts, initiated below the surface. Subsurface fatigue is considered in 2.13.

For jacking gears (self-elevating offshore units) or similar slow speed gears designed for very limited life, the max. static (or very slow running) surface load for surface hardened flanks is limited by the subsurface yield strength.

For case hardened gears operating with relatively thin lubrication oil films, grey staining (micropitting) may be the limiting criterion for the gear rating. Specific calculation methods for this purpose are not given here, but are under consideration for future revisions. Thus depending on experience with similar gear designs, limitations on surface durability rating olhcr than those according to 2.2 - 2 .13 may be applied.

2.2 Basic Equations Calculation of surface durability (pitting) for spur gears is based on the contact str ess at the inner point of single pair 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 < ep < 1, a linear interpolation between the above mentioned applies .

Calculation of surface durability for spiral bevel gears is based on the contact stress a t the midpoint of the zone of contact.

Alternatively for bevel gears the contact stress may be calculated with the program "BECAL". Jn that case, KA and Kv are to be included in the applied tooth force, but not KHfl and KH11 • The calculated (real) Hertzian stresses are to he 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 always represent the real Hertzian stresses.

The corresponding permissible contact stresses <iHr are to be calculated for both pinion and wheel.

17

2.2.1 Contact stress

Cylindrical gears

where:

ZR.n Zone factor for inner point of single pair contact for pinion resp. wheel (sec 2 .3.2).

7.11 Zone factor for pitch point (sec 2.3.1 ).

Ze .Elasticity factor (see 2.4).

Zc Contact ratio factor (see 2.5).

Z~ Helix angle factor (see 2.6).

F" KA, KP K,,, Ki111, Kn0 , sec 1.5 - 1.10.

di, b, u, sec 1.2 -1 .5.

Bevel gears

where:

1.05 is a correlation factor to reach real Hcrtzian stresses (when ZK =I)

Zc, KA etc. see above.

ZM = mid-zone factor, see 2.3.3.

ZK "" bevel gear factor, sec 2. 7.

Fmt> dvi, Uv, see l.2 - l.5.

lt is assumed that the heightwisc crowning is chosen so as to result in the maximum contact stresses at or near the midpoint of the flanks.

2.2.2 Permissible contact stress

- <i Hlim ZN z z Z z Z ()HP - S L v R W X

' H

where:

Endurance limit for contact stresses (see 2.8).

Life factor for contact stresses (see 2.9).

Required safety factor according to the rules.

Oil film in llucnce factors (see 2.10).

Work hardening factor (see 2. I 1 ).

Size factor (see 2.12).

D ET NORSKE VF.RITAS

18

2.3 Zone Factors Zn, Zn,D and Ziv1

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 point and converts the tangential force at the reference cylinder to t11e normal force at the pitch cylinder.

2cosphcosa.wc

cos2u sinu t wt

2.3.2 Zone factors Zn,o

The zone factors, Zu.J..» account for the influence on cont.act stresses ot'the tooth flank curvature at t11e inner point of single pair contact in relation to Z11• Index B refers to pinion D to wheel.

Forr.11'.::: l,ZR,n- I

For internal gears, Z0 = 1

For e11 = 0 (spur gears)

lf Z8 < 1, use ZR= I

If Zn < I , use Zu = 1

For 0 < e11 <1

Za,o = ZR,n (for spur gears) - ep (Z8,D (for spur gears)- I)

2.3.3 Zone factor ZM

The mid-zone factor ZM accounts for the influence of the contact stress al the mid point of the flank and applies to spiral bevel gears.

z _ 2cosPbm tana.v1dv1dv2

M - ( ~d~nl -d~t>I - CaPlnm )( ~d~a2 - d~b2 - l::o:Pbun)

This factor is the product of Zn and ZM-u in ISO I 0300 with the condition that the hcightwise crowning is sufficient to move the peak load towards the midpoint.

2.3.4 Inner contact point

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


May2003

2.4 Elasticity Factor, ZE The elasticity factor, ZE, accounts for the influence of the malerial properties as modulus of elasticity and Poisson's ratio on the contact stresses.

For steel against steel Zr:= 189.8

2.5 Contact Ratio Factor, Zi

The contact ratio factor Ze accounts for the influence of the transverse contact ratio e .. and the overlap ratio ell on the contact stresses.

for ep ~I

4-:; ( ) eµ ZF. = -3 a I - :;~ + -

ea

2.6 Helix Angle Factor, Z~

The helix angle factor, Z1i, accounts for the inllucncc of helix angle (independent of its influence on Z,.) on the surface durability.

Zjl =~COS~

2.7 Bevel Gear Factor, ZK The bevel gear factor accounts for the difference between the real Hertzian stresses in spiral bevel gears and the contact stresses assumed responsible for surface fatigue (pitting). ZK adjusts the contact stresses in such a way 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, O'Htim and Static Strength, c; Hto~, c;810~

crHJim is the limit of contact stress that may be sustained for 5· 10 7 cycles, without the occurrence of progressive pitting.

For most materials 5· l 07 cycles are considered to be the beginning of the endurance strength range or lower knee of the a-N curve. (See also Lifo 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.

crl-f10

s and crHloJ are the contact stresses which the given

material can withstand for I 05 respectively l 03 cycles without subsurface yielding or flank damages as pitting, spalling or case crushing when adequate case depth applies.

The following listed values for <1Htim, crH10

s and o-1110

1 may

only he used for materials subjected to a quality control as

DET NORSKE VERrT AS


May 2003

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%.

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

19

crHlim 5

(jHlO l

0HHI

1650 2500 3100 1500 2400 3100

Nitrided steel of approved grade, gas ni tridcd (surface hardness 700-800 HY): 1250 1.3 O"tJJim J.3 CJHlim

Alloyed quenched and tempered steel, bath or gas nitrided (surface hardness 500-700 HY): 1000 J .3 c;Hlim 1.3 Om;in

Alloyed, flame or induction hardened steel (surface hardness 500-650 HY): o.75 nv 1· 750 J.60Hlim 4.5HV

A lloyed quenched and tempered steel: 1.4 HY + 350 J .6 O"Hlim 4.5 nv Carbon steel: 1.5 HV + 250 J .6 CTHJim 1.6 0"1J1im

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

2.9 Life Factor, ZN The life factor, ZN, takes account of a higher permissible contact stress if only limited life (number of cycles, Ni) 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:

Z, =I o< ZN = ( 5~~7

)'""

T.e. ZN = 0.92 for 1010 cycles.

The ZN= I from 5· 107 on, may only be used when the material cleanliness is of approved high grade (see Rules Pt.4 Ch.2) and the lubrication is optimised by a specially approved filtering process.

(

7

)

0.37 logZ 5 5· 10 Nin

ZN = --Nr.

(but not less than ZN105)

for nitrided steels:

Z, =I orZN = ( 2~~6 f"" Le. ZN = 0.92 for 1010 cycles.

The ZN= 1 from 2· l 06 on, may only be used when the material cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process.

(

6

)

0 .76R6 log Z ~ 2· 10 NIO

ZN = - -N1,

Note that when no index indicating number of cycles is used, the factors arc valid for 5·l01 (respecti vcl y 2·106 for nitriding) cycles.

2.10 Jntluence Factors on Lubrication Film, ZL, Zv and ZR The lubricant factor, ZL, accounts for the influence of the type ofluhricant and its viscosity, the speed factor, Zv, account.s for lhe influence of the pitch line velocity and the rougJmess factor, ZR, accounts for influence of the surface roughness on lhe surface endurance capacity.

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

D ET NORSKE V ER!TAS

20

ZL

Zv

ZR

where:

Surface hardened steels Not surface hardened steels

0 91 0.36 . + ( 2 l.2+134/v40 )

0.83+ 0.68 2

(1.2 + 134 Iv 40 )

0 93 0.14

· + .Jo.8+(32 /v) 0.85+ 0.30

Jo.8+(32/v)

(-3 r08 R zrel

( rlS R~rol

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

For values ofv40 > 500, use v40 --: 500.

R:Grcl = The mean roughness between pinion and wheel (after running iii) relative to an equivalent radi us of curvature at the pitch point Pc = 1 Omm.

Mean peak to valley roughness (µm) (DIN definition) (roughly Rz = 6 Ra)

2.11 Work Hardening Factor, Zw The work hardening factor, Zw, accounts for the increase of surface durability of a soft steel gear when meshing the soil steel gear with a surface hardened or substantially harder gear with a smooth surface.

The following approximation may he used for the endurance limit:

Surface hardened steel against not s urface hardened steel;

z =[l.2 - HB-130][-3-]o.is w 1700 Rz.cq

where: HB .., the Brinell hardness of the soft member For HB > 470, use HB = 470 For HB < 130, use HB = 130

Rzcq =equivalent roughness


R _ R Rn-1 15000

( )

0.66 ( )0.33

Z1'Q- 7H --Rzs V40 V Pc

lfRzeq > 16, then use Rzcq = 16

If Rzeq < 1.5 , then use Rzeq = 1.5

where:

May2003

Rn1 = surface roughness of the hard member before run in.

R:Gs surface roughness of the sofl. member before run in

V40 see 2. J 0.

ff values of Zw < 1 are evaluated, Zw -= 1 should be used for flank endurance. However, tht: low value for Zw may indicate a potential wear problem.

Through hardened pinion against softer wheel:

Zw = 1 + (u- 1)·(0.00898 IIB, - 0.00829] IIB2

F HB, <12 or _ . HH2

For HB, > 1.7 HB2

use Zw = I

use HB, = 1.7 HB2

For u > 20 , use u = 20

For static strength(< 105 cycles):

SLLrface hardened againsl not surface hardened

7,w~• "' 1.05

Through hardened pinion against softer wheel

Zwst = l

2.12 Size Factor, Zx

The size factor accounts for statistics indicating that the stress levels at which fatigue damage occurs decrease with an increase of componcnl size, as a consequence of the influence on subsurface defects combined with small 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 fo llowing subsection 2.13.

2.13 Subsurface Fatigue

This is only applicable to surface hardened pinions and wheels. The main objective is to have a subsurface safety against fatigue (endurance limit) or deformation (static strength) which is at least as high as the safety S11 required for the surface. The following method may be used as an approximation unless otherwise documented.

The high cycle fatigue (>3· 106 cycles) is assumed to mainly depend on the orthogonal shear stresses. Static strength (< 103 cycles) is assumed to depend mainly on equivalent

D ET NORSKE VERITAS


May2003

stresses (von Mises). Both are inl1uenced by residual stresses, but this is only considered roughly and empirically.

The subsurface working slresses at depths inside the peak of the orthogonal shear stresses respectively the equivalent stresses are only dependcnl on the (real) Herlzian stresses. Surface related conditions as expressed by ZL , Zv and ZR arc assumed to have a negligible influence.

The real Hertzian stresses o11R arc determined as:

For helical gears with r.11 ~ l :

For helical gears with cp < l and spur gears:

For bevel gears:

The necessary hardness HV is given as a function of the net depth t~ (net = after grinding or hard metal hobbing, and perpendicular to the Jlank).

The coordinates tz and TTY are to be compared with the design specification, such as:

• for Clame and induction hardening; tnvmin, HY min • for nitriding; l:.ioomin• HV = 400 • for case hardening; t5~min• HV = 550; t400m;n, HV = 400

and t300m;n, HV = 300 (the latter only if the core hardness < 300. If the core hardness> 400, the ~00 is to be replaced by a fictive 4oo = 1.6 · tsso).

ln addition the specified surface hardness is not to be less than the max necessary hardness (at. l~ = 0.S-au). This applies to all hardening methods.

For high cycle fatigue (>3 · I 06 cycles) the following applies:

--0.5

[

tz l HV == 0.4 · o11R ·Su· cos ~I-I · 90° i+o.5

applicable to 1~ 0.5 aH

aH

21

For ~ < 0.5 the value for ~ = 0.5 applies aH an

a =l.2 · cr11R·SH· Pc H 56300

Where a8 is half the hertzian contact width multiplied by an empirical factor of J .2 that takes into account the possible influence ofrcduced compressive residual stresses (or even tensile residual stresses) on the local fatigue strength.

If any of the specified hardness depths including the surface hardness is below the curve described by HV = f(t,,), the actual safety facto r against subsurface fatigue is dctennined as follows:

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

For static strength ( <l 03 cycles) the following applies:

HY =0. 1 9·crHR·S,,·cos[~ - 0.6 ·90°] ~+0.7 a H.s1

applicable to ...!i__ ~ 0.6 aHst

Tn Lhe case of insufficient specified hardness depths, the same procedure for determination of the actual safety factor as above applies.

For limited life fatigue (I 03 <cycles< 3· J 06 ):

For this purpose it is necessary to extend the correction of safety factors to include also higher values than required. I.e. in the case of more than sufficient hardness and depths, the safety factor in the formulae for both high 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 interpolation in a double logarithmic diagram.

logs 6 - logs 3 logs = H3·10 HIO · logN-HN 3.477

0.8628 · logSIJ3.I06 + 1.8628 · logSHIO)

DET NORSKE YERIT AS

22

3. Calculation of Tooth Strength

3.1 Scope and General Remarks

Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurtace initiated) and yielding.

For rim thickness sR ~ 3.5·mn the strength is calculated by means of3.2- 3.13. For cylindrical gears the calculation is based on the assumplion that the highe:,1 tooth root ll-'Tisile stre.ss arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears. The method has, however, a few limitations I.hat are mentioned in 3.6.

For bevel gears the calculation is based on force application at the tooth tip of the virlual cylindrical gear. Subsequently the stress is converted to load application at the mid poinl of the flank due to the heightwise crowning.

Bevel gears may also he calculated with the program BECJ\L. Jn that case, KA and Kv are to he included in the applied tooth force, but not Kr~ and KFn·

In CHse ofa thin annulus or a thin gear rim etc, radial cracking can occur rather lhan tangential cracking (from root fillet to root fillet). Cracking can also start from the compression fillet rather than the tension fillet. For rim thickness sR < 3.5·mn a special calculation procedure is given in 3.15 and 3.16, and a simplified procedure in 3.14.

J\ tooth breakage is often the end () r the life of a gear transmission. Therefore, a high safety SF against breakage is required.

Jt should be noted that this part 3 does not cover fractures caused by:

• oil holes in lhc tooth root space • wear steps on the flank • nank surface distress such as pi Lei, spalls or grey stain ing

Especially the latter is known lo cause oblique fractures starting from the active flank, predominately in spiral bevel gears, but also sometimes in cylindrical gears.

Specific calculation methods for these purposes are not given here, but are under consideration for future revisions. Thus, depending on experience with similar gear designs, limitations other than those outlined in pan 3 may be applied.

3.2 Tooth Root Stresses

The local tooth root stress is defined as the max. principal stress in the looth root caused by application of the tooth force. Le. the stress ratio R = 0. Other stress ratios such as for e.g. idler gears (R ::::: -1.2), shrunk on gear rims (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 measuremenL<; or FE calculations or similar. For both measurements and calculations all details arc to be agreed in advance.


May2003

Normally, the stresses for pinion and wheel are calculated as:

Cylindrical gear~·:

crl:' = bFt Yp Y,'l Yp KA K,. Kv Kpp KFa mn

where:

YF = Toolh form factor (see 3.3).

Y s = Stress correction factor (see 3.4).

Y~ = Helix angle factor (see 3.6).

Ft. KA, Ky, Kv, K"~' Kfo, sec 1.5 -1.10.

h, see t.3.

'Bevel gears:

where:

Y p.= Tooth form factor, see 3.3.

Ysa~ Stress correction factor, see 3.4.

Ye = Contact ratio factor, sec 3.5.

Fm" KA, etc., see 1.5 -· 1.10.

b, see 1.3.

3.2.2 Permissible tooth root stress

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

- Oi:;}.YM YN crpp - Yl\relT YHrclTYx Ye

Sy

Note that all these factors Y M etc. are applicable to 3· l 06

cycles when used in this formula for <1FP· The inlluc11ce of. other number or cycles on these factors is covered by I.he calculation ofYN.

where:

crFE Local tooth root bending endurance limit of reference tesl gear (see 3.7).

YM Mean slrcss influence factor which accounts for other loads than constant load direction, e.g. idler gears, temporary change of load direction, pre-stress due to shrinkage, etc. (see 3.8).

Yl\relT

YRrclT

Life factor for tooth root stresses related to reference test gear dimensions (see 3.9).

Required safety factor according to lhe rules.

Relative notch sensitivity factor of the gear to be determined, related to the reference test gear (see 3.10).

Relative (root fillet) surface condition factor of the gear to be determined, related to the reference test gear (see 3.11 ).

DETNORSKR YERITAS


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Yx Sizefactor(sce 3.12).

Y c Case depth factor (see 3.13).

3.3 Tooth Form Factors Yp, YF~

The lOoth form factors Yr and YFa take into account the influence of the tooth form on the nominal bending stress.

Yr applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears.

Y F& applies to load application at the tooth tip and is used for bevel gears.

Both Y F and Y i:a are based on the distance between the contact points of the 30° -tangents at the root fillet of the tooth 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

Definitions:

6 hre -COS a.Fen

y - mn F - 2

(~:) cos a"

23

6 hFa -COS ClFan

y - mu Fa - ---"--.,---( ~~ J COS O.n

ln the case 0 f helical gears, Y F and Y F'a are determined in the normal section, i.e. for a virtual number of teeth.

Yi:a differs from Y F by the bending moment arm hp8 and C'LFan

and can he determined by the same procedure as Y F wilh exception ofhre and C'Lfan. for hFa and ClFau all indices c will

change to a (tip).

The following formulae apply to cylindrical gears, but may also be used for bevel gears when replacing:

Z,, with Zvn

IXt with Clvt

~with Bm

---------r --------r I I I I I I ,e. I r.

E ~ I I I

I I I I I

E ~ I I I I

with undercut without undercut

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

Tool and basic rack data such as ho>, Pep and Spr etc. are referred to mn, i.e. dimensionless.

3.3.1 Determination of parameters

[

1t Pfr'(l - sin a.n)-spr ] E ::= - - hlPtait an - m 11

4 COS Un

For external gears PIP'= PIP

(x + h _ )' ·95 For internal gears p '= p + 0 IP PIP

IP fP 3.156 · 1.036"0

where

z0 = number of teeth of pinion cutter

x0 = addendum modification coefficient of pinion cutter

hw = addendum of pinion cutter

PtP = tip radius of pinion cutter

G = Prp'-hrp+ x

D ET NORSKE VERIT AS

24

with

7t 'r = - for external gears

3

't = ~ for internal gears 6

&= 2_Q_tan 8-- H Zn

(to be solved iteratively, suitable start value 9 = 2: for exter-6

nal gears and 2: for internal gears). 3

a) Tooth root chord Sfn:

For external gears

Svn · (1t ") r;;3( G ') - = z0 sm --,~ +v j ---Pfp mn 3 cos s

For hevel gears with a tooth thickness modification:

Xsm affects mainly Srn, but also hi:c and Ctfcn· The total influence ofx,m on YFa Ysa can be approximated by only adding 2 Xsm to SFn I m •.

for internal gears

Srn . ( 1t n) ( G ·) - = z0 s1n -- '~ + - --Pt'P m0 6 cos$

b) Root fillet radius PF at 30° tangent:

Pr ' 2G2 - = pfp + ( ) mil cos &\Zn cos 2 S - 2 G

c) Determination of bending moment arm h1,:

dn = Zn nln

e n C:un =--2-

COS ~ b

Pbn = 1t m., cos a,,

dbn = d., COS lln

d a = arccos -.lm...

Cll d en

1 (7t 2 ) . . y., = - - + X tan Ct11 + IOV an - lllV a.en Zn 2


May 2003

For external gears

hFe = J_[(cos 'II -sin 'II tan ex. }den 2 r c 1 e Fen m. m11

- Z n cos( 2: - s)-_G_ +Pr ·] 3 cos s p

ror internal gears

hre 1 [( . ) den - = - cos ye - Sill ye· tan llrcn · - -m11 2 m11

3.3.2 Gearing with ~on > 2

for deep tooth form gearing (2:::;; ean S'. 2.5)produced with a veri fied grade of accuracy of 4 OT hetter, and with applied profile modification to obtain a trape;i:oi<lal load distribution along t11e pat.h of contact, the Y F may be corrected by the factor Y or as:

Y DT = 2.366 - 0.666f:<mfor 2.05 :::;; f: "" ~ 2.50

Y DT = 1.0 for f:un < 2.05

3.4 Stress Correction Factors Y s, Y su

The stress correction fact.on> Y s and Y sa take into account the conversion of the nominal bending stress to the local Louth root stress. Thereby Y s and Y sa cover the stress increasing ellcct of the notch (fillet) and the fact that not only bending stresses arise at the root. A part of the local stress is independent of the bending moment arm. This part increases the more the decisive point of load application approaches the critical tooth root section.

Therefore, in addition to it.s dependence on the notch radius, the stress correction is also dependent on the position of the load application, i.e. the size of the bending moment arm.

Y s applies to the load applil:ation at the outer point of single tooth pair contact, Y sa to the load application at tooth tip.

Y s can be determined as follows:

SF where: L= - " hrc

and q = srn s 2pr

(sec 3.3).

Y so can be calculated by replacing hi:~ with hi:. in the above formulae.

Note:

11) .Range of validity I < q, < 8

In case of sharper root radii (i.e. produced with tools having too ~harp tip radii), Y s resp. Y s• mw;t bt: specially considered.

b) Jn case of grinding notcht:8 (due to iosufiicient protuberance of tht: hob), Ys resp. Ys. can ri8e e(111sidcrably, and must be mul tiplied wilh:

DET NORSKE V ERITAS


May2003

1.3

where:

ls = depth of the grinding notch

Ps = radius of the grinding notch

c) The formulae for Ysrcsp. Ys. are only valid for a.,, =- 20". However, the same formu lae can be used as a lX!fe approximation for other pressure angles.

3.5 Contact Ratio Factor Y2

The contact ratio factor Ye covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears.

The following may be used:

y~ = 0.625

3.6 Helix Angle Factor Y 11

The helix angle factor Y ~ takes into account the difference between the helical gear and the virtual spur gear in the normal section on which the calculation is based in the first step. In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact arc sloping over the flank.

The following may be used (13 input in degrees):

Y II :...: l - Ej1 J3!120

When f.11 > 1, use e~ ~ 1 and when 0 > 30°, use J3 = 30° in the formula.

However, the above equation for Y ~ may only be used for gears with 13 > 25° if adequate tip relief is applied to both pinion and wheel (adequate = at least 0.5 · Ceff• see 4.3.2).

3.7 Values of Endurance Limit, <JFE

Orr: is the local tooth root stress (max. principal) which the material can endure permanently with 99% survival probability. 3· 106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the o - N curve. om is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment). Other stress conditions such as alternating or pre-stressed etc. are covered by the conversion factor Y M·

OFF. can be found by pulsating tests or gear running tests for any material in any condition. If the approval of the gear is

to be based on the results of such tests, all details on the testing conditions have to be approved by the Society. Further, the tests may have to be made under the Society's supervision.

Ifno fatigue tests are available, the following listed values for <iFr: may be used for materials subjected to a quality control as the one referred to in the rules.

Alloyed case hardened steels 11 (fillet surface hardness 58 - 63 HRC):

• or specially approved high grade:

• of normal grade:

- CrNiMo steels with approved

1050

process: 1000 - CrNi and CrNiMo steels generally: 920 - MnCr steels generally: 850

Nitriding steel of approved grade, quenched, tempered and gas nitrided 840 (surface hardness 700 - 800 HV):

Alloyed quenched and tempered steel, bath or gas nitrided (surface hardness 500 720 - 700HV):

Alloyed quenched and tempered steel, flame or induction hardened 2> (incl. entire root fillet) (ti llct surface hardness 500 -650 HV): 0.7 HV + 300

Alloyed quenched and tempered steel, flame or induction hardened (excl. entire root fillet) ( cr8 ,..:; u.t.s. of base material): 0.25 c:rn + 125

Alloyed quenched and tempered steel: 0.4 cr8 + 200

Carbon sleel: 0.25 cru + 250

Note: All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure, sec Pt. 4 Ch. 2 Sec. 3. For rolled steel, the values are to be reduced with I 0%. For blanks cut from forged bars, that are not qualified as mentioned above, the values arc to be reduced with 20%, For cast steel, reduce with 40%.

I) These values arc valid for a root radius

• being tmground. If, however, any giinding is made in the root fillet Nrea in such a way that the residua l stresses may be n.tlectcd, ore; is to be reduced by 20%. (If the grinding alsv leaves a notch, sec 3.4).

• with fillet surface hardness 58 - 63 HRC. In case nr lower surface hardness than 58 I IRC, riH; 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 faccwidth . lf so, the actual tooth root stresses may also be considered along facewidth .)

• not being shot peened. In case o r approved shot peening, O'fE may be increased by 200 for gears where O'fE is reduced by 20% due to root grinding. Otherwise O'FE may be increased hy 100 for

mn 56and I00 - 5(m11 -6) form0 >6.

However, the possible adverse influence on the tlanks regarding grey staining should he 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 ).

D ET NORSKE VERITAS

25

26

3.8 Mean stress influence Factor, YM

The mean stress influence factor, Y M• takes into account the influence of other working stress conditions than pure pulsations (R = 0), such as e.g. load reversals, idler gears, planets and shrink-fitted gears.

Y M (Y M•~ are de fi ned as the ratio between lhc endurance (or static) strength with a stress ratio R .f. 0, and the 1::nduraol;e (or static) strength with R = 0.

Y M and Y Mst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress 0 1, and the permissible stress o FP calculated with Y M or Y Mst·

For thin rings (annulus) in epicyclic gears where the "compression" fi llet may be decisive, special considerations apply, see 3.16.

The following method may be wse<l within a stress ratio - 1.2 < R < 0.5:

3.8.1 For idlers, planets and PTO with ice class

I y M or y Mst = 1 - M

where:

1-R-l+M

R = stress ratio = min. stress divided by max. stress.

For designs with the same force applied on both forwardand back-flank, R may be as.~umed to - 1.2.

For designs with considerably different forces on forwardand back-flank, such as e.g. a marine propulsion wheel with a power take off pinion, R may be assessed as:

_ 1 _2

force per unit faccwidt of p.t.o. force per unit facewidth of the main branch

For a power take off (PTO) with ice class, sec 1.6.1 c.

M considers the mean stress influence cm the endurance (or static) strength amplitudes.

M is defined as the reduction of the endurance strength amplitude for a certain increase 0 r the mean stress divided by that increase of the mean stress.

Following M values may be used:

Endurance Static limit strength

Case hardened 0.8 - 0.15 Y5 I) 0.7

If shot peened 0.4 0.6

Nitrided 0.3 0.3

Induction or flame 0.4 0.6 hardened

Not surface hardened steel 0.3 0.5

Cast steels 0.4 0.6 I)

' = ' For bevel gears, use Y, 2 for determmanon of M.

Classitication Notes- No. 41.2

May 2003

The listed M values for the endurance limit are independent of the fillet shape (Y,), except for case hardening. Jn principle there is a dependency, but wide variations usually only occur for case hardening, e.g. smooth semicircular fillets versus grinding notches.

3.8.2 For gears with periodical change of rotational direction

For case hardened gears with full load applied periodically in both directions, such as side thrusters, the same fonnula for Y M as for idlers (with R = - 1.2) may be used together with the M values for endurance limit. This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3· I 06

.

For gears of other materials, Y M will normally be higher than for a pure idler, provided the number of changes of direction is below 3· 106

. A linear interpolation in a diagram with logarithmic number of changes or direction may be used, i.e. from Y M = 0.9 with one change to Y M (idler) for 3· l 06

changes. This is applicable to Y M for endurance limit. For static strcn!,>th, use Y M as for idlers.

For gears with occasional full load in reversed direction, such as the main wheel in a reversing gear box, Y M = 0.9 may be used.

3.8.3 For gears witb shrinkage stt·esses and unidirectional load

For endurance strength:

y M = I - 2 M (j fit

1 + M cr..-i;

crF'R is the endurance limit for R = 0.

For static strength, Y~1 = l and Ofii accounted for in 3.9.b.

a1;l is the shrinkage stress in the fi llet (30• tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor:

f I 5 2pr

SC fit. = . ---mn

3.8.4 For shrink-fitted idlers and planets

When combined conditions apply, such as idlers with shrinkage stresses, the design factor for endurance strength is:

l YM= 1-M

1- R-l+M

2M otit

(1 + M)· (1 - R) crFF.

Symbols as above, but note that the stress ratio R in this particulllf connection should disregard the influence of or." i.e. R normally equal - 1.2.

For static strength:

1 Y Mst = 1-M

1-R-l+M

The effect of Ofii is accounted for in 3.9 b.

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3.8.5 Additional requirements for peak loads

The total stress range (omax - Omin) in a tooth root fillet is not to exceed:

2.25oy

Sp

SHV Sp

for not surface hardened fillets

for surface hardened fillets

3.9 Life Factor, Y N

The life factor, YN, takes into account that, in the case of limited life (number of cycles), a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles.

Decisive for the strength at limited life is the cr - N - curve of the respective material for g iven hardening, module, fillet radius, roughness in the tooth root, etc. Le. the factors Y oreJT,

YRcrr. Yx and YM have an influence on Y111.

lf no cr - N - curve for the actual material and hardening etc. is available, the following method may be used.

Determination of the o - N - curve:

a) Calculate the permissible stress C>Fr for the beginning of the endurance limit (3· 106 cycles), including the influence of all relevant factors as Sp, Y ~rcrT, Y Rem Y x, Y M and Ye, i.e. OFP == crFL! ·YM .YlircJT ·YR.err ·Yx ·Y c I Sr

b) Calculate the permissible «static» stress ( ~ l 03 load cycles) including the inf\ uence of all relevant factors as SF61i Y &err,.,, Y MS! and Y esi:

cr FPst = ~ (CJ Fst . y Mst . y orelTst . y Cst - CJ fit ) ' Fst

where crF•t is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 990/o survival probabili ty.

27

Or.t

Alloyed case hardened steel 1' 2300

Nitriding steel, quenched, 1250 tempered and gas nitrided (surface hardness 700-800 HV)

Alloyed quenched and tempered 1050 steel, bath or gas nitrided (surface hardness 500 - 700 HV)

Alloyed quenched and tempered 1.8 HV + 800 steel, flame or induction hardened (fillet surface hardness 500- 650 HV)

Steel with not surface hardened 1.8 crR or 2.25 cry fillets, the smaller value of2l

I) This is valid for a fillet surface hardness of58 - 63 HRC. In case of lower fillet surface hllrdness than 58 HRC, CJFst is to he reduced wiLh 30·(58 - HRC) where URC is the actual hardness. Shot peening or grinding notches are not considered to have any significant influence on CJFst·

2) Actual stresses exceeding the yield point (oy or o0.2) will alter the residual stresses locally in the "tension" fillet re-spectively "compression" fillet. This is only to he utilised for gears that lire not later loaded with a high number of cycles at lower loads that could cause fatigue in the "compression" fillet.

c) Calculate YN as:

YN = l or

NL> 3·106 - [ 3 . 106

) 0.01 i.e. Yn = 0.92 for 1010 YN - --NL

The Y N = l from 3· 106 on may only be used when special material cleanness applies, see rules Pl.4 Ch.2.

103<Nt <3· 106 [ 3 .10

6) ~p YN = --

NL

3 _ 0 2876 1 a FP•Jor I 0 cycles exp - . og 6

CJ FP for 3 · 10 cycles

3 y _ cr fP.t for I 0 cycles N -

NL< 103 o rr for 3 · I 0 6 cycles

or simply use crFP.r. as mentioned in b) directly.

Guidance on number ofload cycles N1. for various applications :

• For propulsion purpose, normally N1. == 1010 at full load (yachts etc. may have lower values).

• For auxiliary gears driving generators that normally operate with 70-900/o of rated power, NL == 108 with rated power may be applied.

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28

3.10 Relative Notch Sensitivity Factor, Y 5rerr

The dynamic (respectively static) relative notch sensitivity factor, Ys,c1T (YsrelTsi) indicate to which extent the theoretically concentrated stress lies above the endurance limits (respectively static strengths) in the case or fatigue (respectively overload) breakage.

Y orcJT is a function of the material and the relative stress graclient. Tt differs for static strength and endurance limit.

The following method may be used:

Por endurance limit:

for not surface hardened fillets:

1+(0.135-1.22· 10-4 ·0'0.2~ YorelT = 1.33 - 3. J o-4 . 0'02

for all surface hardened fillets except nitrided:

y - (t + 0.0245~) OrcJT - 1.06

for nitrided fillets:

(1+0.142~1+2 qA ) YsrelT = -'---........:.--"-L

1.347

For static s trength:

for not surface hardened fillets1):

Y .. = t + o.s2(Y5 - t)(3001a0 .. 2 )°'25

&cl l st I + 0.82(300/a o.2)°'25

for surface hardened fillets except nitrided:

Y 6rc1Tst = 0.44 Y S + 0.1 2

for nitrided fillets:

Y &crrs1 = 0.6 + 0.2 Y s

1> These values are only valid if the local stresses do not exceed the yield point and thereby alter the residual stress level. Sec also 3.9b footnote 2.

3.11 Relative Surface Condition Factor, Y Rrcrr

The relalive surface condition factor, Y RrelT• takes into account the dependence of the tooth rool strength on the surface condition in the tooth root fillet, mainly the dependence on the peak to valley surface roughness.

Y RrctT dillers for endurance limit and ~tatic strength.

The following method may be used:

For endurance limit:

Y Rrerr = 1.675 - 0.53 (Ry+ 1 )0· '

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

Y Rrerr = 5.3 - 4.2 (Ry+ 1)0.01

for carbon steels

y &err == 4.3 - 3 .26 (Ry + l)o.aos for nitrided steels


May2003


Y RrclT•• = 1 for all Ry and all materials.

For a fillet without any longitudinal machining trace, R).z. R,.

3.l2 Size Factor, Yx

The sh:e factor, Y x, takes into account the decrease of the strength with increasing size. Y x differs for endurance limit and static strength.

The following may be used:


Yx= 1 form. ~5 generally

y x ""' t .03 - 0.006 1n., for 5 < m. < 30 for not

Yx = 0.85 for mn ;;:: 30 surface hard-ened steels

Yx = l.05 - 0.01 m0 for 5 < m0 2 25 for surface

Yx = 0.8 ror ffin ;:::25 hardened steels


Y xsi = 1 for all mn and all materials.

3.13 Case Depth Factor, Y c

The case depth factor, Y c. takes into account the influence of hardening depth on tooth root strength.

Ye applies only to surface hardened tooth roots, and is different for endurance limit and static strength.

Jn case of insufficient hardening depth, fatigue cracks can develop in the transition £one between the hardened layer and the core. For static strcni,.rth, yielding shall not occur in the transi tion 7.one, as this would alter the surface residual :.1resses and therewith aJso the fatigue strength.

The major parameters arc case depth, sLress gradient, permissible surface respectively subsurface stresses, and subsurface residual stresses.

The following simplifi ed method for Y c may be used.

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

Ye and Y csi may be calculated as given be low, but calculated values above 1.0 arc to be put equal 1.0.


Y _ const. [i 3t ) c - -- +- ---(j'foE pl'+0.2mn

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y = const. [i + 3 t ) Cst O 2 0 rsl Pr+ . mn

where const. and t are connected as:

llardening t= endurance static strength orocess limit consl = cons!=

l550 640 1900 Case hard-cning tioo 500 1200

tJoo 380 800

Nitriding Lioo 500 1200

Induction- or flame tuvmin \.1 HVmin 2.5 HVmin hardening

For symbols, sec 2.13.

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

The max. depth to 550 HV should not exceed

1) l/3 of the top land thickness s.n unless adequate tip relief is applied (see 1.10).

2) 0.25 m 11• Jfthis is exceeded, the following applies additionally in connection with endurance limit:

Ye =l - (ts50m.u -o.2s) mn

3.14 Thin rim factor Y 8

Where the rim thickness is not sufficient to provide foll support for the tooth root, the location of a bending failure may be through the gear rim, rather than from root fillet to root fillet.

Y s is not a factor used to convert t.:alculated root stresses at the 30° tangent to actual stresses in a thin rim tension fillet. Actually, the compression fillet can be more susceptible to fatigue.

Yfl is a simpli fied empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calculation of stresses in both tension and compression lillets are available.

Figure 3.4 Examples on thin rims

Ya is applicable in the range J.75 < sl{/mn < 3.5.

YR= l.l5 · In (8.324 · m,/sR)

(for sR/mn 2: 3.5 , Y 13 =-= 1)

(for sR/mn :S 1.75 , use 3.15)

crF as calculated in 3.2.1 is to be multiplied with Ya when s1tfmn < 3.5. Thus Y 8 is used for both high and low cycle fatigue.

29

Note: This method is considered to be on the safe side for external gear rims. However, for internal gear rims without any flange or web stiffeners the method may not be on the safe side, and it is advised to check with the method in 3. 15/3. 16.

3.15 Stresses in Thin Rims

For rim thickness sR < 3.5 mn the safety against rim cracking has to be checked.

The following method may be used.

3.15.1 General

The stresses in the standardised 30° tangent section, tension side, arc slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sw'mn. On the other hand, during the complete stress cycle of that fillet, a certain amount of compression stresses are also introduced. The complete stress range remains approximately constant. Therefore, the standardised calculation of stresses at the 30" tangent may be retained for thin rims as one of the necessary criteria.

The maximum stress range for thin rims usually occurs at the 60" - 800 tangents, botJi for 1<tension» and «compression» side. The following method assumes the 75° tangent to be the decisive. Therefore, in addition to the a.m. criterion applied at the 30° tangent, it is necessary to evaluate the max. and min. stresses at the 75" tangent for both «tension» (loaded flank) fillet and «compression» (back-llank) fillet. For this purpose the whole stress cycle of each fillet should be considered, but usually the following simplification is justified:

Fr

: <J1 (""'-" nf·;.,,.1.,.,

: 01 .....

Figure 3.5 Nomenclature of fillets

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

crFTmin and crFcm3 x are determined on basis of the nominal rim stresses times the stress concentration factor, Y 75 •

<TFcmin and crnmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75" tangent.

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30

3.15.2 Stress concentration factors at the 75° tangents

The nominal rim stress consists of bending stresses due to local bending momenls, tangential stresses due to the tangential force F" and radial shear stresses due to F,.

The major influence is given by the bending stresses. The influence of the tangential sLrcsscs is minor, and even though ils sln:ss concentration factor is slightly higher than for bending, it is considered to be safo enough when the sum of these nominal stresses are combim.xl with the stress concentration fa.cwr for bending. The inllucnce of the radial shear stress is neglected.

The stress concentration faclor relating nominal rim stresses to local fillet stresses at the 75° tangent may be calculated as:

where P1s is the root radius at the 75° tangent rcr. to mn. Usually p,5 is closed to the tool radius p.0 , and

p75 = pll.()

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

The tooth root stresses of the loaded tooth arc decreasing with decreasing relative rim thickness, approximate1y with the empirical factor

3.15.3 Nominal rim stresses

The bending moment applied to the rim consists of a part of the tooth tilt moment Ft(hr + 0.5 sK) and the bending caused by the radial force F,.

The sectional modulus (first moment or area) which is used for detenninalion of the nominal bending stresses is not necessarily the same for the 2 a.m. bending moments. Tf flanges, webs, etc. outside the toothed section contribute to stiffening the rim against various deflections, th.e influence of these stiffeners should be considered. E.g. an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange. On the other hand, the radial forces, as for instance from the meshes in an annulus, would cause considerable radial deflections that the flange might restrict to a substantial amount. When considering the stiffening of such flanges or webs on basis of simplified models, it is advised to use an effective rim thickness 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 the loaded tooth arc of the same magnitude as right under the applied


May2003

force. This <1ssumption is reasonable for an annulus, but rather much on the safe side for a hollow pinion.

The influence of'Ft on the nominal tangential stress is simpli fied by half of it for compressive stresses (cr1) and the olhcr half for tensile stresses ( cr2). Applying these assumpti"ons, the nominal rim stresses adjacent to the loaded tooth arc:

,... _ - 0.5F,(hp+0.5sR) f rR f(S)_-5..._ v1-

WT WR 2A

C12 = 0.5F,(hF+0.5sR) _ F,R r(s) +_!i_ WT W11. 2A

where e>i, o2 see Fig. 3.5.

A minimum area of cross section (usually bR s1J. the sectional modulus of rim with respect to tooth

tilt moment (usually h11. s~ I 6 ).

the sectional modulus of the rim including the influence of stiffeners as tlanges, webs etc. (Wit 2: WT)·

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 ( S) a function for bending moment distribution around the rim. For a rim (pinion) with one mesh only, the f(S) at the position of load application is 0.24.

For an annulus with 3 or more meshes, f(S) at the position of each load application is approx.: 3 planets f(9) = 0.19

4 planets f(B-) = 0.14

5 planets f( S) = 0.11

6 planets f( 13-) = 0 .09

Tt must be checked if the mait. (tensile) stress in the compression fillet really occurs when the fillet is adjacent to the loaded tooth. Tn principle, the stress variation through a complete rotation should be considered, and the max. value used. The max. value.: is usually never less than 0. For an annulus, e.g. the tilt moment is .:ero in the mid position between the planet meshes, whilsl the bending moment due to the radial forces is half of that at the mesh but with opposite sign.

If these fonnulac arc applied to idler gears, as e.g. planets, the influence of nominal tangential stresses must be corrected by deleting F.1(2 A) for cr1, and using F/A for cr2•

Further, the influence oJ'F, on the nominal bending stresses is usually negligible due to the planet bearing support.

3.15.4 Root fillet stresses

Determination of min. and max. stresses in the «tension» lillct:

D ET NORSK£ VERITAS


May 2003

Minimum stress:

Maximum stress:

where:

0.3 is an empirical factor relating the tension stresses (aF) at the 30u tangent to the part of the tension stresses at the 75° tangent which add to the rim related stresses. (0.3 also talces into account that full superposition of nominal stresses times stress concentration factors from both «sides of the corner fi llet» would result in too high stresses.)

K = KA · Ky · K v • K f~ · K Fo.

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

Mi11imum stress:

Maximum stress:

where:

0.36 is an empirical factor relating the tension slrcsses (crF) at the 30" tangent to the part of the compression stresses at the 75" tangent which add to the rim related stresses.

For gears with reversed loads as idler gears and planets there is no djstinct «tension» or «compression» fillet. The minimum stress crFmin is the minimum of <TFTmin and al'Cmin (usually the latter is decisive). The maximum stress crmnax is the maximum of CTffoax and Gfcmax (usually the former is decisive).

3.16 Permissible Stresses in Thin Rims

3.J 6.J General

The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for soljd gears. The «ordinary» criteria at the 30° tangent apply as given in 3. l through 3.13.

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

3.16.2 For >3·106 cycles

The permissible stresses for the «tension» fillets and for the «compression» fillets are determined by means of a relevant fatigue diagram.

If the actual tooth root stress (tensi le or compressive) exceeds the yield strength to the material, the induced residual stresses are to be talcen into account.

For determination of permissible stresses the following is defined:

R = stress ratio, i.e. crFTmin respectively GFCmin

C>FTmax OfCmax

6.o = stress range, i.e. crFTmAx - crl'Tmin resp. crFCmax - OFcmin·

(J f . (For idler gears and planets R = ~

OFmax

and 6cr = crFmax - crrruin).

The permissible stress range 6.op for the «tensio1m respectively «compression>> fi llets can be calculated as:

For R > - 1 A J.3 L.lOP = 1 + R crFP

1+0.3 -1- R

For - co< R < -1 1.3

!icrp = l + R crr.p 1+0.15-·

1-R

where:

GFr = sec 3.2, determined for unidirectional stresses (YM = I).

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

E.g. if Prcmin l >cry, (i.e. exceeded in compression), the

difference t:, = la r c min I-a >' affects the stress ratio as

R = cr l'Cmio +A = - cry

G l'Cmax + A (J FCmax + 6

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

If the yield strength is exceeded in tensio11, Grnnax >cry, the difference 6 = aFTmax - cry affects the stress ratio as

R = crFTmin - A = crnmin-.6. GFTmax - fi cry

Checking for possi ble exceeding of the yield strength has to be made with the highest torque and with the KAr if this exceeds KA.

3.16.3 For~ 103 cycles

The permissible stress range liap is not to exceed:

ForR> -1 1.5 6 0 p.<1 = 1 + R (J Fl'<I

1+ 0.5 --1- R

For - oo < R < - I 1.5 l+R aPPst

1 + 0.25 I- R

For all values ofR, Aapst is limited by:

2.25 cry not surface hardened:

SF

D ET NORSKE VERITAS

32

surf ace hardened: 5HV Ye· SF .

Definition of /::io and R, see 3 .16.2, with particular attention to possible correction of R if the yield strength is exceeded.

oFPsl see 3 .2, detcnnined for unidirectional stresses (Y M = 1) and <. I 03 cycles.

3.16.4 For 103 <cycles< 3·106

/::icrp is to be determined by linear interpolation a log-log diagram.


May2003

l::i<Jp at NL load cycles is:

l::icr 10' exp= 0.2876 log P

• - 6<JpHO'

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May2003

4. Calculation of Scuffing Load Capacity

4.1 Introduction High surface temperatures due to high loads and sliding velocities can cause lubricant fi lms to break down. This seizure or welding together of areas of toolh surface is tcnned scuffing.

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

Note: Bulk temperature in excess of 120°C for long periods may have an adverse effect on the surface durability, see 2.11.

4.2 General Criteria

In no point along the path of contact the local contact temperature muy exceed the permissible temperature, i.e.:

Ss - S .1 Sl:l ~ °' +Soil Ss

Sn:::; Ss - 50

where:

B-namax

max. contact temperature along the path of contact.

bulk temperature, see 4.3.4.

max. flash temperature along the path of contact, see 4.4.

scuffing temperature, see below.

oil temperature before it reaches the mesh (max. applicable for the actual load case to be used, i.e. nonnally alarm temperature, except for ice cla~i:;cs where a max. expected temperature applies).

required safety factor according to the Rules.

The scuffing temperature Ss may be calculated as:

s, =80+[ o.ss1+112t~: f" .x •• 1}w2x,

where:

Xwn:rr = relative welding factor.

33

XwrelT

Through hardened steel 1.0

Phosphated steel 1.25

Copper-plated steel 1.50

Nitrided steel 1.50

Less than I 0% retained 1.15

Caseharden austcnite

ed steel I 0 - 20% retained austcnite 1.0

20 - 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.

v40 kinematic oil viscosity at 40°C (mm2/s).

Application of other test methods such as the Ryder, the FZG-Ryder R/46.5174, and the fZG L-42 Test 141 /19.5/110 may be i.-pccially considered.

For high speed gears with very short time of contact, S-8 may be increased as follows provided use of EP-oils.

Addition Lo the calculated scuffing temperature .Ss :

Tf tc 2: l 8 µs , no addition

If tc < 18 µs, add 18 · X wre!T · (18- tc}

where

le = contact time ( µs) which is the time needed to

cross the Hertzian contact width.

(µs]

O"H as calculated in 2.2.1.

For bevel gears, use uv in stead ofu.

DETNORSKE VERITAS

34

4.3 lntluence Factors

4.3.1 Coefficient of friction

The following coefficient of friction may apply:

µ = 0 048 w Br n-.o.os R o.2s X ( )

0.2

· ·coil a L , vl:C PredC

where:

w81 specific tooth load (N/mm)

vI:C - sum of tangential velocities at pitch point AL pitch line velocities > 50 mis, the limiting value of vw at v = 50 mis is to be used.

Predc relative radius of curvature (transversal plane) at the pitch point

Cylimlrical gears

WHI = F~t · KA ·K 't ·Kv· K H~·KHa

vi:C = 2 v sin "·iw

PredC: =Pc cos ~b

Bevel gears

Fmhc K Wai = -b-· A·K"l· K,·KHµ·K Ha

(see 1)

(see 1)

110;1 = dynamic viscosity (mPa s) at B-oil , calculated as

_ Voil P TJoil - 1000

where p in kg/m3 approximated a.Cl

P= P1s -(B0il- 15)0.7

and Voil is kinematic Viscosity al. Soil and may be calculated by means of the following equation:

log log(voil + 0.8) = log log(v 100 + 0.8)+

log 373- log(273 + -9-oil) log 373- log 313

(log log ( v 40 + 0.8) - log log (v100 + 0.8 ))

composite arithmetic mean roughness (micron) or pinion an wheel calculated as Ra = 0.5 (R~ 1+ Ra 2 )

This is defined as the roughness on the new llanks i.e. as manufactured.

see 4.2

4.3.2 Effective tip relief C.rr

Ceff is the effective tip relief; that amount oftip relief, which compensates for the elastic deformation of the gear mesh, i.e.

Classification Notes~ No. 41.2

May 2003

zero load at the tooth tip. It is assumed (simplified) to be equal for both pinion and wheel.

Cylindrical gears

for helical:

(see I)

For spur:

(see 1)

Alternatively for spur and helical gears the non-linear approach in 1.11 may be used (taking Cecr.=8) .

Bevel gears

C _ F;nbtKA cff -

c.1b

(see I)

where:

44·€ c =--r.t y 2+G

y

4.3.3 Tip relief and extension

Cylindrical gears

The extension of the tip relief is not to result in an effective contact ratio i:.:(l < 1 when the gear is unloaded (exceptions to this may only apply for applications where the gear js not to run at light loads). This means that the unrelieved part of the path of contact is to be minimum PM· Tt is further assumed that this unrelieved part is placed centrally on the path of contact.

If root relief applic.~, it has to be calculated as equivalent tip relier. I.e. pinion root relief (at mesh position A) is added to c.2, and wheel root relief (at mesh pm;ition E) is added to Cat·

If no design tip rel icf or root relief on the mating gear is specified (i.e. if Ca1+Croot2 = 0 and visa versa), use the running in amount, see 1.12.

Bevel gears

Bevel gears arc to have heightwise crowning, i.e. no distinct relieved/unrelieved area. This may be treated as tip and root rel ief. For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief. lfno resulting tip relieves are specified, the equivalent tip relives may be calculated, as an approximation, based on the tool crowning Co tool (per millc of tool module mo) as follows:

where;

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May2003

C -C . ·(1 - A2+ 2(mo -mn)J2 a2 - •lvol2 lllo

mo

A2 = m0 {l + x2 ) -0.5 ·sin °'vt { ~d~~2 - d~b2 -dvb2 ·tan n v1 )

(rouU = Catooll. mo { 1- ~: r C,oo12 = C.1oo12 • m0 { I -~ )

2

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

Cnleq = Caz1:q = 0.5 ·(sum of calculated Caleqand Ca2cq).

Throughout the following C,.1 and Ca2 mean the equivalent tip relieves Cateq and C a2cq·

4.3.4 Bulk temperature

The bulk temperature may be calculated as:

SMB = s oil + 0.5 XS x mps flaavernge

where:

X,

S flaa veragc

lubrication factor.

1.2 for spray lubrication.

1.0 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed < 5 mis).

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.

= contact factor, xtnp = 0.5 0 + n l')

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

average of the integrated f1ash temperature (see 4.4) along the path or contact.

For high speed gears (v > 50 mis) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission.

4.4 The Flash Temperature 911a

4.4. 1 Basic formula

The local flash temperature Sfln may be calculated as

~2y - -ll

( b /4 1/2 3na =0.325µX corr wl:ll Xry} n I ,,__ __ l/_4 _ ___...:.

Predy

(For bevel gears, replace u with uv)

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

where:

µ coefficient of friction, see 4.3.1.

35

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 fiim and possible shuffling away oil before meshing if insuflicient tip relief.

c. = tip relief or driven member. Xcon is only applicable in the approach path and if Ca < Cefl" otherwise 1.0.

unit load, see 4.3.l.

Xr load sharing factor, see 4.4.3. )

n1 pinion r.p.m.

f y, P1y etc. see 4.4.2.

4.4.2 Geometrical relations

The various radii of flank curvature (transversal plane) are:

PnJy

pinion flank radius at mesh pointy.

wheel flank radius at mesh pointy.

equivalent radius of curvature at mesh point y.

Pty P2y Predy =

P1 y + P2y

Cylindrical gears

t + ry . Piy =~asma1w

u - fy . P2y =~asma1"'

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

DET NORSKE VERIT AS

36

Bevel gears

i+r r . P 1y =--av SID a.vt

1 + Uv

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

At tl1c respective ends, r has the following values:

Root pinion/lip wheel

Cylindrical gears

B evel gears

Tip pinion/root wheel

Cylindrical gears

f E = ~(da1/dbl )2 - 1

tana1w

Bevel gears

At inner point of single pair contact

Cylindrical gears

Bevel gears

At outer point of single pair contact

Cylindrical gears

r 1, 21t

1l = · A+ - - - Z11an a 1w

Bevel gears

21t fn =f'A + - --

Zvl tan <lvt

At pitcl1 point f c = 0.

Classi.llcation Notes- No. 41.2

May 2003

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

ri: = rA +r l:l 2

4.4.3 Load sharing factor Xr

The load sharing factor Xr accounts for the load sharing between the various pairs of teeth in mesh along the path of contact.

Xr is to be calculated stepwise from A to E, using the parameter f r

4.4.3.1 Cylindrical ge.ars with /3 = 0 and no tip relief

1/3

A

Figure 4.1

I 1 f y - r,. x =-+ ----'--r, 3 3 ~ - r K A

Xr =I ,.

1.0

213

B D

4.4.3.2 Cylindrical gears with fJ = 0 and tip relief

E

1/3

Tip relief on the pinion reduces Xr in the range G - E and increases correspondingly Xr in the range F - B.

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

Following remains generally:

Xr =1 y

Xrv=Xro "" l /2

Jn the following it must be dist inguished between Ca< C01t·

respectively Ca> Ccrr· This is shown by an example below where C01 < Ceff and Cai> C•tr·

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May2003

1.0

i ........... ·····

l .............. ············ r

1112 1/2

l :(1- c••) c ~·rc-+--;r---~---~~-...;.------..1' :\ (" lltt'

1 • ..IL . Cd!'/

AA'

Figure 4.2

Note:

F B D D' G E

When Ca> Ccff the path of contact is shortened by A - A' respectively E' - E. The single pair contact path is extended into B' respectively D'. Jf this shift is significant, it is necessary to consider the negative effect on surface durability (B') and bending stresses (B' and D').

Range A-F

For C,.2 $ Cerr

Xr =0 )

.<?al. - 1

With fi\' = rA +{ir -!A) ~eff } --..!l_ _ _

RangeF- B

For Cn1 ::; C.,fT

Xr = l y

Range D-G

ceff 2

Xr =1 y

Cn2 - 1

with r 0 • =in +(10 -r0 ) ~err -1 --..!l_ __

ceff 2

Range G-E

Xr =0 y

4.4.3.3 Gears wilh /3 > 0, buttressing

Due to oblique contact lines over the flanks a certain buttressing may occur near A and E.

37

This applies to both cylindrical and bevel gears with tip relief < Coif· The buttressing Xbutt is simplified as a linear function within the ranges A - H respectively I - E.

Xbutt A

t\ 1.0

A H E

Figure 4.3

XbuttA.~ = 1.3 When £11 ~ l

DET NORSKE VERIT AS

38

Cyli11drical gears

r H - r A = r E - r, = 0.2 sin ~b

Bevel geal's

r H - [A = [ 11 - f I = 0.2 sin ~bm

4. 4. 3. 4 Cylindrical gears with ey ~ 2 and no tip relief

X r is obtained by multiplication of Xr in 4.4.3.1 with r r

Xbuuin 4.4.3.3.

4.4. 3.5 Gears with r:y > 2 and no tip relief

Applicable to both cylindrical and bevel gears.

1 Xr =-

1 ea.

A H F

Figure 4.4

B D G I E

4.4. 3. 6 Cylindrical gears with Cy ~ 2 and tip relief

Xr is obtained by multiplication of Xr in 4.4.3.2 with Xbun 1 y

in 4.4.3.3.

4.4.3. 7 Cylindrical gears with ey > 2 and tip relief

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

Xr is obtained by multiplication of Xr as described below y 1

with xh.,u in 4.4.3.3.

In the Xr example below the influence of tip relief is shown (without the influence ofXm1tt) by means of

Cai > Cetr and Ca2 < Ccrr .

Tip relief > Ce1r causes new end points A 1 respectively E' of the path of contact.


M ay2003

A F

Figure 4.5

RangeA-F

B D G

X _ Cell' - Ca2 fy - r A (ea - l)Cnt + {3£11 + 1 )Cai

r - + ( 1 ca C.,rr fp -rA 2ea s0 + l)C~ff

Xr = 0 1

·thr -r (r r ) (Ca2 - Ceff)2(s(l +1) Wl A' - A + F - A ( ) ( ) ea - 1 C91 + 3 e11 + I Ca2

RangeF-G

Xr =-' + (eu -l)(Cu1 +Ca2) Y Su 2e0 (ea. +l)Ceff

4.4.3.8 Bevel gears with E:y more than approx. 1.8 and heightwise crowning

For Cai = Ca2 = Ceo the following applies:

A M

Figu re 4.6

E'E

E

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May 2003

r M = o.s(rE + rA)

Xr =!2- 6(ry-rMY y 6a e!(ro -rA)2

For tip relief< C.R; Xr is found by linear interpolation be-'

tweeo Xr,(c. =Ce!T ) and Xr~(c, -o) as in 4.4.3.5.

The interpolation is to be made stepwise from A to M with the influence of C82 and from M to E with the influence of Co1· (For c .. 1 -:t: Ca2 there is a discontinuity at M.)

E.g. with Ca1 = 0.4 C.ff and Ca2 = 0.55 C.m then

RangeA - M

Xr = 0.45 Xr (C =O) + 0.55 Xr cc -c } y )' 32 y 12 err

Range M - E

Xr = 0.6Xr(C =0)+ 0.4Xr (C -c ) y 1 1111 y •I - cir

For tip relief> C0 ffthe new end points A' and E' arc found as

rA' = rA + ~(ro -r,,J( C82 - t)

6 ceff

RangeA - A'

Xr =0 ,

Range E' -E

Xr =0 ,

DETNORSKE VERITAS

39

40 Classification Notes- No. 41 .2

May 2003

Appendix A. Fatigue Damage Accumulation

The Palmgren-Miner cumulative damage calculation principle is used. The procedure may be applied as follows:

A.1 Stress Spectrum From the individual torque classes, the torques (T;) at the peak values of class intervals and the associated number of cycles (Nu) for both pinion and wheel arc 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 cyclic variation is such that the same teeth will repeatedly suffer the peak torque, this is a must.)

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

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

A.3 Damage accumulation The individual damage ratio D; at i1h stress level is defined as

o. = NLi I Np;

where:

Nr.; = The number of applied cycles at ith stress.

NF; The number of cycles to failure at i0' stress.

Basically stresses oi below the permissible stress level for inf mite Ii fe (if a constant ZN or Y N is accepted) do not contribute to the damage sum. However, calculating the actual safety factor s.c1 as described below all the cr; for which the product Sa; is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the detern1ination of Soc1. The final value of S is decisive.

(Nr; can be found mathematically by putting the permissible stress op; equal the actual stress o;, thereby finding the actual life factor. This life factor can be solved with regard to load cycles, i.e. Nri.)

The damage sum r.Di is not to exceed unity.

If 1:0; t 1, the saiety against cumulative fatigue damage is different from the applied demand safety factor. For determination of this theoretical safoty factor an iteration procedure is required as described ill the following flowchart:

Reduce S

Yes

Stress spectrum

Multiply all stresses C1\

\\lith a factor S

Calculate the new minersum tOi

Output S

Increase S

Yes

S is correction factor with which the actual safety factor Snc1 can be found.

Sid is the demand safety factor (used in determination of the pennissible stresses in the o - N - curve) times the com~ction factor S.

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

Note: T f alternating stresses occur in a spectrum of mainly pulsating stresses, the alternating stresses may be replaced by equivalent pulsating stresses, i.e. by means of division with the act11al mean stress influence factor YM.

D ET NORSKE V ERITAS

Classification Notes- No. 41.2 41

May2003

Appendix B. Application Factors for Diesel Driven Gears

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

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

B.1 Definitions

Normal operation KAnonn ==

where:

To

Tvnorm

rated nominal torque

vibratory torque amplitude for normal operation (see rules Pt.4, Ch.3, Sec. I G30l for definition of "normal" irregularity)

Misfiring operation

where:

T remaining nominal torque when one cylinder out of action

vibratory torque amplitude in misfiring condition. This refers to a pennissible misfiring condition, i.e. a condition that does not require automatic or immediate corrective actions as speed or pitch reduction.

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

•

The misfiring o?eration is assumed to last for a limited duration, such as I 0 cycles.

B.2 Determination of decisive load Assuming life factor at 1010 cycles as Y N = ZN= 0.92 which usually is relevant, the calculation may be performed only once with the combination having the highest value of application factornife factor.

For bending stresses and scuffing, the higher value of

KA norm alld KA misf

0.92 0.98

For contact stresses, the higher value of

K K . f KA . f A nonn d A mis (b t mis c. 'tr'd d ) --- an u ior m 1 e gears 0.92 1.13 0.97

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

Z-1 T=-·T z 0

where Z == number of cylinders

Tvnorm = ~ (Tvmisf - Tv idea1)+Tv idc1<l

where T v ideal = vibratory torque with all cylinders perfectly equal. See also rules Pt.4, Ch.3, Sec.I, 0300

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

T v idea1 / T0 is close to zero for engines with few cylinders and using a suitable elastic coupling, and increases with relative coupling stiffness and number of cylinders.

T v m;,1/T0 may be high for engines with few cylinders and decreases with number of cylinders.

This can be indicated as:

T vidcal Z --- ::=: -- d T vmlsf O 4 Z an--- ::::: . --

T0 80

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

KA misf ::. 1.12 - 1.l 8

KA norm::::: 1.10 - 1.15

Since KA nom1 is to be combined with the lower life factors, the decisive load condition will be the nonnal one, and a K,.. of 1.15 will cover most relevant cases, when a suitable e lastic coupling is chosen.

DET N ORSKE VER IT AS

42 Classification Notes- No. 41.2

May 2003

Appendix C. Calculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units ar.e open !!:ears that are subjected to wear and tear. With nonnal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out, however, with exception of surface hardened pinions where case crushing has to be considered.

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

C.1 Pinion tooth root stresses

Since 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·m,,.

Y Fa and Y sa replace Y F and Y s because load application at tooth tip has to be assumed for such inaccurate gears.

Pinions often usc a non-involute profile in the dedendum part or the flank, e.g. a constant radius equal the radius of curvature at reference circle. For such pinions sr-n and hFa arc to be measured directly on a sectional drawing of the pinion tooth.

Due to high loads and narrow facewidths it may be assumed that KFri2 = KHri = 1.0. However, when b1 > b1, then KF11 1 >1.0.

lfno detailed documentation ofKFrii is available, the following may be used:

~l = 1 + 0.15·(b1/b2 - 1)

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

_ O'Fstl y CT !'Pl - -- ' llrclTst

SF

Tl1e 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 stresses

The actual stress is calculated as:

Ft O'F2 =--- · Yra · Ysa

b2 ·mn

See C. l for details.

The permissible stress is calculated as:

_ O'Fst2 y O FP2 - -- ' l\rclTst

Sp

For alloyed steels (Ni, Cr, Mo) with high toughness and ductility the value ofYsrerrsi may be put equal to Y Sa·

C.3 Surf ace hardened pinions

For 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, bul when the theoretical Hertzian stress exceeds the approximately 1.8 times the yield strength of the rack material, plastic deformation will occur. This will limit the peak Hertzian stress but increases the contact width, and thus the penetralion of stresses into the depth.

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

DF.T NORSKE VERITAS

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