Power Transmitting Gears

Design and materials selection for power- transmitting gears

Abstract - The great major i ty of a l l power gears are manufactured in hardened or case-carbur ized s tee l with a sma l le r number manufactured from cast i ron and from non-meta l l ic mater ia ls such as thermoplast ics and laminated bonded wood, fabric and paper materials. The choice of mater ia l depends on the power to be t ransmit ted and the runn ing speed. Fo l lowing an introduct ion to the pr inc ip les of gear design, the select ion of appropr ia te gear mater ia ls - thei r fat igue strength and permiss ib le stress levels - is discussed.

In gear design the objective is to define a gearbox which can be manufactured at minimum cost, which will transmit the design torque (or power) at the design speed, at a specified ratio, and will have appropriate rel iabil i ty during the life of the transmission system. Constraints may be imposed on the design by space, weight or cost limitations, and by avai lable manufacturing facilities.

To design rel iable gearboxes which will not fail, the possible failure modes must be understood. Gearboxes can break or stop working due to failures of the gears, or other components, that is:

Shaft seals (leakage, ingress of dirt) Bearings (wear, breakage, seizure) Lubrication (loss of lubricant, entrainment of dirt) Shafts (bending fatigue failure, particularly at keyways and splines, fretting at bearing seats, wear under seals).

In many types of machinery the most common gearbox failures are bearing failures resulting from ingress of dirt due to seal failure. In dirty environments, inadequate gearcase breathers can lead to severe dirt contamination of the lubricating oil, and resulting bearing and gear failure. Inade- quate gearcase design, without oil-weirs to channel oil to bearings, or without oil drainage holes from the closed end of bearing housings, can lead to bearing-oil starvation or overheating and seal failures.

Poor assembly, especial ly too much play or excessive pre- load in bearings, can lead to rapid bearing failure. In extreme cases, excessive bearing pre-load can result in roll ing element bearings overheating sufficiently to weld up.

Shaft breakage, both inside and outside the gearcase, due to sharp shoulders, keyways and other stress raisers can cause more gearbox downtime than breakages of the gear elements themselves. Attention to detail, and fol lowing proper design rules, are important to avoiding lubrication, bearing and shaft failure, and can often contribute more to achieving high gearbox reliabil i ty than improving the design of the gear elements.

The gears themselves can fail in service primari ly in any of three ways:

• Tooth breakage, that is, the breaking of part of (or a whole) tooth, under operating conditions when the bending stress exceeds the fracture strength or the bending fatigue strength of the gear material.

• Surface failure, that is, pitting or sub-surface cracking, when the contact stress exceeds the fatigue strength or the crushing strength of the gear material. (Surface pitting can initiate tooth bending fatigue failure.)

• Scuffing, scoring or abrasive wear, caused by oil film failure, inadequate lubrication or dirt contamination.

In this article the design of spur and helical gears is discussed in relation to tooth breakage and surface failure. Scuffing occurs primari ly in high-speed, high-power gearing, but is of less significance in normal industrial gearing. Vari- ous scuffing criteria have been proposed but the accuracy and validity of the procedures is limited. Abrasive wear occurs primari ly in low-speed gears, where the theoretical oil film thickness is less than 0.1 ~m (i.e. peripheral velocity

less than 0.5 m/s). Abrasive wear can be reduced by increasing the oil film thickness, that is, by increasing the viscosity of the oil at working temperature. If wear cannot be avoided, wear rate can be calculated.

The procedure and the stress analysis outlined should result in an adequate gear design. The approximate analysis proposed is based on the current gear standards BS 436: 1986 and DIN 3990: 1987. However, an adequate design does not guarantee good performance. Gear failures can still occur due to sub-standard material, inaccurate manufacture and inadequate heat treatment.

The design of gears follows a normal design route, with the fol lowing steps:

(1) The duty for the gearbox is specified, and corresponding design life and reliabil i ty estimated, from which the equivalent design load and appropriate safety factors are determined.

(2) The most appropriate material for the gears is selected from a range of steels and non-metall ic materials, and permissible stress levels established.

(3) The type of gear is selected (spur or helical) and basic design decisions taken on ratio split, facewidth ratio and gear-manufacturing route and accuracy.

(4) The gears are roughly sized using a simplified, non- iterative calculation for pinion diameter and module.

(5) Once the gear size is known, a more accurate stress analysis is applied to refine the design.

Specification, design life, reliability Before a design or a stress analysis can be carried out a realistic specification for the gear pair must be established. In some cases this may be easy - a gearbox used to drive a centrifugal pump running at a a known flow rate and pres- sure is subjected to a well-defined torque. In other cases (for example, in the drive to a stone crusher or a tunnelling machine) the gearbox will be subjected to high dynamic forces which are difficult to determine.

The gear specification must, at a minimum, include:

• The nominal design torque or power (T or P) and the application, or

• The torque-t ime characterist ic, or load histogram, for the drive and

• Input speed, n (rpm) • The transmission ratio, i • The design life in hours or shaft revolutions • The required reliabil i ty

In addition, one or more of the fol lowing may be specified:

• Centre distance or overall gearbox size or weight • Pinion and wheel material and heat treatment • Manufacturing route and associated accuracy • Safety factors • Application factor

The gear ratio is governed by the characterist ics of the prime mover (engine or motor) and the needs of the driven

230 MATERIALS & DESIGN Vol. 13 No. 4 1992

Table 1 Application factor KA

Technical report

Examples of driven machines with different working characteristics Application factor KA for different prime movers

Driven machines Character Electric Multi- Single- motor cylinder cylinder

turbine engine engine

Generators, uniformly loaded belt or platform conveyors, worm conveyors, light elevators, packag ing machines, feed gears for machine tools, ventilators, light centrifuges, centrifugal pumps, mixers for light fluids or constant-density materials.

Uniform 105 125 15

Non-uniformly loaded belts or platform conveyors, main drives of machine tools, heavy elevators, turning gears of cranes, industrial and mine ventilators, heavy centrifuges, centrifugal pumps, mixers for high-viscosity or variable-density materials, multi-cylinder piston pumps, feed pumps, extruders (general), calenders, rotary furnaces, rolling mills (continuous zinc strip, aluminium strip as well as wire and bar rolling mills).

Moderate 13 15 1.75 shock

Extruders for rubber mixers with interrupted operation for rubber and plastics, ball mills(light), Medium shock 1.5 I 75 2.00 wood working (mills, saws, lathes), billet rolling mills, lifting gear single cylinder piston pumps

Excavators (bucket wheel gears, multi-bucket gears, sieve gears, power shovels), ball mills (heavy), Heavy shock 18 200 2.25 rubber dough mills breaker (stone ore) metallurgical machines, heavy feed pumps, rotary drilling apparatus, brick-moulding presses, braking drums, peehng machines, cold strip-rolling mills, bri- quette presses.

mach ine ry . The des ign life and re l iab i l i t y shou ld be spec i f ied by the user of the gearbox , but, in pract ice, must of ten be es t imated by the gear eng inee r .

Des ign life wi l l va ry f rom a few hours for low- ra t io gears in au tomo t i ve g e a r b o x e s to 2000 or 3000 hours for s o m e mob i l e equ ipment , to 100 000 hours for p rocess p lant run 24 hours per day. For ca lcu la t ion purposes, des ign l i fe is exp ressed as NL in n u m b e r of p in ion shaft revo lu t ions .

The re l iab i l i t y requ i red f rom a g e a r b o x wi l l depend upon the consequences of fa i lure. In no rma l indust r ia l dr ives, a gea r re l iab i l i t y s im i l a r to the re l iab i l i t y expec ted in ro l l ing e l e m e n t bear ing des ign may be appropr ia te , w h e r e a/310 life is g e n e r a l y used, i.e. a 10% p robab i l i t y of fa i lu re wi th in the des ign life. For cr i t ical p rocess plant, w h e r e d o w n t i m e is ve ry expens i ve , or in d r i ves w h e r e the consequences of fa i lu re cou ld be ca tas t roph ic , such as the fa i lu re of a hel i - cop ter ro tor gearbox , a much h igher re l iab i l i t y must be sought , and a p robab i l i t y of fa i lu re of 1 in 103 to 1 in 105 in the des ign l i fe is appropr ia te .

Design load The des ign load or equ i va l en t t o rque must be de te rm ined as accu ra te l y as poss ib le to avo id over - or under -des ign . The loads to wh ich the gea r ing is sub jec ted are a funct ion of the charac te r i s t i cs of both the d r i v ing and d r i ven mach inery . In all but the s imp les t cases, the t ransmi t ted to rques shou ld be accura te ly ca lcu la ted or m e a s u r e d and an equ i va len t to rque ca lcu la ted. When this is not poss ib le , an app l i ca t ion factor, KA, can be used as an a p p r o x i m a t e w a y of account ing for d y n a m i c o v e r l o a d s in a pa r t i cu la r type of gea r dr ive .

The to rque be ing t ransmi t ted by a g e a r b o x may be g rea te r or s m a l l e r than the nomina l p o w e r of the d r i v ing mo to r or eng ine. If it can be a s s u m e d that the mo to r has been cor- rect ly s ized to supp ly the mean p o w e r requ i red by the d r i ven mach ine, the ef fect of d y n a m i c o v e r l o a d can be a l l owed for by an app l i ca t ion factor, KA, app rop r i a t e to the type of p r ime m o v e r and the c lass of d r i ven mach ine . KA for typ ica l p r ime move rs and a range of d r i ven mach ines is shown in Tab le 1. Using an app rop r i a te va lue of KA, the des ign to rque To is:

To= To.G

w h e r e 7-, is the nomina l mo to r torque. It shou ld be noted that the app l i ca t ion factor KA a lso makes a l l o w a n c e for the typ ica l ope ra t i on of the mach ine , that is, the n u m b e r of s tar ts per hour and the u t i l i za t ion of the mach ine .

However , to repeat , the equ i va l en t des ign load is best

based on a measu red or ca lcu la ted t o r q u e - t i m e charac ter is - tic, wh ich takes account of al l ope ra t i ng cond i t ions , acce le- rat ion, va r i ab l e speed running, brak ing, and the sys tem iner- t ias. When the dynam ic load character is t ic , as shown in Fig. l (a) , has been de te rm ined , this can be reduced to a load h is tog ram by ca lcu la t ing the t ime spent at each to rque leve l ?'1, 7-2 ..... T, and runn ing speed, n, and s u m m i n g this for the life of the gearbox .

To a l l ow for d i f fe rent runn ing speeds, the load h is tog ram is best based on the total numbe r of p in ion revo lu t ions (AM, AN2 ..... AN~ at each load leve l 7-1, 7-2 . . . . . T, for the des ign life of the gearbox , as shown in Fig. l (b) . The equ i va len t des ign torque, TD, for a va r i ab l e - t o rque dr ive as in Fig. 1 can be ca lcu la ted using the l inear cumu la t i ve d a m a g e hypothes is of Pa lmgren and Miner . For a ser ies of to rque levels, L , 7-2 . . . . . T~, occur r ing for load cyc les AM, AN~ ..... AN~, the des ign to rque To for a des ign life ND < N, is g iven by

TD = [AN~T~q + AN2T2q + ...ANff?]4 No (1)

w h e r e q is the s lope of the l oad - l i f e curve.* Cumu la t i ve d a m a g e theo ry assumes that there is no fa t igue d a m a g e at s t resses be low the endu rance l imi t (occurr ing at N~) so that in the ca lcu la t ion of TD, app l ied to rques at a leve l l ower than 7", must not be cons ide red in the ca lcu la t ion. S imi la r ly , for a total des ign life g rea te r than N~, N~ must be in p lace of ND in equa t ion (1).

Fig. 2 shows the typ ica l l oad - l i f e curves for bend ing fa i lu re and sur face fa i lu re for w rough t gea r steels, through- , surface- or case -ha rdened . It shou ld be noted that the nomina l fa t igue l imi t for sur face s t rength (pi t t ing) is 5.107 to 109 load cycles, wh i le that for bend ing s t rength is on ly 3.106 . The va lues of No, N, and q f o r d i f ferent gear mate r ia l s are g iven in Tab le 2.

6ear reliabilily The re l iab i l i t y of a g e a r b o x must be seen in the con tex t of the re l iab i l i t y of the who le d r ive sys tem or plant, and the impl i - cat ion of gear fa i lu re assessed in re la t ion to total p lant re l ia- b i l i ty and total cost. For examp le , if the g e a r b o x in a la rge p lant accounts for on ly a smal l p ropor t ion of total capi ta l

*For root bending stress, the slope of the S - N curve q* is identical to that of the load-life curve q. For contact stress, however, q* = 2q.

MATERIALS & DESIGN Vol. 13 No. 4 1992 231

Technical report

#_

a

T O

T 1

%

%

r4

T 5

%

T 7

T 8 Time

%

>

F o

I - -

T : . . . . . . .

T5 . . . . . .

T 6 . . . . . .

T 7

AN1 i, I

L L

AN 2_ - AN 3 _,_ AN 4 - _ AN 5 -

b Load cycles (revolutions - NL}

Fig 1 (a) Load-time characteristic; (b) load histogram

A N ~_22_o

300, E

c

"0

"0

~oo -o 90

-~ ao ~ 70

u .

Fig 2

24O

2OO 180 160

140

120

\\ ~, %. ~ \ \ \ ' \ ".,.

\ " \ ",.

I I

(N I) * ( Nl°)* I I I I I I I

102 103 104 105 106 107 108 109 1010

Load cycles, N L

Idealized load-life curves. (a) Case-carburized steel, bending failure; (b) direct-hardening steels (including induction hardening), bending failure; (c) direct- and case-hardening steel, contact failure - no pitting; (d) direct- and case- hardening steel, contact failure - some pitting. *No and N~ for (a)

cost, yet a gear fai lure can result in an expensive shutdown, then an increase in gear cost to improve rel iabi l i ty would be justif ied. If, on the other hand, the gearbox represents a very large proport ion of the total cost - as, for example, in a tractor - and considerable cost would be incurred for a

marginal increase in gear reliabil i ty, th is may not be justi- fied.

Few data are avai lable on the rel iabi l i ty of industrial gear drives, and the target rel iabi l i t ies which should be assumed at the design stage. The design rel iabi l i ty wil l depend on:

• The target rel iabi l i ty of the whole system; • The attainable rel iabi l ty for shafts, bearings, seals and

other parts of the gearbox; • The consequences of fai lure - both in financial terms and

in relation to safety (e.g. potential risk to life and limb); • The rel iabi l i ty of competing products.

As a rough guideline, gears for mobi le plant and non-crit ical industrial drives should have a rel iabi l i ty of 90-95% within their l ifetime. Critical plant, where the consequences of gear fai lure are expensive, should be designed with a rel iabi l i ty of 95-99%, whi le plant which is safety-crit ical should have a rel iabi l i ty better than 99.9%. Gearing where fai lures are potential ly fatal should be designed for even greater rel iabi- lity.

The re lat ionship between stress, strength and re l iabi l i ty The fatigue strength of a gear depends on the gear material and heat treatment, and on the manufacturing process, par- t icularly the surface finish and the residual stresses induced by grinding, shot peening or other surface treatments. There wi l l be variabi l i ty of fatigue strength due to variat ions in material composit ion, he~t t reatment and finishing, the mag- nitude of this variabi l i ty depending on the quality control exercised in producing the steel and forgings and in the


Table 2 Index q for load

Technical report

Gear steel and heat treatment

Thro-Hard wrought steel, CI perlitic and bainitic, ductile iron

Induction and flame hardened steel

Case-carburized steel

Gas and plasma ni t r ided nltr idingsteeI, C l a n d d u c - tile iron, ferritic

Alloy steels bath and nitro carburized

Bending failure

N o N~ q

10 4 3.10 6 6.2 2

104 3.106 6.22

10 3 3.10 6 8.7

103 3,106 17.0

103 3.106 84.0

Surface failure

No pitting

N o' N,' q'

105 5107 6.6

10 s 5.107 6.6

105 5.107 66

10 ~ 210 G 57

105 2106 15.7

Some pitting

No" N~ " q "

6.10 5 1 ()~ 7.9

6.105 109 7.9

610 ~ 109 7.9

Min. stress

Max. strength 'Max.' probable stress

99% confidence limit strength

Frequency

i / ! Stress Strength

Fig 3 Distribution of stress and strength. I Stress, high-accuracy gear, II stress, low-accuracy gear, III strength, high-quality heat treatment, IV strength, low-quality heat treatment

subsequent manufacture and heat treatment. There is thus no unique value of fatigue strength, only a probabi l i ty that the fatigue strength wil l not fall beneath a given value.

The stress induced in a part icular gear in a part icular appl icat ion wil l depend on the way the load is distr ibuted between the flanks of mating gear teeth, that is, on the profile, pitch and lead variat ions of the gears, and the mis- a l ignment between them. The smal ler the variat ion in manufacturing tolerances and mounting (that is, the more accurate the gear), the smal ler wil l be the variat ion in stress.

By definition, fai lure cannot occur if the fatigue strength is greater than the fatigue stress. When, however, there is a range of possible strengths and stresses, the probabi l i ty of fai lure depends on the statistical distr ibution of stress and strength. This is d iagrammat ica l ly i l lustrated in Fig. 3, show- ing the probabi l i ty distr ibution of stress and strength for a part icular gear application, for high- and low-accuracy gears, and high- and low-quali ty heat treatment.

In Fig. 3 it is seen that for the high-accuracy gear (curve I) the stress hardly exceeds the strength of the high-quality steel with high-quality heat t reatment (curve III). There is thus a very low probabi l i ty of failure. As gear accuracy and material quali ty are lowered (for example, to the stress and strength distr ibution represented by curves II and IV) there is an increasing number of occurrences where the stress

exceeds the strength. For a low-accuracy gear, with low- quality heat treatment, the probabihty of fai lure is related to the area of overlap W.

It should be noted that for any size of gear there is a minimum theoretical stress which wil l be attained with geo- metr ical ly perfect gears, perfectly aligned. The mean stress wil l then vary with the accuracy of the gears, the lower the gear accuracy, the higher the mean stress. Similarly, there wil l be a maximum attainable strength, and the mean strength wil l decrease with decreasing material quality and heat treatment.

If the stress and strength distr ibution were known, the factor of safety, that is, the ratio of strength to stress, should ideally be calculated on the basis of mean strength and stress. In BS 436 and ISO-DIN 3990 the calculation procedure is based on calculating a maximum probable stress, whi le the fatigue strength data are given for 1% probabi l i ty of failure. The stress and strength as calculated to these standards is indicated in Fig. 3 for curves II and IV. For the example shown, the 'worst ' stress distr ibution (curve II) results in a maximum probable stress which is greater than the 99% confidence l imit for the material strength. That is, for strength and stress distr ibutions II and IV, and the probabil i ty of fai lure represented by the shaded area W, the factor of safety as defined in BS 436 and ISO-DIN 3990, where

SF, SH = Fatigue strength to standard Fatigue stress to standard

would be SH, & < 1.0. While academic purists may baulk at the concept of a

safety factor less than unity, the adopted procedure is con- ceptually sound and the best which can be used whi le the stress and strength distr ibutions are not known, and whi le the factor of safety is still largely a 'factor of ignorance'.

Safety factors for bending stress SF and surface stress SH The required reliabil i ty, in terms of percentage probabi l i ty of survival or fai lure within the design life at design loads, should be defined in the specif ication or selected as indicated above. The material fatigue strength is given in Table 3 for 99% survival. If the fatigue stress were known with absol- ute certainty, then, when the fatigue stress is equal to the fatigue strength (at 99% survival) the probabi l i ty of fai lure should be only 1%. As discussed above, the actual stressing depends not only on the applied load but also on the particular tolerances of manufacture and al ignment of a part icular gear pair.

The factors of safety necessary to achieve a design rel iabi- lity al lowing for the uncertainty in stress, the uncertainty


Technical report

Fig 4

-r"

1.7

1.6

1.5

1.4

1.3

1.2

1.1

1.0

0.9

0.8 0.01 0.1 1

Probability of failure (~o)

Approximate relationship between probability of failure and SH and S~

10 50

3.0

2.8

2.6

2.4

2.2

2.0 LL

U%

1.8

1.6

1.4

1.2

1.0

associated with the appl ied load and the uncertainty in material strengths are not accurately known. Since the 'spread' of fatigue strength, the 'spread' of component tolerances and the uncertainty associated with the loading can vary widely, there is no simple relat ionship between factors of safety and gear rel iabil i ty.

Fig. 4 indicates the ranges of factors of safety for bending stress, SF, and surface stress, SH, for varying probabi l i t ies of failure. S~ is the ratio of permissib le bending stress (~FP to the actual bending stress, CrF, i.e.

S F = (3"Fp/(~" F

SF is the ratio of permissib le contact stress ~HP to the actual contact stress cr,, i.e.

S H = ~=Hpl~* H

When the duty cycles and drive system dynamic loads are only estimated, a larger factor of safety is required to achieve a part icular rel iabi l i ty than when the load spectrum is known accurately from measurement. Conversely, improving gear accuracy (a smal ler spread of tolerances and hence a smal ler range of stresses) and improving quality control for heat t reatment and metal lurgy, a l low the factor of safety to be reduced.

Select ion of gear mater ia l Gears can be manufactured from virtual ly any material. His- torically, instrument gears had steel pinions and brass wheels, whi le gears for transmitt ing power (water mills, engines, etc.) started off being made from wood, then iron and wood, later steel and then surface-hardened and case- carburized steel.

This art icle is concerned with the design and stress analysis of power-transmit t ing gears. The great major i ty of all power gears are manufactured in hardened or case-carburized steel, with a smal ler number manufactured from cast iron and from non-metal l ic mater ials such as thermoplast ics and laminated bonded wood, fabric and paper materials. The choice of material wil l depend on the power to be transmitted and the running speed. Non-metal l ic gears are used pr imar- ily when powers and speeds are low, especial ly when lubri-

cation is a problem, or noise levels must be reduced. The selection of appropr iate gear materials, and their fatigue strengths and permissible stress levels are discussed in the fol lowing sections.

Steel gears The majori ty of gears are manufactured from forged steel blanks or rol led steel stock and fewer from cast steel and cast iron. The choice of gear steel and heat treatment depends on the proposed manufacturing route. The highest bending and surface fatigue strength is achieved with surface-hardened gears; these are thus the smallest, and usually the most economical, for a wide range of appli- cations. In certain cases (for example, where large centre- distances are specif ied relative to the power to be transmitted) through-hardened gears in steel or iron are economical.

Heat t reatment of steel gears Gear steels can be subdivided according to heat treatment into the fol lowing classes:

(1) Soft gears: made from carbon or al loy steels, heat- treated to a hardness of 100-360 HB before gear hobbing or shaving. Gear cutting becomes difficult if the blanks are harder than 280 HB, and a l imit of 320 HB is usually set for conventional HSS tools.

(2) Surface-hardened gears: made from direct-hardening steels, with carbon content in the range 0.3-0.45% with or without al loying elements (Cr, Ni, Mo). Ti~ese are surface hardened after the gears are cut by heating the surface and then quenching the gears. Surface heating is by electric induction or f lame, either tooth by tooth (con- tour hardening) or on the whole gear circumference (spin hardening). Heating is fol lowed by quenching in oil or aqueous emulsions and immediate tempering at 180- 200°C to avoid quenching cracks. The surface hardness can lie in the range 550-650 HV. Since only the teeth are heated and quenched, distort ion of the gear is less than in case-carburizing. Large-module, slow-speed gears may be run in the 'as-heat-treated' condition, with attainable accuracy grade 8 or 9 (to BS 436: 1970). For best performance, gears should be hard hobbed (skived) or ground after heat treatment, to improve lead, pitch and profi le accuracy.


Technical report

Fig 5

E 131 ¢.

~E ~n r "

r~

t ' . ,,t u_

8

o

"T3 t -

r -

E

E c

7 6

5

4

0.7

0.5

0.4

0.3

0.2

0.1

-Root case thickness ~ ~ / ~ ~ A90h

-induction or spin ~ ~ - ~ ~ ~ ~ T o o t h by tooth flame ha r d e n ~ ~ / ~ ~ " "\ - 3 ~ "~////~>,>" inductive or

~ ~ x x ~ ~ ~ / _ / / ' ~ ~ flame hardening

2 _ 30h

~O~f~, ' r l time at 930°C

~O~f~, / // Carburizing -~5h

I _ ~ i i ~ o ~ , Ga s ~ O ~ x ' ~ ' 7 1" nitriding

,~x ~ --~50h - Approximately 2-5h

ath n,tr,d,n ~ 8 h a t 5 7 0 ° C

I I I I I I I I I I I

I 2 3 4 5 6 7 8 910 20 30 40 50 6070 I00

Normal module, mn(mm)

Required case thickness

(3) Case-carbur ized gears: After the gear teeth are cut (hobbed, shaped or planed, or formed by precision forging), the carbon content of plain and alloy steels is increased by carburization at 850-950°C in a carbon-rich environment to achieve a case carbon content of 0.9- 1.1% in low-alloy steels, and 0.6-0.9% carbon in higher- alloy steels. After carburization, the gears are reheated and quenched and subsequently tempered at 150°-210°C to a surface hardness of 650-750 HV. The thickness of the hard case is critical to performance, and a case thickness at least as great as that shown in Fig. 5 must be achieved. Distortion is substantial, due to both carburization and quenching (transformation of the case from aus- tenite to ferrite). In general, gears need to be finish ground (or skive hobbed) after hardening. Where identical gears are produced in very large quantities under carefully controlled conditions (e.g. automotive gears), al lowance can be made for heat-treatment distortion when the gears are cut. Acceptable lead and profile tolerances can then be achieved without finishing after heat treatment.

(4) Nit r ided gears: Gears are nitrided after gear cutting. Nitriding of plain carbon, alloy and nitriding steels is carried out in ammonia at 500-550°C (gas nitriding), or in a cyanide salt bath at 520-580°C to form a nitrogen-rich case with a surface hardness up to 850 HV. Due to the lower process temperature and slow cooling (no quenching) distortion is less than with other hardening pro- cesses, and can be almost eliminated with good process control. To achieve good fatigue strength, the gear tooth core strength ~B should be as high as possible, typically in the range 1000 < oB < K1400 N/mm 2. The case is much thinner than in case-carburized gears (see Fig. 5) so that overload capacity is less than in gears with heat treatments (1)-(3) above.

Fatigue strength of steel gears Typical strength and fatigue properties for 1% probabil ity of failure are given for a range of typical gear steels to BS and DIN standards in Table 3. The fatigue endurance limits for

contact stress, (~,~,r,, and for bending, ~FE, are based on the results of gear testing. Similarly, the UTS (ultimate tensile strength) is the apparent breaking strength of gears, based on the root bending stress or at rupture. The material properties will, however, vary very substantially by up to + 20% for a particular gear, depending on the steel cleanliness, grain size and details of heat treatment, surface hardness, case/ core residual stress gradient, and core strength (see below for the material quality factor Mo).

It should be noted the given strength values for case- carburized gears cannot be achieved in large-diameter, large-module gears, with steels such as 20 MoCr 4, or 20 MoCr 5, because of the low hardenabil i ty of these lower-al loy steels. The fatigue strength data in Table 3 presupposes full case depth (after grinding) to Fig. 5.

Mater ia l cost for steel gears Costs for different steels vary widely with the alloy content (see Table 4). However, some steels are much easier to machine than others, so that the relative machinability, R, must also be considered (Table 4). Hobbing and shaping costs are approximately proportional to dbm/R, or, for geo- metrically similar gears, to d3/R (where d is the reference diameter, b the facewidth and m the module). Grinding costs are proportional to dbz x (grinding allowance), where the grinding al lowance is a function of d, b and B. For geometri- cally similar gears, grinding costs can be taken to be proportional to d 2s. For medium-sized gears (20-400 mm dia.), grinding costs are typically 1.5-3 times the hobbing costs. For accurate, medium-sized gears, produced in small batches, material costs are usually a small part of total manufacturing cost. In smaller, mass-produced gears (e.g. automotive gears, electric motor gearheads), material, including forging, may account for 60% or more of the total cost, and cheap, low-alloy steels should be used wherever possible.

Permiss ib le stress for steel gears The most suitable gear material and heat treatment, according to Table 3, is selected, depending on the size of the gear,


Technical report

Table 3 Steel gear material properties

Material type

Typical specification Hardness

BS BS DIN Surface Core

Surface Bending fatigue fatigue limit limit UTS (~Hlim oFE (N/mm 2) (N/mm 2) (N/mm 2)

Comments

Cast iron and cast steels Cast iron BS 821 - DIN 1691

Ductile iron BS 2789 - DIN 1692

SG iron BS 2789 - DIN 1692 Cast alloy steels BS 3100

As core

200 HB 330 100 380

220 HB 460 360 880

180 300 He 480-620 240~440 590-1100 220-320 HB 560-700 420~450

Very cheap material with low fatigue strength. Good for complex wheels with alloy rims. Suitable for small, high-quantity lighty stressed gears. Suitable for complex shapes, as alternative to carbon and alloy steels. About 75% fatigue strength of equivalent wrought steel.

Through-hardened BS 970 BS 970 DIN 17200 steels Carbon steel En 8 080 M40 St 50

En 9 070 M55 St 70

Alloy steel En 19 709 M40 34 CrMo4 En 19C 708 A42 42 CrMo4 En 24 817 M40 40 NiCrMo6 En 26 826 M40 34 CrNiMo6

As core

160-210 HE 450%50 440~420 560-710

220-320 HB 600-740 500~80 800-1580

Suitable for lightly stressed drives and positional drives. At 210 11B ruling section t00 mm. Traditional gear materials, difficult to machine at higher hardness, Careful heat treatment necessary to achieve good fatigue strength.

Surface-hardened BS 970 BS 970 DIN 17200 steels (alloy steels. En 19 709 M40 34 CrMo4 560 flame or induction 42 CrMo4 610 HV hardened) En 24 817 M40 40 NiCrMo6

En 26 826 M40 34 CrNiMo6

200-280 HV 1160 680 1850 Through Through root root hardening hardening

Relatively cheap method of obtaining high surface durability. Less distortion than carburizing large gears, but grinding generally required after heat treatment. Large gears and large modules require tooth by tooth through root induction hardening, with careful quality control, very low distortion. Danger of quench cracking.

Carburized steels BS 970 BS 970 DIN 17210 (hardening and 15 MnCr5 650- tempered) 20 MnCr5 750 HV

En 34 665 M17 15 CrNi6 17 CrNiMo14

En 36 655 M13 10 NiCrMo14 En 39 20 MoCr4

659 M15 14 NiCr18

270-360 HV 1500 920 2300 Highest possible load capacity. Distortion significant, requires finish grinding on all but smallest gears. Case thickness as Fig. 5. Compressive residual stresses reduced by grinding, in extreme cases grinding burns can lead to tensile residual stress in the case. Low-alloy steels are much cheaper than En 34.36 without great loss in fatigue strength.

Nitrided steels BS 970 BS 970 DIN 17211 En 40 722 M24 14 CrMoV6.9 700- En 40 905 M31 800 HV

270-360 H~ 1250 760 1250 Very low distortion achievable. particularly suitable for internal gears. Relatively thin case. less suitable for shock loads. Nitrided steels substantially more expensive than carburized steels.

Note: The bending fatigue limit for idler gears and others subjected to reverse bending is 0.7 x C~E.

Table 4 Steel costs

Cast iron Cost (at 1990 prices) R Cast discs (p 150 x 150 % cost Relative (i.e. finish cast blanks) £ per tonne relative to machinability (%) Steel En 8 (En 8 = 100%) Forging stock 100 x 150

T h e p e r m i s s i b l e b e n d i n g and s u r f a c e s t ress , CFp and (~HP a r e

t hen c a l c u l a t e d us i ng a s i m p l i f i e d f o r m of t he c a l c u l a t i o n

p r o c e d u r e in BS 436: 1986, w h e r e

(]FP = CrFE. YN. Yx.MQ ( p e r m i s s i b l e b e n d i n g s t ress ) (2a)

Cast iron. BS 821 600 125 Ductile iron, BS 2789 900 100 Carbon steel, En 8 432 100 100

En 9 432 100 100 Alloy steel. En 19 539 125 70

En 24 694 161 55-70 Carburizing steel. En 36 773 179 40-50

En 39 1000 231 30~.0 16 MnCr5 519 120 55 20 MoCr4 526 122 55

Nitriding steel. En 40 40-50

*'Finished' Blanks. not directly comparable with forging stock

the w e i g h t and w h e t h e r t he g e a r is to be g r o u n d a f t e r ha r -

d e n i n g o r not. The m e a n b e n d i n g f a t i g u e s t r e n g t h , (~FE, and t he s u r f a c e f a t i g u e s t r e n g t h , C%ir., a re t a k e n f r o m T a b l e 3.

and (~H, = (~.,r,.ZN. ' / M o ( p e r m i s s i b l e s u r f a c e s t ress ) (3a)

w h e r e t he f a c t o r s m o d i f y i n g b e n d i n g s t r e n g t h a re

YN = l i fe f ac to r fo r s t r e n g t h , to t a k e a c c o u n t of t he h i g h e r f a t i g u e s t r e n g t h in the f i n i te l i fe r e g i o n at less t han

3.10 e load c y c l e s (see Fig. 6),

Yx = s i ze fac to r , to t a k e a c c o u n t of t he r e d u c t i o n in f a t i g u e

s t r e n g t h as g e a r s i ze i n c r e a s e s (see Fig. 8),

MQ = m a t e r i a l q u a l i t y fac to r , to t a k e a c c o u n t of t he q u a l i t y of t he f o r g i n g s and h e a t t r e a t m e n t (see Fig. 9),

and the f a c t o r s m o d i f y i n g s u r f a c e s t r e n g t h a r e

ZN = l i fe f ac to r fo r p i t t ing , to t a k e a c c o u n t of t he h i g h e r f a t i g u e s t r e n g t h at less t han 5.107 o r 109 load c y c l e s

(see Fig. 7).


3.0 2.6 2.4 2.0 1.8

1,6

#1.4 1.2

1.0

Fig 6

I I I I l I I l l I I I t l l l l l I l l l l I l l ~ I I I t l l l l 03 10 q 105 106 107

Number of tooth load cycles, N L

Life factor for bending stress, Y~. (1) Through- and direct surface-hardened steels; (2) case-carburized steels; (3) gas-nitrided steels; (4) bath-nitrided steels

2.0

1.8

1.6

Z 1.4-

1.2-

1.0-

lO 4

Fig 7

I I I I i 105 106 107 108 109

Number of tooth load cycles, N L

Life factor for contact stress, Z~. (1) Through-, induction- and case-hardened steels - some pitting permitted," (2) through-hardened steels - no pitting permitted; (3) case- and induction-hardened steels - no pitting permitted; (4) gas-nitrided steels; (5) bath-nitrided steels

X >-

Fig 8

1.0

0 .9

0 .8

0 .7

I I I 10 20 30 40

m n

Size factor bending stress, Y,~. (1) Through-hardened steel; (2) surface-hardened stool, (3) cast iron

Technical report

X

1.3

1.2

1.1

1.0

0.9

0.7

I/// / J J

/

No quality Average Best possible control quality control quality control

Increasing quality of forgings and heat treatment

Fig 9 Material quality factor, Mo, for surface- and through- hardened steel gears. (1) All surface-hardened gears, (2) all through-hardened gears

Non-meta l l ic gears Size for size, the load capacity of non-metal l ic (plastic) gears is less than a tenth that of steel gears. The advantages and disadvantages of these gears are:

• Low noise levels; • Low modulus of elasticity so that lower accuracy is

required; • Advantageous frictional behaviour so that dry running or

lubrication with water is possible; • High hysteresis, low conductivity (1/100th that of steel) so

overheating is a problem; • High coefficient of expansion, large backlash required; • Thermosplast ic gears can be injection moulded to

achieve adequate accuracy at very low part piece cost; • Complex features such as splines or clutches can be

moulded into gears with little extra cost; • Gears can be moulded as parts of links or other mechani-

cal features

With these characteristics, inject ion-moulded plastic gears are attractive for small gears in domestic appliances and in textile, food processing and business machines. Larger machine-cut gears are used on low-power drives in industrial appl icat ions where noise is a problem. The com- monly used materials are:

Polyamide - PA (e.g. Nylon) Polyacetal - POM (e.g. Deirin) Synthetic Resin Bonded Fabric - SRBF (e.g. Tufnol)

Physical proper t ies of non-meta l l ic gears The physical propert ies of non-metal l ic materials are fairly well documented, but the fatigue strength when used in gears is not well researched, and the values given below

1 must be considered as very approximate only. The coefficient of thermal expansion of non-metal l ic materials such as the thermoplast ics are much greater than for steel, and addi- tional backlash must be al lowed, especial ly when these are

2 mounted in steel gearcases. The physical and mechanical propert ies are summarized

3 in Table 5. The surface and bending fatigue strength at 20°C is given for the pairing plastic pinion mating with plastic wheel without lubrication.

Permiss ib le stress for non-meta l l ic gears The permissible stress for plastic gears depends not only on the required fatigue life but also on the temperature and the lubrication conditions. Using the same form of expression as


Technical report

Table 5 Properties of non-metallic gear materials

Characteristic property (at 20°C) Material Nylon 12 Nylon 12 fibre Nylon 66 Nylon 66 fibre Delrin Tufnol

reinforced reinforced

Density, g/cm 3 1.01 1.23 1 14 1.4 1 42 1.3 Max. operating temperature (°C) 100 100 100-120 100 120 100 110 Thermal conductivity W/(M*K) 0.29 0.29 Q23 0.23 0.31 0.23 Coefficient of expansion 10 6/°C 110 30 85 110 20 Dynamic E modulus 103 N/mm 2 1.6 3.1 3,2 - 3.5 7.8 Surface fatigue strength (]H,~m N/mm2 (at 108 load cycles) 20 18 25 22 25 50 Bending fatigue strength O-FE N/mm 2 (at 108 load cycles) 40 50 60 70 70 100 Slope of S Ncurve between 10 s and 108

For surface, qH 7 7 7 7 7 8 For bending, qF 10 15 10 15 10 7

At 100°C dynamic E modulus 103 N/mm 2 0.3 0.85 1.0 - 1.3

for steel gears , the pe rm iss ib le s t ress for bend ing and surface st ress are g iven by:

(3"FP = O'FE" YNP" Yr. YLP (2b)

and

~Hp = CSH,,m . ZNP.Zt. ZLp (3b)

w h e r e YNP, ZNp are life factors, Yt, Zt are t e m p e r a t u r e factors, to take account of reduced s t rength at h ighe r t empera tu res , YLP, ZLp are lubr icant factors, to take account of the gea r mat ing mate r ia l and lubr icat ion, for bend ing and sur face stress, respec t ive ly .

With the ind ices q~ and qH f rom Tab le 5, the l i fe factors are

YN~ = r 1°8]~ LNoJ

Z.p = [1°81~ LNoJ

w h e r e the des ign l i fe is No > 10 s. The t e m p e r a t u r e factors for Ny lon and Delr in at a wo rk i ng

t e m p e r a t u r e tw (°C) are ve r y a p p r o x i m a t e l y

Y, = 1 - yF(tw -- 20°C)

w h e r e YF = 8 "10-3 and

Zt = 1 - yH(tw - 20°C)

w h e r e YH = 4-10-3. The lubr icant factors to take account of the bet ter pe r f o rmance of non -meta l l i c gears when g rease and oi l lubr ica ted are

YLP = 2 for oi l and g rease lubr ica t ion ZLp = 1.5 for g rease lubr ica t ion ZLp = 1.75 for oi l lubr ica t ion

Acknowledgement Mater ia l for this repor t has been ex t rac ted f rom a th ree- v o l u m e Teach ing Pack on G e a r T e c h n o l o g y (see Append ix ) p roduced and pub l ished by the Br i t ish Gear Assoc ia t ion , The Edi tor is gratefu l for the con t r ibu t ion of the BGA in the prep- aration of this ar t ic le.

Appendix. The Teaching pack on Gear Technology Prepara t ion and pub l ica t ion of the Teach ing Pack has been sponsored f inanc ia l l y by the Br i t ish Gear Assoc ia t ion (BGA), to fu r ther a w ide r unders tand ing of gear ing , and to help c o m p e n s a t e for a dear th of up- to-date Eng l i sh - l anguage text- books on the sub ject of gear ing des ign. Six modu les com- pr ise a total of mo re than 350 pages of tex t /notes, ove r 200 d i a g r a m s su i tab le for rep roduc t ion as ove rhead pro jec t ion sl ides, th i r ty 35-mm sl ides, a PC disk and gu idance on teaching the subject . They cover , respect ive ly : Dr ive Systems; Gear Sys tem Design; Gea r Geomet ry ; Des ign and Stress Ana lys i s of Spur and Hel ica l Gears; G e a r b o x Design; Manu- fac ture and Me t ro l ogy of Spur and Hel ical Gears.

Three UK gear ing des ign exper ts , wi th e x p e r i e n c e both of indust r ia l consu l tancy and teaching, have each comp i led two of the modu les that make up the Teach ing Pack. They are Die ter Hof fman of the Design Unit at the Un ivers i t y of New- cast le, He l lmuth Koh le r of Shef f ie ld Un ivers i ty and Bob Munro, a consu l tan t at Hudders f ie ld Poly technic , who proposed the p repa ra t i on of such an educa t iona l aid some years ago. All t h ree es tab l i shments a re c lose ly invo lved wi th BGA act iv i t ies as assoc ia te members .

The pack is ava i l ab le f ree of cha rge to app rop r i a te h igher- educa t ion es tab l i shments in the UK and may be f ree ly used and cop ied for educa t iona l purposes. Vers ions of these teach ing mate r ia l s for use by c o m m e r c i a l o rgan i za t i ons wi l l be sepa ra te l y ava i l ab le at r easonab le cost f rom the BGA in due course.

Fur ther deta i ls can be ob ta ined f rom J im Hewitt , Director , Br i t ish Gear Assoc ia t ion , St James ' s House, Freder ick Road, Edgbaston, B i r m i n g h a m B15 1JJ UK. Tel: + 44 (0)21 456 3445, Fax: + 4 4 (0)21 456 3161.


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