CA1145974A - Low noise gearing - Google Patents

Abstract

Abstract of the Disclosure A gear tooth form is disclosed that maximizes the profile (transverse) contact ratio for coplanar axis gearing.
As the tooth form allows teeth two or three times as fine as those of conventional involute gearing to be used without loss of torque capacity, the face (axial) contact ratio for helical or spiral bevel gearing is also maximized. The total contact ratio (profile plus face contact ratios) is therefore maximized, and the torque load is divided over the maximum number of teeth. This minimizes the transmission error, which in turn minimizes operating noise and vibration. The gearing employs a minimum pressure angle and a path of contact that includes a curved portion.

Description

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This invention concerns -th~ profiles of gear teeth, and speci~ically it concerns the shaping of these teeth so that the maximum possible number or them will be in contact during operation. The invention comprises an improvement ovex the disclosures incorporated in a technical article entitled "Maximum-Conjugacy Gearing" published in the April, 1979,issue of "Power Transmission Design".

The noise and vibration of gears varies greatly with the gear type. I1lorm gears, including both the edge engaging and face-engaging ("Spiroid") types nearly always run quietly.
This is partly because worm-type teeth mesh at a very oblique angle, so the relative velocity of engayement is small, and partly because worm-type gears can be designed for very large contact ratios. ~See for example "Gear Design and Application", edited by Nicholas P. Chironics, McGraw~Hill, 1967; pp. 48, 69-77~. The reason a large contact ratio is effective in minimizing noise and vibration is that dividing the torgue load over a larye number of tee-th reduces the "transmission error", as the minute variations in effective speed ratio are sometimes called.
Worm gears are not suited for most gearing applica-tions, however, because the range of speed ratios is so high ' ' ..~, ?7~

1 and the e~iciencies are so low. For most low and medium ratio gear sets, gears having coplanar axes are used, even though they do not run as smoothly. Tooth engagement velocitie~ normal to the tooth sur~ace can be reduced to some extent by using moderate helix angle~ (or, in bevel gears, spiral angles), but if these angles are too large, torque capacity is limited. Further, with conventional involute gear~ -the contact ratio in the direction of motion is limi~ed to the range from about 1.3 to 2, and this lead~
10 to considerable transmission error even in gears that are carefully machined.
The ob~ect o~ the present ~nvention is there-~ore to devise a tooth form that ~ill afford the maximum poss~ble profile (transverse) contact ratio for gears having smaller ratios than those of ~orm gears. The transverse cont~ct rat~o ~or such gearing cannot be as great as in the case of worm-type gears because of the more rapid convergence (and divergence) of the pltch circles on either side of the pltch point, but the invention discloses a method o~ mak~ng the 20 most of what geometric po~sibilities do exist. In general, the proposed tooth ~orm allows the transverse contact ratio to be approximately doubled, typically to a range ~rom about

2~5 or 2.75 to abou~ 6.
The essential method o~ the invention is to maintain tooth contact over a path that i~ e~ceptionally long relative to the tooth height, by u~ing as small a pressu~e-angle as possible, then eliminating the profile interference that occur0 in lo~ pres~ure-angle involute gears by ~sing suitahle curves in the tooth contact path. I~ these curves are co~rectly 30 chosen, teeth two or three t~mes flner than conventional involute teeth may be used without los~ of any tor~ue capacity 9 allowing tooth num~0rs to be ~ncreased proport~onately and, i~

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1 a helix or spiral angle is present, givlng much higher ~ace (a~ial) contao~ ratios. The "to~al con~act ra~o" (pro~ile plus ~ace contact ratios), which represents the average number of teeth that share the load, i~ there~ore ma~imized, and the operating noise and ~ibration is minimized.
The prior art disclose~ a number of gear types that employ curved path~ o~ contact~ Some o~ these gear types also have a low pre~sure-angle in the vicinity o~
the pitch point, ranging ~rom 0 to 10, a~ ~or example 10 the British '1Double Circular Arc" system, o~ Standard 978 Part 2-1952, A~dendum No. 1-1959; or the no~ obsolete 14-1~2 "composite system." These gears are among the many that :
employ cycloidal pro~ile portions in order to allow use o~
small nu~bers o~ teeth on the pinion. Minimum tooth numbers are governed by the requirement that i~ the tooth pro-~iles are not to have cusps, all portions o~ the path of contact must have normals that pass between the gear centers. The reasons ~or this are explaine~ by Buckingham ("Analytlcal Mechanics o~ Gear~," Dover, 1963, p. 48). To meet thi~
20 requirement, con$act paths ~or pinions with small tooth numbers muæt have large ~lopes at the ends o~ the path that are o~ten 30 or larger. ~owever, it i~ of historical ln$eres~ to note that to a great e~tent these cycloidal-type gear systems hav~ been supplanted by the 20 (and 25~ i~volute sy~te~, which with th~ aid o~ adde~dum modi~icatlon permit~ pinions with only 8 or 9 teeth to be made without undercutting (cusping~. These æmall tooth numbers are especially u~e~ul in applications re~uiring large reductions per ~tage~ as ~or example gear trains for cloc~ mechanism~.
The ob~ectives o~ the too~h pro~lles di~clo~ed in the present ~pecification are es3entially just the opposite o~ those of cycloida~-type gear~. In~tea~ o~ making lt

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1 pos~ible to use small tooth nu~bers, the objective is to make large tooth numbers practicable. 'l`o minimi~e nolse and vibration it is desirable that the pinion have at least 30 to 36 teeth, and ~henever possible 50 or 60. To be able to employ such ~ine teeth wlthout loss oY torque capaclty it is necessary to divide the load over as many teeth as possible, and this requires that the curvature introduced into the ends of the contact path be as small as possible rather than as large as possible. Only in this way can the 10 tra~sv0rse contact ratio be maximized. Typically the variation in path slope is there~ore limited to one~third to on~-tenth o~ that employed in cycloidal-type gears, whlch means that in all cases it wlll be le~s than ~0.
Other types o~ prior art gearing that employ curved paths, and the purposes of the curvature (in parentheses),~re as follo~s: U.S. Patent No~ 3,937,098 (increased permissible suriace load); U.S. Patent No. 3,251,236 (reduced tooth i~pact3; U.S. Patent No. 3,631,736 (reduced varia~ion in Hertz stre~es); and U.S. Patent No. 3,946,621 Sfluid 20 entrapment, utilizing contact path~ having average pressure-angle~ o~ 20 to 50 ). Buckingham (op. cit.) also shows a large number o~ curved contact paths, a~ in hl~ k'igs. 1-2, 1-3, 1-5, 1-8, 1-10, 1-11, 1-12, 1-14, and 1-15 (instructional, to illustrate the principal of gear prof ile a~alysis ln a more general way than is possible i~ only the straight line involute path 1 s considered).
It is a primary objective o~ the prese~t inven*ion to diselose the means by which curved paths o~` contact can be made practicable in power tran~mis~ion gearlng. It 30 will be evi~nt that all curved pa~h~ o~ contact, including tho~e disclosed by ~uc~ingham and the prior art patents li~ted above, must by their natur0 involve a v~rying pressure 7~
1 angle a~d therePore giYe rise ~o tooth ~orce~ that vary ln direction durlng the meshin~ cycle. I~ the transverse contact rat10 is small, these ~luctuating load ~omponents nor~l to the mean pressure line cause nolse and bearing vlbration. But if the transverse eontact ratio is large7 a~ in the gearing disclosed here~n, these ~luct~ating components will be phase-summed to produce a resultant that is negligibly small, especially i~ the path curvature is made as small as possible.
The means to achieve the special objects and advantages of the invention will be evident ~rom the drawings as explained in the speci~ication that ~ollow~. :
Fig. 1 is a partial ~ection of a pair o~ mating helical gears taken perpendicularly to the common pitch element (i.e., "transversely") and showing m~ting tooth profile~ embodying the invention.
~ig. 2 is a diagram showing the path o~ co~tact o~
the teeth o~ Fig. 1 and also the basic rack profile as~ociated with that path, enlarged to twice the scale o~ Fig. 1.
Figs. 3,~:4~:~and 5 are diagrams o~ the transYer~e plane area lying between the addendum circle~ o~ pai~ of mating gears embodyin~ the invention and showing alternative paths of contact together with the basic rack pro~iles associated with those pathæ.
In detall, and re~erring to Fi~. 1, typical teeth 11, 13 embodying the invention are shown in transverse section engaged at pitch point P. Tooth 11, at right, 1 on the smalIer gear 12 (pinion), which has its center at ~1 and tooth 13, at le~t, is on the larger gear 14, ~hich has 30 its center at 2 (~ the drawing). Other parts o~ pinion 12 and gear 14, such as hubs, webs, rim~, key~ays, etc.~ are standard and are o~tted in the interest o~ clarity.

1 In the embodlment illustrated in Fig. 1, pinion 12 ~s driving in the counterclockwise direc~lon and Gont~ct between the mating teeth ta~es place over a curved p~th that starts at point Sl on the addendum circle 15 of ~ear 14, passes through the pitch point P, and ends at point Fl on the pinion addendum circle 16. (In a speed increaser the direction of movement o~ the point o~ contact along the pa$h is o~ course reversed.) The path segment SlI is concave toward the pinion 12, but the main portion of the path, IPFl, lO is straight~
A straight line 17 ~oining Sl and Fl makes an angle with the common tangent plane. This plane is ~;hown in Pdge view as line 18, which is also the line tangent to the pitch circles (not shown) o~ ~he pinlon 12 and the gear 14.
In order to maxi~ize the tr~nsverse contact ratio, the angle ~, which in this speci~ication will be desi~nated as the "average pre~sure angle," must be made smaller than 14 .
The optlmum angle ~ that will still allow the use o~ a "con~tant pro~ile" (sharpenable) hob i~ ~rom 7 or 8 to 20 about 10. In so~e cases, such as gears having ~ ground ~inish, average pressure angles a~ small as 5 or 6 may be ~ound to bs practlcable.
The angle through which a pinion or gear turns while a given tooth ls in contact with its mate is called the "angle o~ action" or "roll angle." In Fig. 1 the pinion roll angle is the angle S101~. Fig. 1 al~o sho~s ~he pin~on p~tch angle, which is angle PO~Q ~here Q is a point on the pinion pitch circle ~or the pro~ile o~ the ~irst tooth to the rlght o~ tooth 11). Th~ quotient o~ the roll angle SlOlÆland 30 ~he pitch angle POlQ is the transverse contact ~atio. It i$
this ratlo that is maxlmirzed by mln~mizing the pre~ure angle 1 Fig. 1 also shows why it is advantageous, when the pressure ~ngle ls ~mall, to use a path o~ contact that has a curved portion rather than one that is entirely ~traight. It will be noted ~rom Fig. 1 that the normal lg to the path o~
eontact at Sl intersects the llne o~ centers 20 at a point 21 just inside the pinion center at 1~ (In the case illustrated the distance 21-0~ is substantially less than a ~ourth of the distance 21-P.) I~ the point-21 lay outside o~
l, this would mean that somewhere between P and Sl there must 10 be a point where the normal to the path passed exactly through l~ I~ the path extended beyond such a point (called the "interference point") it wou~d have to contain points equidls-tant from l that to obey the Law o~ Gearing would have to have di~ferent pressure angles (one negati~e and one positive).
In other words, a single point on the tooth profile would have to ma~e two dif~erent angles with respect to the radius ~ -vector. As this is not possible the hob simply pro~uces a cusp on the tooth pro~ile and no contact with the mating tooth occurs beyond this "interference polnt." The lnterference 20point for involute pro~iles is shown in ~ig. 1 as polnt I.
In gearing which employs involute pro~iles alone, the path o~ contact cannot extend beyond the point I. This mean~
ths maxlmum possible approach roll angle is limited to the angle lOlP, which is tAe pressure angle for the involute segmQnt o~ the path IP~'l. Since both torque capacity as tr~nsverse contact ratio are dir~ctly proportional to the roll angle, this l~mitation is highly undesirable. The solution, ~ccording to the present invention, is to increase the roll angle by extending the contact path with a curved llne I~l, all normals 30to whlch pass above l (such as the normal at S1 whlch interseets line P0~ at poin~ 21, as noted). ~ circular arc with it~ center at point 23, where th~ normal to the path at 7~
1 Sl intersect~ IVl, is one o~ several curves that may be employed Yor the segment ISl. In practice a non-uni~orm radius curve that is associated with a uni*orm radius curve on the hob pro~ile is ~ound to be preferable. The reasons ~or this will be explained in connection with Fig. 2.
Other ~eatures shown in Fig. 1 include a tooth ~la~k portion 24 that has a radius of curvature of at least one tooth module, which is considerably longer than the tooth root radius 25. The purpose o~ this long radius ~lank portion 10 24 is to minimize the tooth root stress concentration factor, and at the same time increase the tooth depth and hence its ~lexibility, so torque load will be distributed as equitably as possible. The whole depth of the teeth is characteristically at least 2.~ modules, and pre~erably 3 or more. When a~dition-al bending strength is needPd, these deep tooth roots may be shot peened or nitrided.
Fig. 1 also shows the normal 26 to the path at Flo When the gear ratio is larg~ enough, there is no inter~ere~ce problem on the resess side o~ the path, because the normal 20line 26 will intersec~ the line joining the pitch polnt P to the center o~ gear 14 at a point below 2 even though there ls no curvature in the recess segment PFl o~ the path. In e~ect, the reces~ path segment PFl is a typical i~volute path, except ~or the very low pressure-angle.
In Fig. 2 the path SlIPFl o~ the gears 12,1~ shown in Fig. 1 has been dlagram~ed separately to show its relation to its basic rack pro~ile Sl'I'PF~ hich is in e~ect a foreshortened version o~ the path rotated gO . As those skilled in the art will be ~ware, the e~tablishment o~ the path o~ cont~ct o~ a pair o~ con~ugate gears ~ully determines the basic rack pro~ile ~or the palr (Buckingham, op. ~it., p. 4).
Consequently the speci-~ication o~ the path o~ contact ~or a ~ ~l45~

1 pair of conjugate ~ears completely speciPies the shapes o~
the gear tooth pro~iles.
I~ the standard basic equations for finding the basic rack pro~ile are appli~d to the path SlIPFl, it will ~e Iound that the straight portion of the pa$h IPFl produces a straight rack portion I 'PFl ', and the curved ~gment SlI produces a curved rack portion Sl ' I ' . In practice it is easiest for the hob maker to produce a circular arc curve, in preference to other curve forms. I~ the hob 10 is made ln this way, the segment Sl ' I ' will be a clrcular arc i$ the gear set has spur teeth, or a ~egment o~ an ellip~e if the gears are helical. In the latter case, the standard equations ~or tra~sforming normal plane pro~iles to transver~e plane prof iles and vice versa are used.
(Buckingham, op. cit., pp. 143-146; as Buckingham notes, on pg. 142, the spur and helical gear equations ~or paths o-~
contact and basic rack pro~iles are the same i~ the analysls is made in the transverse plane. This is ~hy all the ~igures shown herein are tr~nsverse pla~e views.) Although Fig. 2 shows the basic rack pro~lle S1'Il'PFl' superimposed on the path SlIlPFl at the pitch point P, lt will be evident that in gener~ting the conjugate prQ~iles that are to contact along the giYen path, the basic rack pro~ile translates laterally, with Sl' and Fl' remaining on the parallel lines ~7, 28 starting at a position in which ~li coincides wlth Sl and ending with a pos~tlon in which Fl' coincide~ with Fl. (For any conjugate gear pair, the path for pro~ile generation is identical to the path :~or meslbing. ) There is one further requirement, however, which is that a pair of mating gears will have conjugate action only ~I they are generated by oppo~ite sides o1F the same baslc rack pro:eile. Thus the hob or generat:~ve ~rincl~ng whe~l for the _9.

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1 gear 14 must have t~eth 29 shaped to the ~ran~verse plane pro~ile rorm 31-Sl'I'PFl'-32, ~hile that ~or the pinion 12 must have teQth 30 shaped to the pro~ile form 33-Fl'PI'Sl'-34.
The latter profile form may be perceived more readily by turning Fig. 2 upside down, or by noting that the transverse plane hob working tooth profile for the pinion 12 corresponds to the inverted hob tooth space Yor the gear 14.
Fig. 2 also shows a fea~re that is particularly advantageous with low pressure-angle gearing. I~ thP gear 10 set is a spur gear set, the curved s~g~ent I'Sl' is a circular arc of radius Sl'T(or I'T) which is typically larger than four tooth modules. Be~ween the curve I'Sl 7 and the small ra~ius 35 o~ the hob tip, an intermediate radius 36 (or 37) ~s interposed. Because it has a length of at least one tooth module, which is several times that of the tip radius 35, this intermediate radius 36 (or 37) affords a tooth root stress concentration factor that is substantially smaller than that ~or conventional involute gearing.
Fig. 3 is a transYerse plane di~gram of an embo~iment 20 O~ the invention best suited ~or gear ratios of unlty or slightly larger. Thi~ ~igure shows a flat S-shaped path S3F~
that stre~ches bet~een the addendum circles 39,4~ of the mat;lng gears and lncludes a straight portion m-n that cD~nects the interference points for involute p~ofiles mJn. Beyond m and n respectively are curved portions mS3 and nF3 that h~ve normals such as 4]. and 42 that intersect the l~ine o~ centers 20 at points between the gear centers at 0~ and 2 (o~f the drawing~.
As in the case o~ the embodiDIent o~ ~igs.l and 2, the additlon of curved portions m~3 and nF~ greatly increases the gear set's roll angle and th~reby it~ torque capaclty and tran~verse contact ratio as well~
~uperimposed on the contact path ~3mnF3 in Fig. 3 is 37~

1 it~ basic r~ck protile S3'm'n'F~I. This rack pro~ile ~ again a foreshortened version o~ the path, in that it has a straight portion m'n' aajac~nt to the pitch point ~ e~-tending into curved ~egmenis ~3'm' and F3'n' at its ends. If de~ired, the path and it~ ~ssociated basic rac~ pro~ile may be made symm~trical with respect to the pltch point, in which case the same h~b may be used to generate both members of a gear pair, even if they hav~ different numbers of tee1th.
Figs. 4 and 5 show modifications of the S-shaped and J-shaped paths and basic racks o~ Figs. 2 and 3 respectively.
In both cases th~ ~urved segments, S4~, S5Y and F5P, run from ghe respectlve addendum circles 15, 16, 39, 40 all the wa~ to the pitch po~nt P and pr~auce corresponding co~inuous curved portions in the basic racks, S4'P, S5'P ~nd F5'P. The main advaniages of these modified path~ i9 that they simpli-fy the hob manu~acture slightly and allo~ for tooth pro~ile rclief that run~ all the way to ~he pitch point and i~
there~ore ef~ective at part load as well as full load. In 20 Fig. 4 $he path segme~t FlP and the baslc rac~ segment Fl'P
are straight, a~ in the embodiment of Fig~. 1 and 2.
It should be noted that the figure~ shown and described herein are for geometrical~y con~ugate gear~. h~o~t lnvolute gear~ ~ith tran~mitted lo~d~ in exce~s o~ about 1000 lbs. per lnch o~ ~ac~ width (17.86 kp. per ~m) are giv~n "profile modific~tion" (al~o called "~iguring") to correct ~or tooth de~lection under load and machining errors. ~uch modi~ications introduce sligh$ ~eviation~ from ~tralghtnes~
in both the path o~ ~ontact and the basi~ rack profile.
30 These deviation~ are introduced ~or entirely di~erent purpo~e~ than those ~ho~n in the accompanying drawing~ and are distingui~hable ~rom them in ~everal ways: (1) optimization --11~

1 of the gearing herein dl~closed requlres larger deviatlons than those employed for tip, or tip-and-root~ relie~ o~
conventlonal lnvolute gears~ ~hich involve varlation~ in path slope o:E about 1.5 at most; (2) involute pro~ile relie~ is obtained by makin~ the generating tip, or tip and ~lank, of the basic rack slightly concave, so extra material is removed ~rom the tooth at the top, or top and bottom, o~
the working proile, whereas in the case of the ge~ring herein disclosed the basic rac~ pro~lle ~or at least one of 10 the mating gears has a convex portion (pinion profile 33-Fl'PI'Sl'-34 in Fig. ~, segment m'S3' or n'F3' in Flg. 3, segment PS4' in Fig. 4~ and segment PSs' or P~s' in Fig. 5); and (~) conventional ~ethods of relleving standard lnvolute gears may be applied to the gearing herein disclosed, by adding a ~ight amount of material to the tips and/or roots of the ba~ic racks, the ef~ects o~ which would be superimposed on the already non-straight paths and basic rack profiles shown.
A number of further observatiolls may be made with respect to the herein disclosed gearing: (a) The gearing 20 obtain~ ver~ large transverse contact ratios without requiring the rad~al preloading called ~or in prior art patent U.S.
No. 4,149,431. (b) As in the ca~e o~ conventional high-ratio involut~ gear sets, unequal addenda may be utilized. The inequality (about 5~ in ~i~. 13 may be i~cluded in the hob deslgn or may be obtained during cutting by advan¢ing the hob ~or the gear and retracting that ~or the pin~on. In either case, the inequality, ~hich may ~ake the gear addendum only a third or less of the pinion addendum, permits the curved segme~t o~ the hob I~Sl' to be shorter but has the 30 disadvantage o~ reducing the total roll angle and greatly increasing the wear and scoring hazard at the end o~' the contact path: (c~ The roll angle (e.g., an~le ~lOlElin Fig~ 1) 1 is increased not only by minimizing the average pr~ssure angle ~ but also by maximizing the addendum heights of both members o-f the mating pair. It is there~ore generally advantageous to use addendum coefficients that have a sum greater than 2.0; (d) The system i~ applicable not only to spur and helical gears but also internal gears and straight and spiral bevel gears. In the case of bevel gears, the sur~ace in which characteristic meshing action occur-s is nofc a plane, as it is in par~llel-axis gear sets, but a 10 spherical sur~ace. In applying the present invention to bevel gearing, the transverse plane views Oe the dra~ings should therefore be construed as pro~ections of this spherlcal transverse surface onto a plan~ normal to the common pitch ele~ent. The common pitch element is shown in end view as the pitch ~oint P, as in the case o~ spur and helical gearingJ
but ~or bevel gearing the line 20 represents an edge view of the plane containing the gear axes; ~e~ It is possible to replace part or all of the straight path segment PFl 1n Figs. 1,2 and 4 by a curved segment that is concav0 to~ard the 20 pinion 12. This would give the p~th a ~lat C-shape lnstead o~ the J-shapes and ~-shapes illustrated ~n the various ~igures. A C shaped path would have adv~ntages only i~ the gear ratio is quite large and the minimum permissible pressure a~gle *or the hob decreases with distanee from the pltch point; ~:e) I~ desired, the segments shown as curved in Figs. 1, 2, 3 and 4 may he replacQd by thelr chords, in which case the paths and basic rack pro~iles illu~trated could be made up o~ sets o~ two or three interconnected non-colinear straight lines.
To clari~y the scope of the ens~ng~ clalms, the following definitions are provided: "not preloaded"means that no components oi ~orce are urgin~ the gears to~ard each ~5~'7~

1 other ~hen no torque is being transmitted; "tools" means hobs, shaper cutters, shavers or generative grindin~ wheels (Rei~hauer~ used to rough cut or ~inish gears embodying the invention; "non-straight" means including a curved portion or a portion made up o~ a pair o~ non-colinear straight lines.

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Claims (16)

I CLAIM:

1. A pair of mating gears that are not preloaded, are mounted on coplanar axes, and have conjugate teeth formed to make contact, when said gears are transmitting torque, over a transverse plane path of contact that affords a ratio of roll angle to pitch angle of at least 2.5, said path being non-straight and having an average pressure-angle smaller than 14°.

2. A pair of mating gears according to claim 1 wherein the number of teeth on each of said gears is at least 30.

3. A pair of mating gears according to clam 1 wherein the number of teeth on each of said gears is at least 36.

4. A pair of mating gears according to claim 1 wherein said teeth are slantingly disposed with respect to the common pitch element of said pair.

5. A pair of mating gears according to claim 1 wherein said ratio is at least 2.75.

6. A pair of mating gears according to clam 1 wherein said average pressure-angle is smaller than 10°.

7. A pair of mating gears according to claim 1 wherein said average pressure-angle is smaller than 8°.

8. A pair of mating gears according to claim 1 wherein said path has an S-shape.

9. A pair of mating gears according to claim 1 wherein said path has a J-shape.

10. A pair of meting gears according to claim 1 wherein one of said pair is smaller than the other and said path has at least one segment that is concave toward said one of said pair.

11. A pair of mating gears according to claim 1 wherein said path includes a straight portion adjacent to the pitch point.

12. A pair of meting gears according to claim 1 wherein the whole depth of the teeth of one of said pair is greater than 2.6 tooth modules.

13. A pair of mating gears according to claim 1 wherein the transverse profiles of the teeth of one of said pair include a dedendum segment that lies between the working profile and the minimum radius root curve, said segment haveing a minimum radius of curvature of at least one tooth module.

14. A pair of mating gears according to claim 1 wherein the sum of the addendum heights of the teeth of said pair is greater than 2Ø

15. A pair of mating gears according to claim 1 wherein the normal to said path at one of its ends intersects the shortest line between the pitch point and the axis of one of said gears at a point four times closer to said axis that to said pitch point.

16. A pair of mating gears according to claim 1 wherein said teeth have unequal addendum heights.