Note: Descriptions are shown in the official language in which they were submitted.
~5~3~7'~
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
'
- 1 -
'
..~,
?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~
7'~
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
3 7 ~L
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
--4--
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.
--5--
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
--6--
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--
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
--8--
~ ~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.
7'~
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
-10--
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.
~14-
