Paper
67-WAIDE-8 P. C. RENZO Director of Engineering.
S. K A U F M A N Chief Engineer, Coupling Division.
D. E. DE ROCKER Chief Engineer, Product Developmen!.
Gear Couplings disc ussio n and mathema tical an al ysi s of of the operatio n of of gear couplings at angula misalignment; transmission of uniform motion, tooth separation, both load distribution , coupling load load cap acit y, tooth bearing, and special tooth tooth forms.
Sier-Bath Gear Co., Inc. North Bergen, N.
T H E R E are dilferences of of opin ion a s t o how flexible flexible gear couplings act, whether they transmit true uniform motion at substantial angles, how many teeth are in contact, and what kind of crown should be appli ed. We shall now now try to establish establish th e f acts pertinent to this subject. Gear couplings are usually arranged in pairs, each individual coupling coupling compr comprisin ising g a sleeve sleeve with str aight teeth on it s inside, and a hub with teeth crowned crowned to cooperate with th e sleeve t the range of of angl es specified. sleeve, and hub, H, are shown in aligned position in a fragmentary cross section in Fig. and in an axial section in Fig. 2. Generally, the sleeve teeth have involute profiles, am, rising from a base circle, as on conventional gears. gears. Adjacent in
should should also have a const ant distance from each othe r in the direcdirection of of t he surface normals to match th e sleeve sleeve teeth. This rerequirement is no different on cor~plingstha n it is is on gears. gears. Teeth with unequal normal distance p could not be brought to match and take over load smoothly from one another. of t his requireme nt, the t ooth profiles f the hub, in As a result of planes perpendicular to the hub axis, should change increasingly ingly with increasing distance from from t he hu b axis, at least when the coupling is designed for a subst ant ial runni ng angle. angle. This will be further described. Fig. shows shows a conventional conventional uniformly crowned crowned hub. The hub looks like an excessively crowned gear. As on a gear, its shape is best defined by the shape of of t he rac k teeth with which which it can mesh and run so th at its entire tooth sides get into contact. This rack can be considered an extremely large, infinitely large, anywhere, taken in the direction of of t he surfa ce normal q. gear. hob produces th e stra igh t profile f t he involute rack in consequence, consequence, the adj acen t crowned toot h surfaces f th e hub the midplane G. now the hob is fed about an axis as if turned about this axis, it will produce the sam e straight profi profile le in Contributed by the Design Engineering Division for presentation all planes containing axis and envelop the rack tooth shape. at the Winter Annual Meeting, Pittsburgh, Pa., November 12-17, Each toot h surface of of the rack contains straight profil profiles es that THE AMERICAN SOCIETYOF MECHANICAL ENQINEERB. 1967, intersect axis at the same point and that have a constant inManuscript received at ASME Headquarters, July 26, 1967. Paper clination to axis In other words, the surface that would be No. 67-WA/DE8. produced on th e rack is a conical surface with with axis C.
Fig.
Fig.
Fig.
In its feed motion about axis C, the rotating hob, so to say, represents a moving rack t ha t is the count erpar t of th e rack or infinitely large gear produced by the hob. Fig. shows part of the represented rack with axis C in perspect,ive. Th e straigh rack profiles show up in all nxial sect,ions, as in midplane G. They come to a point or apex on axis C. Th e cone apex of side SI is at CI, the'one f side SZat CZ Adjacent sides S,, are identical conical surfaces merely displaced alon g the cone axis C They have a constant distance from each other everywhere in the direction of t he surface normals. The tooth sides f t he hu b are produced in a generati ng motion whereby the rack represented by the hob rolls on the hub. This rolling motion is as if a cy lindrical pi tch sur face f th e hu would roll on the pitch p lane of t he rack in con tact therewit h. The pitch surfaces intersect the t ooth sides in curves called pitch lines. As the Pitch surfaces roll on each othe r without sliding, the pitch lines f t he h ub ar e as if prin ted on its cylindrical pitch surface by the pitch lines f th e rack. In develop ment of the cylindrical pitch surface to a plane, the p itch lines f th e hub are identical with the pitch lines of t he rack. At any intersection point Q of a rack pitch line with t he contact line Q of th e pitch surfaces, the co ntacting t ooth surfaces have the same direction. The y have a common surface normal Q whose inclination to axis C is constant and equal to the (Fig. pressure angle in the midpla ne G. Normal QC remains normal t o th e conical i.ack tooth sides in its fixed position even as t he rack moves along axis C. In each rac positioii, it intersects t,he rack-tooth side a t a poirit of c ol ~t acwit,h the hub meshing therewith. t is a path of contact As this is true for all points Q and t heir s urface riormnls, these suila ce riormals determine and d efine the surfa ce f prog ressive conta ct. As in gearing, this surface can be used to establish the required tooth shape of th e hub. As the conical rack-tooth sides, su ch a s St, St, have a constant normal distance from each other, the so-generated hub-tooth sides also have a con stant normal dis tance from each other, as required for smooth power transmission. At an y point Q, th e profile inclination o r pressure angle is con sta nt in plane QC (Fig. th at contains the cone axis C Hence it is bound lo be different in plarie th at is parallel to midplane and perpendicular to the hu b axis. t can be shown th at the profile inclinat.ion 4' or pressure angle in plane is related to the pressure angle in midplane and to angle QCP as follows: an The t,erm
is
4'
an
smaller than $.
cos
It
(1
drop s o f first very slowly e.
Fig.
To describe th e coupling action, we shall first look at the too th contact without appreciable load and without ally elastic deflecThe effect of tion of th e contacting too th surfaces and teeth elastic yielding will be introduced later. We shall also show tha involute gear couplings can transmit true uniform motion, even t very large coupling angles.. Fig. is an axial section showing a sleeve and a hub at an angle i. Here the tooth contact has shifted toward the tooth ends. Fig. is an axial end view of the sleeve that has straight involute teeth with base circle Th e uniform motion of th e sleeve and hu b is like the motion of a pair f b evel gears whose axe s coincide with t he sleeve and h ub axes ., a, that intersect at 0. Th e sleeve and hu b move as if two imagined conical pi tch su rfaces of t he gear members roll on each othe r without sliding. These rolling surfaces contact along the instant axis th at bisects t he angle between th e coupling axes. The instant axi is very useful t o show up th e relat,ive moliol~ f oiie member with respect to th e ot.her. At any one instant, tllc hu b moves briefly as if turned abo ut the i nsta nt axis relatively 1.0 th e sleeve. Thi s defines the direction of relative motion of each point of the hub. Th e velocity of this motioli depe nds on the turning velocity abo ut the ins tant axis. t is known to be obtainable from the turning velocity f t he sleeve and hub by geometric addition, as expressed in formula (2):
ui
sin
(s)
'The relative velocity of a ny po int is the product of its distance At a poirit of to oth contact, i from the instant axis and of wi. defines the sliding velocity. t increases with increasing angle i. As the instant, aneous relative motion is a turn ing motion about the i nstant axis, the surface normal at, any point f tooth contact is bound to intersect the instant axis I. With involute sleeve teeth , a surface normal fixed ill space stay s a surface normal at all turn ing angles of t he sleeve. Uniform motion is tran smit ted.
It
becomes a path of contact.
Th e contact point moving along
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Fig.
Fig.
Fig.
its straight path describes a line on the t ooth sides as the cor~pling turns. This line coincides with an invo lute profile on the sleeve teeth. A pat,h of cont act lies in a plane perpendicrllar to the sleeve a, t also intersects the instant axis I, N,. The crown f the hu b teet h can be determined so as tJo place the path of conta ct at a desired axial distance from the intersection f th e axes, t the design misalignment angle i. Th foregoing requirement defines the location of th e con tac t point at the pitch circle k. generally does not lie directly on the vertical t,hrough but close enough to i t th at its distance fro the instant axis does not differ much from pitch radius R. this reason, the sliding velocity at mean contact point be put down as
6
(approx)
Fo can
3)
where is the peripheral velocity R . w , an io is the coupling angle measured in degrees. Each of t he two sides f the t eet h ha s two diametrically opposite paths of contact. One is along normal that intersects the N,. The other is along normal n' that interinstant axis sects instant axis on the opposite side from Th cont,act normals n, intersect t he cylindrical insi de surface of t he sleeve teeth and a spherical outer s~trfacef t,he h11b te et h. Th path of contact. is between the two intersection points. It s length determines the durat ion of co ntac t. it were exactly equal to the normal distance (Fig. f adj ace nt too th sides, then each tooth start s contact when the preceding tooth leaves off. As on gears, profile overlap is desired, a l ength 1.2p to 1.6p or more. This length and the duration of c ontact depe nd on the too th depth and on the pressure angle or profile i~lclination. After passing through the contact, a tooth separates from its mate, to contact it again at a different spot after about half a
Tooth Bearing Fig. is a fra gmen tary axial section of a sleeve, wherein involute tooth profiles appear as straight lines. In the aligned position of t he sleeve and hub , t he to oth c onta ct is along profile at zero load, all the tee th contacting simultaneously. On both sid es f lin the contacting tooth surfaces gradually separate from each other a t a r ate depending on t he amount of crowning. They separate a t first very slowly. The cross-hatched area around point has a separation within a fixed, very small amount z' such as 0.001 in. Such a separ atio n might be overcome by elastic deflection under heavy loa d. The area then becomes tooth bearing area. At a coupling angle the cont act hasshifted away from central position to two mesh zones. A tooth contactsonly a t onepoint at a time, at zero load, a t a point such as in one turning position; and after half a turn at point '. The cross-hatched elliptical or is oval area has a separation within a given small amount z'. As the smaller than the cross-hatched area around point P. coupling turns, the contact point moves along profile y. Th rectangular area around or Q' is within a separat ion z' of getting into tooth contact a t zero load. Under load, it may become the area swept by tooth coritact. With the conventi onal uniform crowning the width of these areas aror~ndpointas a ~ Q, ~ Q' d is approximately equal. When the cor~plingruns at an angle, however, there a re fewer teeth in contact, o111ytwo at times a t th e maximum design angle, and these fewer t,eeth have less int ima te cont act. Moreover, sliding increases with increasing angle i. 111 consequence, the sustained load capacit y a t the design angle is only a smal l fraction of t he capa city of t he coripling in alignment or near-alignme nt. Fig. shows th e kind of t ooth bearing obtained a t a substantial angle when the profile inclination of t he hub tee th is constant in planes g, Fig. 2, perpendicrllar to the hub axis rather than being constant in planes QC containing axis The profile inclination is then too large in planes g, so th at the toot h bearing is displaced toward t he top of the sleeve teet h when the co uplh g runs at the
turn. The maximum separation attain ed depends on the coupling
design angle.
angle i.
motion and causes early wear.
Journal of Engineering for Industry
Th is affects the smoothness f the transm itted
ih4 Fig.
Fig.
Contact Cycles and Backlash Let us look at the relative motion f the hu b with respect to a sleeve maintained statio nary. Instead of turning both the sleev and hub on their axes, an opposite turning motion about the sleeve axis is added to the system comprising sleeve and hub, so tha t the sleeve turning motion is cancelled out and the sleeve stands still. The hub axis then describes a conical surface about t,he sleeve axis. s apex is at th e intersection point f axes. We shall first consider the case where the crowning axis intersects the hub,&is, at -0 , and look a t a spherical surface thrdugh mean pointlP.of the centered at O and Fig. is a radial view taken in direction PO. Fig. is a si de view tak en in the direction of t he sleeve axis a,. Th e spherical 'surface intersects the hub-tooth surfa ce passing through substantially in a circle ( k ' ) centered a In projection, Fig. 8, aDpears as a straight line that coincides with the hub axis ah The same sphere intersects the contacting tooth surface of the sleeve in a curve whose mean curvature radius in projection, Fig. 8, can be shown to amoun t to ctn on curves having only a small distance zu from The circle and curve p, contact or nearly contact a t point P'. In the relative motion, the hub axis describes a conical surface f t he hu b axis describes about sleeve axis a#, whereby a point circle k'. At a turning angle 0, point reaches a position H. 90 deg and 180 deg, And a t turning angle reaches positions an H' respectively. n the view in Fig. 8, th projected hub axis appears inclined a t an angle to sleeve axis Tan can be readily computed as a, ta
tance
ct
from P. cos The distance of curvature center from projected circle is R ctn f o u n d to amount to cos and the distance of circle cos from curve
(*%'
R ctn
cos
more general cases where the distance R, of mean point, from sphere with radius Re is considered. The foregoing formula for applies also when R, is substituted At small angles as in common use, th formrlla can be t,ransformed into
In th
from the crowning axis C-C (Fig. 2)
l/,R, ctn The maximum separation 1. Hence sin 20
20
t a n 2 sin2
is attained when
I/1Re ct
is 90 deg where
tan2
The coupling runs a t minimum backlash a t the maximum angle The backlash is increased by Ab when th e coupling is set in alignment, whereby th e separation zo is added on each side:
The foregoing figures apply to uniformly crowned teeth.
cos
At a turning angle of 90 deg, when is at ', the circle k' the hub-tooth side again appears projected a s a straight line in Fig. 8, a line coinciding with the projected hub axis O H ' . And 180 deg, it appears projected as a at a turning angle of straight line . t appears in Fig. as if swinging abou t rad ial line PO between end positions an OH'. We shall now compute the distances between circle k' an curve as if circle k' would swing abo ut radial line PO whereby its plane always contains th e hub axis. Although this assumptio is not exactly fulfilled, it provide s a close enough result a t the ~ n o d e r ~angles le now considered. The curv-ature center of projected curve p, lies in t he plane
passing through
t right angles to the sleeve axis a,, at
dis-
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The variation of separatio n z with th e turn ing angle 8 is directly shown in diagram, Fig. 10. t shows a circl k, with center Oo and diameter zo The term z shows up as the distance 0'0'. of any point Q' of th e circle from th e straight-line element X-X tangent to the circle at 0'. Point Q' corresponds to a turning angle 8 QfO'O'. t can also be obtai ned by plottin g an angle 28'f rom center Oo. Sepa ration is seen to vary harmonically with the double turning angle 28. Example: With 0.0010 in. from (4).
20 deg,
1 deg and
21/2
in .;
This is a quantity small enough that it compares with the elastic deflection f the teeth under load. Und er load, then, more teeth get into simultaneous conta ct, especially a t small angles i. We shall now determine th e number of t eeth in con tact under load.
Load Distribution The number of teeth tha t carry the load depends on the angle between the coupling axes, on the tooth design, and also on the load. _-
In operation t an angle at very small loads, each tooth gets into contact, separates, and contact s again after half a turn f th coupling. We have given th e maxim um too th sepa rat ion zo in
Journal of Engineering for Industry
formula (4 for no appreciable load and uniformly crowned teeth. The separation z t any turning angle 8 from contact position is defined in formula (5). To esti mate the number f teeth in contact, we consider average conditions, wit hout such irregularities in contact pattern as may occur when a new tooth gets into contact or a tooth gets out of contact. In th e case considered here, moreover, the individual tooth load is in direct proportion to the elastic tooth deflection, th e added deflection f t he sleeve tooth and hu b tooth, both surface deflection and bending. Thi s proportionality is at least approximately fulfilled. is smaller than Those teeth are in contact whose separation the deflection z, f the te eth t ha t carry th e largest individual load where
cz, The proportionality factor depends on the material used and on the tooth design. Its computation is involved and omitted t can also be determined reliably by test. In thet est, all here. bu t two diametrically opposite teeth of th e test hub are removed. Cont act with t he sleeve teeth is established a t zero coupling angle. Then to rque is applied which results in a slight relative turni ng displacement. Th e displacement is measured close to the
is the proportion of the tooth contacting teeth. to the displacement t the pitch point
load applied
The elastic deflection z, f loaded tee th allows adja cen t mating teeth to move toward each other. The teeth whose,separation was equal to th e elastic deflect,ion z, of the m ost loaded t eet bheri move in to cont,act wit,h each ot,her carry no load. The t,eeth whose separat,ion z was less than z,areloaded proport.ion lo t.he difference (z, z) Their load is
The term z is plott ed in Fig. 11 in ternis of th e a~ ig le0, the 0. angular distance from the mean contacting tooth where z It is the vertical dist ance from X-X f any poi nt (Q ) whose horizontal distance from is proportional to angle 0. Th curve so obtained is a sine-curve. t repeats with every hal turn. After determining z, for a load that, can be carried by a single tooth with a margin of safet y, z, is plotted n Fi g. 11 from X-X up, and a line is drawn parallel t o X-X t a distance z, therefrom. z, should be plotted a t the scale used for 20, th e maximum separat.ion at t he now considered coupling angle i. Line B-B cuts of the bottom of curve s, between end points
9,'. All the teeth within the spread QrQel carry some load, that fades out and beromes zero at these end points. The load t any point is proportional t o th e vertical distance within t he cross-hatched area at tha t point. The tota l load carried in on engagement zone is proportional to th e cross-hatched are a,f below the spread QrQ,'. all the teeth within 90 deg to both sides of would carry the maxim\\m load P,, then the tot al loa within th at range would be proportional t o the a rea of the rectangle 1-2-3-4. The said total load amounts to
or double th at a mount all around t he periphery, N tooth number. The safe load th at call be actually carried on perfectly accurate couplings is a fraction q f the load N P, , where q is the rat,io f th e cross-hatched ar ea o th e area of re ctang le 1-2-3-4. These areas can be readily computed. Using radian or arc measure for the angles, ar ea can be shown to am ount t,o ' / ~ z ocos 20. (t an 20,
arc 20,)
1/2z0 cos 20, inv 20,
(7)
while the are a of r ed an gl e 1-2-3-4 is in ar c measltre nz,
zo s in z 0,
1/2nzo(l
cos 20,)
Hence proportion q amounts t
Tz,
=
cos 20, (tan 20, n(1
arc 20,)
r
cos 20.)
o
The load cap acit y of an act ual coupling comes t.he closer to the figure qNPi at the co~~pling angle i; the more accurate it is, the closer its tolerances. We may intro duce a coefficient to express the tolerances; 1 for absolutely accurate couplings. I t somewhat smaller tha n 1 on commercially accu rate couplings, the difference from 1 increasing with increasing tolerances. Thu the load capacity P, at the coupling angle is
Proportion
is'plotted in Fig.
in ter ms of th e propor tion
Line B-B in Fig. 11 corresponds to 0.500.
If z.
zo, then q
z.
['/ZZO
0.250. (z
If z,
20,
zo)]
z.
Zo
then
A fe ik xa mp le s will illus trate t he use f th e q-graph in Fig 12. Example 1: In the example previously given for coupling 20 angle 1 deg, 20 was computed 0.001 (in.). z, is deter-
mined from the maximum safe loa per t ooth, based on surfac stress as well as bendi ng stress. If factor is already known, t.hen 2. Pi/ C. Let i t be assumed that, z, was determined as z,
0.00026.
Then
Zo
0.250 a
1 deg.
Read from th e q-graph, Fig. 12, 0.218. the maximum safe tooth load at a given tooth pitch and crowning is, for ins tan ce, 1 O lb, a nd the hu b has 50 tee th whil the accuracy factor is 0.80, then the coupling could cariy a tot al load of 40,000 Ib when in a lig nme nt, and (40,OOO lb 8720 lb when running a t an angle 1 deg. Example 2: I/ half the angle f Exa mple 1, deg? is the same as before, and remains 0.00025. 20,however, is smaller, as tan in formula (4 ) is appro ximat ely one half f th former amount, and tan2 is one quart er thereof. Hence 20 0.00025 in close approximat ion, and Z" Zo
1.
Read from the q-graph, 0.500. Thi s results in a total toot,h load of q(40,000 lb 20,000 lb that the coupling can safely carry a t a coupling angle deg. Example 3: How m~lchload can the same conpliiig carry at an angle double th at of t he first example, at 2 deg? Here zo is app rox ima tel
0.004 and
Zo
0.0625.
Th e graph
shows q 0.107, so th at the to tal t oothlo ad figuresq(40,000lb) 4280 1b. These figures are based on contact at a mean tooth depth. At the larger augles i, profile action should be considered as well. Elastic tooth deflection then decreases in importance. The couplings act more and more like gears, with increasing angles We are aware of bu t have not direc tly introduced tooth sliding in the foregoing computations, which increases with increasing angles i; nor inti macy of t oot h conta ct, which decreases with increasing angles and thereby increases the surface stresses. elastic tooth deflection. There will hardly be any disagreement with ishe general oonclusion that the usual couplings can carry much more load at, small running angles than t large ones.
The Vari-Crown1 To ma ke the d rop in load ca pacit y less drastic, Sier-Bath has developed the Vari-Crown. Fig. 13 is an axial section taken throug h a hub. n conventional crowning, as produced by hobbing, th e feed path of the hob and th e tooth bottom prot may be centered a duced thereby is ordinarily a circular arc. on the hub axis, or at or C' dep end ing on how much f a crown is required. Othe r methods also aim to produce the same kind of crowning. Wit h the Vari-Crown (T rade M ark ), the feed path of the hob and the tooth bottom B' are more curved in the midplane and less curved furt her tow ard the to oth ends. n axial section f the tooth surface through pitch point shows a curve almost identical with t he feed pat h there . The varying curvature f the t,ooth surfaces widens the tooth contacts with increasing distance from mid pla ~~e while at t,he midplane it has the smallest width. t ten ds t,o more equalize t.he load capacity a t different angularities, taking aw ay from the excessive load capacity at zero allgularity and dis tributing this excess to larger angularities. Thus the ca pacity a t the largest angularity s more than twice as large as with the uniform crown.
End Round n some cases, a r ange of angu larit ies is specified for action .under load, and an additional range for action without appreciable load. There a re two common ways f meeting this Made under patent No. 2,922,294;other patents pending.
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specification. One is designing the coupling for the whole range of angularities. This means increased crowning and increased stress. The other is to design th e coupling for the angularities under load, and letting the tooth contact shift to the ends and edges of the teeth for th e no-load angularities, allowing sufficient backlash so that the coupling can take the largest angularity
without binding. Neith er of these ways is quit e satisfactory. For such cases, Sier-Bath introduces the End Round. I t i s illustrated in Fig. 14. Crowning is sharply increased a t the tooth ends, a E. The main crowning is figured for the angularities under load. At the maximum load-angular ity, th e center of the tooth contact is placed at where t would be normally placed. The End Round starts further out, at a distance from about half the width of the contact area. t is sufficiently curved to prevent contact t o shift to th e end edges, keeping it on the tooth-side surfaces. Thi s is tang ent contact capable f carrying moderate load, withou t any tend ency to chew up th e sleeve tooth sides.
Extended Contact Design With spindle-type couplings, sometimes large angularities are specified, bu t each coupling of a gro up is run only within a restricted angular range which is different for different couplings. n general, all these couplings are made alike and are designed to run up to the largest angularity. For simplicity f replacement, this arrangement is ideal. Bu t we can make much stronger couplings and multiply the coupling life perhaps 10 times if the couplings do no t all have t o interchange with each other. For instance, if the range of adjustment of a coupling with a maximum angularity of deg is cut down to deg, so that the coupling may be designed to run only between and deg angularity, the load capacity based on surface stress can be increased three to four times, and the coupling life at a given load increased many more times. Th e tooth contact can be muc widened. s the ex tra simplicity f t he present setup worth wasting coupling life th at much? t is not difficult to have two or thr ee different coupling designs to t ake t he place of a singl design for th e whole range. The y could be marked, for in stance, in different colors.
Wearing In and Wearing Out Fig. 13
t might be thought t ha t the couplings wear into the right shape anyhow, and th at ther e is no need for refinements or even for accuracy. Bu t this we consider no more valid for couplings than it is for gears. Impr oper s hape causes excessive surface stresses and tends to damage the tooth surfaces. t is the first stage f wearing out . We suggest tha t the tooth shape should be as nearly correct and adequate as possible.
History and Conclusion
Fig. 14
Journal of Engineering for Industry
Gear co~~plings started out with straight teeth on the hub. And t hey worked t th e small angularit.ies where they were used. As the need for large angularities arose and grew, it was recog11izedth at the too th ends of th e hub had to beeasedoff. Crowning was inve~~ted. t was also recogt~izedthat a t the larger i~ngularitiesthe tooth contact is confilled to two diametrically opposite zones and t ha t th en centeri ng was desirable or required. A spherical outside surface was introduced on the hub, centered on the hub axis, to let the hub teeth bear against the tooth bottoms of th e sleeve for centering. Th e contact between the spherical outside surface and the cylindrical inside surface of the tooth bottoms provides accurate centering a t all angularities. This may have been suggestive of the thought t ha t the side of the hub t eeth should also be crowned about center (Fig. 2). A while late r came t he realization th at there was no compulsion or natural law for crowning about center 0. Other crowning centers were used a s well t o achieve different amounts of crowning. And now we know th at we do not even need a center fo crowning, that crowning may be varied along the tooth. The stepwise progress was accompanied by a n increased ~~nderstanding of how the teeth act. At large angularities, they are like gea n with internal mesh. Like gears, they transmit true uniform motion. The y have, however, the peculiarity th at a toot h gets int o cont act in two zones of mesh. Between these
z o ne s, t h e t e e t h s e p a r a t e . T h e i r s e p a ra t i o n i s m u c h s m a l le r t h a n on gears even with internal mesh. And a t small angularities, it is very small, so small that the elastic deflection of the teeth plays an important part. I n t h i s a r t ic l e , we h a v e t r e a t e d c o u p li n g s li k e g e a rs , s t a r t i n g o u t by assuming rigid bodies, and formulated the separation of the teeth for rigid bodies A n d t h e n w e h a v e c o n s i d e r ed t h e e l a s t i deflections of the contacting teeth, obtaining modified results
t h e re f o r. W e h a v e c o n f i r m e d m a t h e m a t i c a l l y t h a t t h e n u m be r f teeth in contact increases with decreasing angularity and have described a simplified an d approxim ate co mputation of th e number of te eth in c ontact a t different angularities.
Acknowledgment We express our thanks to Ernest Wildhaber, our consultant, for his valuable advice an d help in preparing this paper.
Printed in U. 9. A.
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the ASME
