Nutating drive mechanisms having roller driving elements

A torque transmitting gearing system of the nutating type is equipped with a nutating idler member which is in torque transmiting engagement with both a stator and an output gear. Torque transmission between the respective elements is achieved via respective series of rolling, torque transmitting elements in the form of tapered rollers. The rollers are maintained in substantially continuous contact with both their respective driving and driven raceway surfaces, which are formed with trochoidal curvature. Within a given pair of coacting gear surfaces, one surface will be formed with epitrochoidal curvature, and the other with hypotrochoidal curvature.

BACKGROUND OF THE INVENTION 
Nutating type torque transmitting systems are well known for their utility 
as speed reduction mechanisms. Typically in these systems, an input member 
is provided with means for initiating nutating movement on the part of an 
intermediary idler member. To obtain a speed reduction, the nutating 
member is normally held rotationally stationary with respect to a stator, 
while the intermediate member engages an output via gear teeth, rotating 
the same at a speed determined by the relative numbers of teeth on the 
idler and the output. 
Of the nutating devices known in the prior art, all have employed teeth as 
a mechanism by which torque is transmitted. Applicant has found that 
nutating mechanisms of this sort are disadvantageous, owing to the extreme 
precision which must often be observed in manufacturing the gear faces in 
the member initiating the nutation. If such mechanisms are of conventional 
gearing material, substantial expense is entailed in machining the gears 
to acceptable tolerances. Further, since precision is at a premium, it is 
often difficult to form the gears out of extrudable materials such as 
plastic. 
Conventional nutating gear mechanisms also are incapable of transmitting 
large torque loadings, as 100 percent of the torque is transmitted at any 
given time by only a small percentage of the total number of gear teeth. 
This is an inherent problem in prior art nutating mechanisms, due to the 
fact that the wobbling intermediate member could not be in contact with 
more than a few teeth of the stator and/or the output member at any given 
time. This distinct disadvantage is overcome in the present invention by 
means of a unique departure from the use of gear elements as the mechanism 
by which torque is transmitted. In particular, the stator, idler, and 
output member are formed with surfaces which constitute raceways for two 
or more series of rolling elements provided in the form of tapered roller 
bearings. Owing to the novel curvature of the respective members and the 
relationship between coacting grooves, the rollers are maintained in 
contact with both the driving and driven surfaces at all times, while they 
rollingly transmit torque between the respective members. 
All known prior art nutating mechanisms have employed coacting teeth as at 
least part of the torque transmitting means. The only known example of a 
prior art system which does not exclusively employ teeth for this purpose 
is disclosed in the patent to Vallance, U.S. Pat. No. 1,748,907. This 
patent discloses a speed reduction mechanism in the form of a nutating 
gear system, wherein an input shaft initiates wobbling motion of an 
intermediate member 7, via the engagement of a portion 9b of the 
intermediate member with an angled or canted portion of the input shaft 2. 
Radially outwardly on the member 7 are disposed a train of teeth 10 which 
engage stator teeth 11 formed on a portion of the stationary housing 5. 
Inside of the cup-member 7 are arranged a number of hemispherical recesses 
7b, in which are fixedly seated a like number of balls 8. These balls are 
in turn in engagement with a continuous curved groove 6b formed in an 
output member 6. As with other known nutating systems, the engagement 
between stator teeth 10, 11 prevents the intermediate member 7 from 
rotating during nutation, so that output rotation is effected solely by 
means of the engagement between the fixed balls and the groove. As the 
idler member 7 nutates, the balls 8 will successively cam the element 6 
rotationally by engaging the walls of the curved groove. 
Although being of interest for the feature noted above, the patent to 
Vallance nonetheless suffers from the several deficiencies noted 
previously. In particular, torque transmission via the system is limited 
by the small number of teeth engaged at the stator, and strict 
manufacturing tolerances must still be observed both in the manufacture of 
the teeth and the groove itself. In any event, the teachings of Vallance 
by no means approach the present system, wherein at least one series of 
rollers rollingly transmit torque between respective elements of the 
device. 
Other prior art nutating mechanisms employing rolling elements in some 
capacity are disclosed in U.S. Pat. Nos. 3,139,772, 2,913,911, 3,525,890, 
and 3,094,880. In these patents, the rolling elements are normally used in 
connection with the means initiating motion on the part of an intermediate 
member, which is provided with the usual teeth. 
SUMMARY OF THE INVENTION 
The present invention represents a radical departure from known prior art 
nutating gear mechanisms employing teeth as the means of torque 
transmission. Input rotation is converted into the nutational movement of 
an intermediate member, which is coupled to both a stator and an output 
member by the intermediary of separate series of tapered roller bearings. 
Grooves or raceway surfaces are formed on the respective elements, and the 
roller elements constantly engage both surfaces as they rollingly transmit 
torque. Within a pair of coacting elements, one of the raceways will be 
formed with epitrochoidal curvature, while the other is formed with 
hypotrochoidal curvature. The surfaces of these grooves undulate, and can 
be thought of as resembling "lobes". The member having the hypotrochoidal 
groove cut therein is normally thought of as the "outer" member, as this 
element will have two more teeth or lobes than will the conjugate 
epitrochoidal groove. By "conjugate" is meant that the curvature of the 
epitrochoidal grooves and the hypotrochoidal groove are related in such a 
way that the rollers will be in substantially continuous contact with the 
surfaces of both grooves. One method of producing conjugate epitrochoidal 
and hypotrochoidal surfaces is taught in copending patent application Ser. 
No. 313,442, by the present inventors, the disclosure of which is hereby 
incorporated by reference. 
If desired, any number of gear reduction stages may be obtained 
consistently with the invention by merely adding the additional requisite 
elements.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
Referring now to the drawing figures, and in particular to FIG. 1, a two 
stage reduction nutating gear mechanism is illustrated, wherein within a 
cylindrical housing generally designated at 10 are coupled a spaced pair 
of gearbox end pieces 14, 16. Within the enclosure formed by these 
elements are housed portions of an input shaft 20, a stator 30 suitably 
keyed or otherwise attached to the end piece 14, an intermediate nutating 
or wobbling idler member 40, and an output member 50 coupled to an output 
shaft 60. 
The input shaft 20 is journaled for rotation within the stator member 30 
and the output member 50 by means of a pair of suitable ball or rollers 
bearings 52, 54. Between the two bearings 52, 54, the input shaft 20 is 
equipped with a pair of opposed counterweight members 56, 58, and an 
canted cam member 70. The counterweights 56, 58 serve to compensate for 
dynamic imbalances within the rotating system in a manner to be disclosed 
in more detail hereinfter, while the cam member 70 serves to initiate 
wobbling or nutating movement on the part of the intermediate idler 40. As 
seen in FIG. 1, the idler 40 is journaled for rotation with respect to the 
canted cam 70 by means of roller bearing 72. In this manner, the 
intermediate idler member 40 will follow the wobbling motion directed by 
the cam 70, while being able to rotate with respect thereto. 
As can be seen in FIG. 1, the intermediate idler member 40 is provided on 
either side thereof with undulating or lobed surfaces 42, 44. These two 
surfaces are similar in that they are both formed with the same 
(hypotrochoidal) curvature, but are different in that the number of 
umdulations or lobes on one side, in the present instance the side 42, is 
larger by at least 1. 
Opposing the intermediate idler gear surfaces 42, 44 are, on the one hand, 
an epitrochoidal surface 55 formed on the output member and an 
epitrochoidal surface 35 formed on the stator, respectively. In both 
instances, the hypo- and epitrochoidal grooves coact through the 
intermediary of a series of tapered roller elements 80,82, respectively. 
It should be noted here that the designation of the surfaces 35,55 as 
epitrochoidal and surfaces 42,44 as hypotrochoidal is arbitrary. The 
reverse configuration would likewise result in a workable device, it being 
important only that the surfaces entraining rollers be in matched "sets" 
of epi- and hypotrochoidal form. The hypotrochoidal gear is provided with 
the greater number of lobes. 
Accordingly, the epitrochoidal surface 35 on the stator is formed such that 
there are two less undulations or lobes on this surface than on the 
coacting hypotrochoidal surface 44. In this manner, a configuration can be 
achieved wherein the rollers 82 are all simultaneously in substantially 
continuous contact with both the epitrochoidal surface 35 and the 
hypotrochoidal surface 44. Where appropriate, epi- and hypotrochoidal 
surfaces which are related in such a manner will be termed "conjugate" 
curves or surfaces. One manner of generating such surfaces, albeit in a 
substantially planar fashion rather than in three dimensions, is disclosed 
in the aforementioned patent application Ser. No. 313,442, by the present 
inventors. 
Like the hypotrochoidal surfaces 42, 44, the epitrochoidal surfaces 35, 55 
are similar, but nominally, they will be provided with different numbers 
of lobes thereon. As is well known in this type of gearing, the speed 
reduction obtainable is often dependent upon the relative numbers of 
"teeth" on the coacting gears, and the present construction is no 
exception to this general rule. Specific formulas for determining the 
speed ratios obtainable will be given following the present discussion. 
The trochoidally cut surfaces 42 and 55 of the idler member and the output 
member, respectively, are also formed as conjugate surfaces, there being 
two less lobes on the surface 55, with the intermediate rollers 80 being 
in substantially continuous rolling contact with both of the surfaces. In 
this manner, the rollers are made to rollingly transmit torque from the 
nutating and rotating idler to the output member 50. 
The two series of rollers 80, 82 are identical to one another, and are 
maintained spaced at even intervals by means of cage or carrier elements 
90. These elements are provided generally in the form of spherical shell 
segments having radially inwardly directed tabs or protrusions 92. The 
space between adjacent tabs is sized and shaped so as to accommodate a 
single tapered roller member. The cages 90 will generally follow the 
wobbling motion of the intermediate member 40, due to the progressive 
engagement therebetween as the idler member nutates. For example, in FIG. 
1, the upper peripheral portion of the idler 40 is in contact with the 
left-hand carrier 90, while the lower peripheral portion of the idler is 
in contact with the righthand carrier. In other words, the carriers are 
"spherically trapped" between the coacting gear elements. 
Of course, the two carriers 90, although being substantially identical, may 
be of slightly different cone angles since they nominally will "house" 
differing numbers of rollers, this difference being at least one. Of 
course, the number of rollers operating between any two respective 
elements is dictated by the number of epitrochoidal and hypotrochoidal 
lobes formed on the coacting surfaces, as mentioned previously, the number 
of rollers being the number intermediate the number of epitrochoidal and 
hypotrochoidal lobes, respectively. 
In use, the device illustrated in FIG. 1 operates as follows: input 
rotation via the input shaft 20 is translated into wobbling or nutating 
motion on the part of the intermediate member 40, due to the presence of 
the canted cam member 70. This nutating movement will be at the same speed 
or frequency as the input rotary speed, the idler being free to rotate 
independently of the wobbling motion. 
Due to the engagement between the stator 30 and the intermediate member 40 
via trochoidal surfaces 35, 44 and rollers 82, the nutating member will 
also be made to rotate, at a speed determined by the relative numbers of 
lobes provided on the surfaces 35, 44. In particular, the nutating member 
will be made to rotate at a speed which may be determined by the following 
formula: 
##EQU1## 
Where: n.sub.1 =number of lobes on the stator 30, and 
n.sub.2 =number of lobes on the idler surface 44. 
As previously noted, the number of lobes on the stator surface 35 is less 
than the number of hypotrochoidal lobes on the idler surface 44 by 2. The 
number of rolling torque transmitting elements 82 will thus be equal to 
the number between n.sub.1 and n.sub.2. For example, if the epitrochoidal 
stator surface 35 is provided with 10 lobes while the hypotrochodial 
surface 44 is provided with n.sub.1 +2 or 12 such lobes, there will be a 
total of 11 tapered rollers 82 operating between these two members, and 
the rotary speed reduction between the input shaft 20 and the idler 40 can 
be calculated to be equal to 6:1. 
It should be noted that the cage or assembly housing the rollers will 
rotate at a speed determined by the following equation for the case of 1 
speed reduction: 
##EQU2## 
where N.sub.R represents the number of rollers in the cage, this speed 
being about one half of the rotational speed of the idler. 
For two speed reduction, the rotational speed of the cage for the second 
set of rollers is given by: 
##EQU3## 
where N.sub.R2 is the number of rollers in the second cage. 
As can be observed from equation (1), the speed ratio is positive when the 
idler member has the greater number of lobes, as in the present case. 
However, negative speed ratios could be obtained if the epi- and 
hypotrochoidal surfaces 35, 44 were reversed such that the hypotrochoidal 
surface (with the larger number of lobes) would lie on the stator, and 
vice versa. 
As noted previously, due to the conjugate configuration of the 
epitrochoidal surface 35 and the hypotrochoidal surface 44, the rollers 82 
are maintained in substantially continuous surface contact with both 
surfaces as they rollingly transmit torque between the respective 
elements. 
Torque is transmitted from the intermediate member 40 to the output member 
50 in a similar manner, with the rollers 80 rollingly transmitting torque 
therebetween. Owing to the conjugate epi- and hypotrochoidal surfaces of 
the output member and the intermediate member, a further speed reduction 
is obtained between these two elements, such that the overall transmission 
may be termed a double reduction gear. The speed ratio of the overall 
transmission may be easily calculated via the following formula: 
##EQU4## 
Where: n.sub.1 =number of stator lobes, 
n.sub.2 =number of lobes on the idler surface 44, 
n.sub.3 =number of lobes on the second idler surface 42, and 
n.sub.4 =number of lobes formed on output race 55. 
For example, if the number of lobes on the epitrochoidal and hypotrochoidal 
surfaces of the stator and intermediate member surfaces 35, 44 are 10 and 
12 as before, and the number of lobes on intermediate surface 42 and 
output surface 55 are 11 and 9, respectively, the reduction ratio of the 
overall FIG. 1 device can be calculated to be -54:1, where the minus sign 
indicates that the output rotation is in the direction opposite that of 
the input. This reduction ratio may be altered quite easily within the 
confines of the invention by merely changing the relative numbers of lobes 
on the several members, keeping in mind the constraint that, as between 
conjugate epi- and hypotrochoidal pairs, the difference in numbers of 
lobes must be two. 
From equation (2) given above, it is evident that it is possible for the 
speed ratio to approach infinity when the product of n.sub.1 and n.sub.3 
approaches that of n.sub.2 and n.sub.4. The effect of such a condition 
would be to cause the intermediate idler 40 to rotate in the same 
direction as or opposite to the input at a rate of speed such that the 
output member would be driven neither forwardly nor reversely. In this 
regard, it is noted that the speed and direction of rotation of the idler 
member 40 is controlled by n.sub.1 and n.sub.2. From equation (1), it is 
evident that the idler member 40 will rotate in the same direction as the 
input shaft 20 if 
EQU n.sub.1 /n.sub.2 &lt;1, 
and the idler member 40 will rotate in the opposite direction if 
EQU n.sub.1 /n.sub.2 &gt;1. 
In addition, it should also be noted that if the idler member is prevented 
from rotating and only allowed to nutate, the output member would be 
caused to rotate reversely to the input direction of the input shaft 20, 
at a reduction rate of -4.5:1, using the numbers of lobes given in the 
example above. By causing the idler member to rotate forwardly by means of 
the engagement thereof with the stator, the output member is made to 
rotate reversely at slower speeds (higher reduction ratios), or stop 
completely when the above equation becomes indefinite. 
As noted previously, a pair of counterweight members 56, 58 are arranged on 
either side of the idler and the cam member 70. Although the device as 
illustrated in FIG. 1 is staticly balanced, dynamic imbalances arise due 
to the fact that the idler 40 is always "tilted" with respect to the 
gearing axis, where this "tilt" progresses circumferentially as the member 
nutates. At high speeds, the mentioned nutational motion will tend to rock 
the gear box back and forth. 
In order to compensate for and prevent dynamic imbalance, the left-hand 
counterweight 56 can be thought of as compensating for the leftward "tilt" 
of the upper portion of the idler 40, while the right-hand counter weight 
member 58 compensates for the tilt of the lower portion of the idler 
member. It will be noted that the idler 40 necessarily nutates at the same 
speed as the input rotation, and thus the counterweights remain at the 
same position relative to the tilted idler regardless of the speed of 
rotation. 
In FIG. 3 is illustrated a modification of the invention, which is 
basically identical to the device illustrated in FIG. 1, but is 
characterized by a simplified design. In this embodiment, the housing 300, 
which may be made in several parts, may incorporate the stator member 310 
and the end pieces which were employed in FIG. 1. As is evident from FIG. 
3, the housing is formed directly with bearing surfaces, such that the 
several internal parts may slide upon the internal surfaces of the housing 
itself. In particular, the output shaft 320 bears against an inner bearing 
surface provided in the left-hand side of the housing, while the input 
shaft 330 bears against that part of the housing now incorporating the 
stator. Also, roller bearings are eliminated between the idler 350 and the 
cam portion 340 of the input shaft. 
Although the embodiment of FIG. 3 obviously entails more friction than the 
FIG. 1 embodiment, the FIG. 3 device has a number of advantages in that it 
is quite inexpensively produced, and may be made of molded materials such 
as hard plastics. The two series of rollers used in this embodiment are, 
however, made of usual materials, such as steel. As was the case with the 
embodiment of FIG. 1, a number of inexpensively produced interchangeable 
parts can be provided for this gear box, such that the number of gear 
ratios obtainable is quite wide. Changes in the reduction ratio may most 
easily be made by replacing the idler 350 and output member 320 with 
similar members having different numbers of coacting lobes. 
Turning now to FIG. 4, there is illustrated a further embodiment of the 
device wherein a three stage gear reduction is achieved in a manner 
analogous to the double reduction gearing mechanism of FIG. 1. As is seen 
in FIG. 4, a pair of nutating idler members 410, 420 are provided, rather 
than the single idler of the prior embodiments. 
In FIG. 4, a housing 425 bearingly supports an output shaft 500 for 
rotation, while an input shaft 400 extends within the housing through an 
aperture formed in a threaded end member 435 and a stator 430. Between the 
stator 430 and the first idler member 420 there is arranged a first series 
of caged tapered rollers 460, which cooperate with the stator and the 
idler member similarly as in previous embodiments. Specifically, the idler 
member 420 and the stator 430 are provided with conjugate pairs of 
epitrochoidal and hypotrochoidal surfaces 432, 422, respectively, such 
that a first rotary speed reduction is obtained between the input shaft 
400 and the first idler member 420. 
Operating between the idler members 410, 420 there is arranged a second 
series of rollers 470, which, although resembling the series of rollers 
460, contain rollers of about twice the size. Accordingly, there are only 
about one half as many of the large rollers 470 as there are rollers 460, 
for example. The idler 410 is provided with a surface 412 of trochoidal 
curvature on the side thereof facing the idler 420, while this latter 
mentioned member is provided with a conjugate trochoidal surface 424. The 
number of lobes on the idler surface 412 is either greater or less than 
the number of lobes on the idler surface 424 by two, with the surface 
having the greater number of lobes being the hypotrochoidal surface having 
the lesser number of lobes being the epitrochoidal surface. A second 
rotary speed reduction is obtained between the idler 420 and the idler 410 
due to the roller engagement between the conjugate trochoidal surfaces 
412, 424, although it should be noted that these members mutate at the 
same speed. As with prior embodiments, it should be understood that the 
assignment of the epitrochoidal lobes to the idler 410 and the 
hypotrochoidal lobes to the idler 420 can be appropriately selected 
depending on the reduction ratio desired. 
In FIG. 4, a cross-section of the device is shown, such that at the bottom, 
one roller 470 is seen fully because at this position this roller abuts 
the crests of the lobes on both surfaces 412, 424. A cage member 434 
maintains the rollers 470 in spaced relationship in a manner similar to 
the carriers of the previously described embodiments, and at the top of 
FIG. 4, this member 434 alone is seen. At this point in the circular 
rolling movement of the rollers 470 as shown at the top of FIG. 4, the 
roller will engage the "troughs" of either surface 412, 424, while these 
surfaces actually abut either side of the carrier member 434. This 
configuration of carrier may be used in the FIG. 1 embodiment as well, to 
replace the "spherically trapped" carriers thereof. The carriers of the 
present embodiment may aptly be termed as "pinched" carriers, since they 
will be progressively circumferentially engaged by portions of both the 
idlers 410, 420. Rollers may also be cased by pins through hollow rollers. 
It should be noted that the idlers 410, 420 are arranged in mirror-image 
fashion, such that the nutating motion of one idler mirrors the movement 
of the either. However, although the idlers do not nutate with respect to 
one another, they are capable of differential rates of rotation. By way of 
analogy, the motion of the two nutating idlers may be compared to that of 
a coin spinning on a mirrored surface, where the coin may rotate with 
respect to its mirror image. 
An advantage of this configuration lies in the fact that no counterweights 
are required, as the device is dynamically counterbalanced. Basically, the 
two idlers 410, 420 balance each other's movement as they nutate 
oppositely, or in mirror image fashion, with respect to each other. 
Between the idler 410 and the output member 500 are arranged the third 
series of rollers 480, which are maintained separated by a pinched carrier 
404, as are the rollers 460. The idler 410 is provided with a surface 492 
of hypotrochoidal curvature, while a conjugate epitrochoidal surface 502 
is formed on the output member. Thus, a third rotary speed reduction is 
obtained between the idler 410 and the output 500. The overall speed 
reduction of the device may be easily calculated from the following 
formula, which, as can be seen, is merely an extension of equations (1) 
and (2) presented previously. 
##EQU5## 
Where: n.sub.1 =number of stator lobes, 
n.sub.2 =number of lobes on first idler surface 422, 
n.sub.3 =number of lobes on first idler surface 424, 
n.sub.4 =number of lobes on second idler trochoidal surface 412, 
n.sub.5 =number of lobes on trochoidal idler surface 492, and 
n.sub.6 =number of lobes formed on output trochoidal surface 502. 
As can readily be verified by plugging-in sample values for the several 
lobe numbers, the reduction ratio obtainable with the device of FIG. 4 can 
easily reach several thousand to one. Differing ratios may be easily 
obtained, as was the case in earlier embodiments by merely placing the 
operative gear components with like components having different numbers of 
lobes. Also, within the confines of the present invention, it is possible 
to obtain any number of stages of speed reduction by suitably adding 
additional coacting element pairs having conjugate trochoidally formed 
surfaces as described hereinabove. 
FIGS. 5-8 are two-dimensional and three-dimensional illustrations of 
geometric models which can be utilized for generating the undulating or 
lobed gear surfaces used in a nutating device constructed according to the 
present invention. Specifically, these models can be used to generate both 
epi- and hypocycloidal surfaces, as well as epi- and hypotrochoidal 
surfaces. 
Referring initially to FIGS. 5 and 7 which illustrate a two- and 
three-dimensional model, respectively, for generating epicycloidal and 
epitrochoidal surfaces, a sphere 21 has a spherical surface S on which a 
fixed inner cone 22 is disposed, the cone 22 having a fixed Z axis which 
passes through an apex O of the inner cone 22. A lower circumferential 
line 27 of the inner cone 22 extends over an arc 2.gamma..sub.i on the 
spherical surface S. An outer, movable cone 24 is disposed on the surface 
S around the inner cone 22, and the outer, movable cone 24 has a movable 
Z' axis which passes through the apex O, which is also the apex of the 
outer cone 24. The movable Z' axis is offset from the fixed Z axis by a 
predetermined angle .epsilon.. A lower circumferential line 28 of the 
outer, movable cone extends over an arc 2.gamma..sub.O on the spherical 
surface S. The outer, movable cone 24 is capable of nutating about the 
fixed inner cone 22 so that the movable Z' axis of the movable cone 24 
revolves about the fixed Z axis of the inner cone 22 along the circle 
indicated generally by the symbol .theta. in FIG. 5. As the outer cone 24 
nutates, varying portions of its lower circumferential line 28 are 
maintained in contact with varying portions of the lower circumferential 
line 27 of the inner cone. In FIG. 7, the outer cone has been nutated 
about the Z axis along the circle .theta. by approximately 
100.degree.-120.degree. from the position shown in FIG. 5. The outer, 
movable cone 24 has a leg portion AB which lies on the spherical surface 
S, and the outermost point A of the leg portion AB is displaced from a 
point B on the lower circumferential line 28 of the outer cone 24 by angle 
.gamma..sub.AB. 
Referring to FIG. 7, the outer, movable cone 24 nutates so that its Z' axis 
revolves a little more than once around the Z axis of the fixed, inner 
cone 22, the point B on the lower circumferential line 28 of the outer 
cone 24 moves along a path portion BB'B" of an epicycloidal line 29. The 
epicycloidal line 29 is located on the spherical surface S. As the outer, 
movable cone and its Z' axis continue to undergo numerous nutations and 
revolutions, respectively, the point B will slowly trace out a complete, 
connected epicycloidal line 29. As FIG. 7 readily illustrates, the 
epicycloidal line 29 has a plurality of interconnected lobes comprising a 
plurality of "loops" which are connected together at a plurality of 
"nodes" 41. In addition, as the outer cone 24 undergoes the 
above-described nutations, the outermost point A of the leg portion AB of 
the outer cone will move along a path portion of an epitrochoidal line 31, 
only a portion of which is illustrated in FIG. 7. The epitrochoidal line 
31, which is traced out by the point A, has the same number of lobes as 
the epicycloidal line 29, which is traced out by the point B. However, the 
epitrochoidal line 31 is smoother than the epicycloidal line 29, and the 
"nodes" interconnecting the "loops" of the epitrochoidal line lie on 
smooth, curved line portions rather than on sharp points, such as the 
points 41 on the epicycloidal line 29. Curved line portions are more 
advantageous than sharp points because the rollers undergo smaller 
accelerations and decelerations and, hence, smaller velocity changes when 
traveling in races having smooth surfaces rather than races having sharp 
points. Therefore, the use of trochoidal surfaces results in a smoother 
running gear than does the use of cycloidal surfaces. 
Once a first epicycloidal line 29 or a first epitrochoidal line 31 is 
formed, it is possible to form a corresponding epicycloidal or 
epitrochoidal surface from these lines. One method for doing this would be 
to form a second epicycloidal or epitrochoidal line which utilizes a 
larger or smaller sphere and additional inner and outer cones which have 
lower circumferential lines extending over identically sized arcs, as used 
in the inner and outer cones which are used for generating the first 
epicycloidal line or epitrochoidal line. The epicycloidal or epitrochoidal 
surface is then formed by connecting the second epicycloidal or second 
epitrochoidal line with the first epicycloidal or epitrochoidal line, 
respectively. 
The same epicycloidal or epitrochoidal surface could also be traced out by 
fixing a tapered roller bearing 80 or 80', shown in FIGS. 2A and 2B, to 
the points A or B of the outer cone 24, so that the longitudinal axis and 
the point of convergence of the tapered roller bearing 80 or 80' passes 
through the common cone apex O and, thereafter, nutating the outer cone 
until the roller undergoes one complete revolution about the Z axis of the 
inner cone. The barrel shaped roller 80' of FIG. 2B has advantages in that 
rollers having such barrel shapes can be more easily trapped in the races 
or grooves of the nutating gear than can tapered cylindrical rollers. In 
practice, the above-described epicycloidal and epitrochoidal surfaces can 
be generated using machines which function in the manner described for the 
cones shown in FIGS. 5 and 7, and by attaching a tapered mill, having a 
desired roller shape only two of which are shown in FIGS. 2A and 2B, to 
the points A or B so that the desired surface configuration is formed. 
FIGS. 6 and 8 illustrate two- and three-dimensional models, respectively, 
for generating hypocycloidal and hypotrochoidal surfaces. In these 
figures, an outer cone 34, which has a fixed Z axis, is fixed to a sphere 
21', which has a spherical surface S'. A lower circumferential line 38 of 
the fixed outer cone 34 extends over an arc 2.gamma..sub.O ' on the 
spherical surface S', and a lower circumferential line 37 of an inner, 
movable cone 32 extends over an arc 2.gamma..sub.i ' on the spherical 
surface S'. The inner, movable cone 32 is disposed within the outer fixed 
cone 34, and the inner, movable cone has a movable Z' axis which passes 
through the common apex OO of the inner and outer cones 32, 34. The 
movable Z' axis is offset from the fixed Z axis by an angle .epsilon.' 
shown in FIGS. 6 and 8. The sum of the angles .epsilon.+.epsilon.' 
represents the nutating angle created by the canted cam member 70 of the 
input shaft 20 in FIG. 1. The inner, movable cone 32 is capable of 
nutating about the fixed, outer cone 34 so that the movable Z' axis of the 
movable cone 32 revolves about the fixed Z axis of the outer cone 34. As 
the inner cone 32 nutates, varying portions of its lower circumferential 
line 37 are kept in continuous contact with varying portions of the lower 
circumferential line 38 of the outer cone 34. The inner, movable cone 32 
has a leg portion CD which lies on the spherical surface S', and the 
outermost point C of the leg portion CD is displaced from the point D, 
which is located on the lower circumferential line 37 of the inner cone, 
by an angle .gamma..sub.CD. 
The generation of hypocycloidal and hypotrochoidal surfaces which utilize 
the cones shown in FIGS. 6 and 8 is similar to that described in the 
discussion related to FIGS. 5 and 7 for generating the epicycloidal and 
epitrochoidal surfaces, except that, in FIGS. 6 and 8, it is the inner 
cone which nutates rather than the outer cone, as is the case in FIGS. 5 
and 7. As the inner cone 32 nutates, the point D traces out a 
hypocycloidal line, while the point A traces out a hypotrochoidal line. 
Hypocycloidal and hypotrochoidal surfaces can readily be generated from 
these lines, as described in the discussion relating to FIGS. 5 and 7. 
The actual parametric, mathematical formula for determining the optimum 
dimensions of the epicycloidal, epitrochoidal, hypocycloidal and 
hypotrochoidal surfaces are quite complex because three-dimensional 
surfaces are being generated. However, one criterion which must be 
satisfied is that the number of lobes which are generated by the points A 
or B as these points revolve around the fixed cone axis Z must be an exact 
integer so that the point A or B returns to its exact original position 
after undergoing one complete revolution around the circumferential line 
of the fixed cone. Regarding the generation of the epicycloidal or 
epitrochoidal surface, it can readily be shown that, to satisfy the 
condition that the number of lobes formed be an exact integer, the 
following relationship must be satisfied: 
##EQU6## 
where (n-1) represents the number of lobes of the epicycloidal or 
epitrochoidal surface. 
Regarding the hypocycloidal or hypotrochoidal surface, to satisfy the 
above-mentioned condition, the following relationship must be satisfied: 
##EQU7## 
where (n-1) represents the number of lobes of the hypocycloidal or 
hypotrochoidal surface. 
It should also be noted that the sum .epsilon.+.epsilon.' should equal the 
nutating angle of the shaft 20 of the assembled gear shown in FIG. 1. As 
described in copending Application Ser. Nos. 313,442and 362,195, filed on 
Oct. 20, 1981 and Mar. 26, 1982, respectively, the disclosure of which is 
incorporated herein by reference, the balls or roller which are disposed 
between the conjugate pairs of epi- and hypocycloidal surfaces have a 
maximum velocity as they travel through the "loops" of the epi- and 
hypocycloids and have a minimum velocity which is, in fact, 0, when they 
pass through the "nodes" adjoining adjacent "loops". However, the 
velocities and accelerations of the balls or rollers is less extreme in 
the case where conjugate pairs of epi- and hypotrochoidal surfaces are 
utilized. In addition, it is possible to form conjugate pairs of epi- and 
hypotrochoidal surfaces which aid the acceleration of the balls or rollers 
by varying .epsilon.+.epsilon.'. Accordingly, in practice, the effect of 
the acceleration of the rollers should be considered in determining the 
optimum value of the various angles shown in FIGS. 5-8. However, the 
actual size of the lobes should not be so large that it allows the nearest 
surfaces of the conjugate pairs of surfaces to contact each other when the 
assembled gear is operated. An additional criterion which must be 
satisfied is that the generating point A of FIG. 5 and the generating 
point C of FIG. 6 should be of identical radii in order to form conjugate 
pairs of races. In this case, the following criteria must be satisfied: 
EQU .gamma..sub.O +.gamma..sub.AB =.gamma..sub.i +.gamma..sub.CD. 
Computers can be used to analyze the above-mentioned considerations to 
generate numerical and discrete solutions for determining optimum values 
of the angles shown in FIGS. 5-8. 
While the foregoing embodiments are presently preferred, it is understood 
that numerous additional modifications may be made by those of skill in 
the art, and it is intended to cover in the appended claims all such 
modifications as fall within the true spirit and scope of the invention.