United States Patent |
6,382,038 |
Angeles , et al.
|
May 7, 2002 |
Transmission device
Abstract
A transmission mechanism for transmitting motion with a uniform speed
transmission factor between first and second moveable elements comprises a set
of cams adapted to rotate with the first moveable element about a revolving
axis, and corresponding arrays of spaced-apart rollers connected to the second
moveable element for movement therewith. The cams cooperate in relays with the
corresponding arrays of spaced-apart rollers to continuously communicate motion
between the first and second moveable elements. The cams are in rolling contact
with the corresponding arrays of rollers, whereby torque and force transmission
can be performed smoothly. Furthermore, the transmission mechanism allows for
the reversal of both the direction of the input speed and the roles of the first
and second moveable elements.
Inventors: |
Angeles; Jorge (Singapore, SG);
Gonzalez-Palacios; Max Antonio (Westmount, CA) |
Assignee: |
McGill University (Montreal, CA) |
Appl. No.: |
816129 |
Filed: |
March 26, 2001 |
Current U.S. Class: |
74/84R; 74/89; 74/567;
74/569 |
Intern'l Class: |
F16H 025/04; F16H 025/18 |
Field of Search: |
74/84 R,89,567,569
|
References Cited [Referenced
By]
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2986949 |
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Lancaster et al. |
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3170333 |
Feb., 1965 |
Umbnricht. |
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3525268 |
Aug., 1970 |
Kenny. |
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4576099 |
Mar., 1986 |
Makino et al. |
104/287. |
5116291 |
May., 1992 |
Toyosumi et al. |
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5123883 |
Jun., 1992 |
Fukaya. |
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5167590 |
Dec., 1992 |
Kratochvil et al. |
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5176036 |
Jan., 1993 |
Harris. |
|
5222922 |
Jun., 1993 |
Takahashi et al. |
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5247847 |
Sep., 1993 |
Gu. |
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5286237 |
Feb., 1994 |
Minegishi. |
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5431605 |
Jul., 1995 |
Ko. |
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5470283 |
Nov., 1995 |
Seidou. |
|
Foreign Patent Documents |
135239 |
Oct., 1902 |
DE. |
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346944 |
Dec., 1904 |
FR. |
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370492 |
Dec., 1906 |
FR. |
|
2102532 |
Feb., 1983 |
GB. |
|
Other References
Roth, K.: "Evolventenverzahnungen mit extremen
Eigenschaften" Antriebstechnik, vol. 35, No. 7, Jul. 1, 1996, pp. 43-48.
|
Primary Examiner: Herrmann; Allan D.
Attorney, Agent or Firm: Renault; Swabey Ogilvy, Mitchell; Robert
Parent Case Text
RELATED APPLICATIONS
This is a continuation in part of U.S.
patent application Ser. No. 09/518,241 filed on Mar. 3, 2000, now abandoned,
which is a continuation of International PCT Application No. PCT/CA98/00831
filed on Sep. 1, 1998, which claims the benefit of U.S. Provisional Application
No. 60/057,490 filed Sep. 4, 1997.
Claims
We claim:
1. A transmission mechanism for producing uniform
speed transmission between first and second moveable elements, comprising a set
of conjugate cams adapted to rotate with said first moveable element about a
first axis, and corresponding arrays of spaced-apart rollers connected to said
second moveable element for movement therewith, said set of conjugate cams being
adapted to alternately cooperate with said spaced-apart rollers of said
corresponding arrays of spaced-apart rollers to communicate continuous motion to
one of said first and second moveable elements in response to a driving action
of the other of said first and second moveable elements, wherein each said cam
has a fully convex cam profile.
2. A transmission mechanism as defined
in claim 1, wherein said cams are shifted in phase by a prescribed angle, and
wherein said corresponding arrays of spaced-apart rollers are connected to said
second moveable element with a prescribed phase difference via a carrier, each
said corresponding array of spaced-apart rollers including a plurality of
uniformly distributed rollers which are configured to be engaged by
corresponding cams, said carrier including a single support element to which all
said corresponding arrays of spaced-apart rollers are mounted.
3. A
transmission mechanism as defined in claim 2, wherein said cams have respective
fully convex contoured cam surfaces configured to be in rolling-contact with
said rollers of said corresponding arrays of spaced-apart rollers to achieve a
prescribed speed transmission factor 1/N between said first and second moveable
elements, N being an integer.
4. A transmission mechanism as defined in
claim 1, wherein said first and second moveable elements, respectively, include
first and second shafts having parallel axes, said cams being disposed at
axially spaced-apart locations on said first shaft for rotation therewith about
said first axis, each of said cams being in a common plane with one of said
corresponding arrays of spaced-apart rollers, said rollers of each said
corresponding arrays of spaced-apart rollers having respective axes lying on a
circular cylinder coaxial with said second shaft, said circular cylinder having
a radius a.sub.3, each said roller having a cylindrical configuration and being
axially disposed relative to said second shaft, and wherein each said cam has a
fully convex contoured cam surface generated by a vector r.sub.c which is
defined as follows: ##EQU31##
wherein:
.psi.: is an angle of
rotation of the first shaft;
.phi.: is an angle of rotation of the
second shaft;
a.sub.1 : is the distance between the first and second
shafts;
a.sub.4 : is the radius of the rollers;
k.sub.i : are
temporary variables, where i=1,2,3 . . .
.lambda.: is a real number
defining one specific point along the contoured cam surface;
N: an
integer producing the speed reduction as 1/N, with N being equal to the number
of rollers on each side of the second element;
and with r=a.sub.3
/a.sub.1 satisfying the following condition ##EQU32##
when said first
axis is located outside the circular cylinder, and the following condition
##EQU33##
when said first axis is located inside the circular cylinder.
5. A transmission mechanism as defined in claim 4, wherein said first
and second shafts are mounted to distinct moveable parts of a frame structure
for providing a preloading of said rollers in contact with said contoured cam
surfaces.
6. A transmission mechanism as defined in claim 4, wherein
said arrays of spaced-apart rollers are assembled to a unitary carrier member
fixedly mounted on said second shaft.
7. A transmission mechanism as
defined in claim 4, wherein said set of cams includes two cams which are out of
phase by an angle of 180 degrees and which operate in relays with the rollers of
two corresponding arrays of spaced-apart rollers to provide a continues torque
transmission between said first and second shafts, said arrays of spaced-apart
rollers being shifted in phase by an angle which is equal to the quotient of 360
degrees by the number of rollers.
8. A transmission mechanism as defined
in claim 1, wherein said first and second moveable elements respectively include
first and second shafts having orthogonal axes, said cams being disposed at
axially spaced-apart locations on said first shaft for engaging corresponding
arrays of spaced-apart rollers, said rollers of each said arrays of spaced-apart
rollers having a frusto-conical shape and a rotating axis which is angled with
respect to said second shaft, said rotating axes forming a circular conical
surface, and wherein each said cams has a fully convex contoured cam surface
generated by a vector r.sub.c which is defined as follows: ##EQU34##
wherein:
.psi.: is an angle of rotation of the first shaft;
.phi.: is an angle of rotation of the second shaft;
.alpha..sub.1 : is the angle between the axes of the first and second
shafts, said angle being valued between 0.degree. and 180.degree.;
.alpha..sub.3 : is the angle between the axis of the second shaft and
the axis of rotation of the rollers;
.alpha..sub.4 : is an angle of the
roller cone;
k.sub.i : are temporary variables, where i=1,2,3 . . .
.lambda.: is a real number defining one specific point along the
contoured cam surface;
N: an integer producing the speed reduction as
1/N, with N being equal to the number of rollers on each side of the second
element;
and in which .alpha..sub.3 satisfies the following condition:
.alpha..sub.3 =arctan (1/N).
9. A transmission mechanism as
defined in claim 8, wherein said set of cams includes first and second cams
disposed at axially spaced-apart locations on said first shaft for respectively
engaging first and second corresponding arrays of spaced-apart rollers, said
rollers of said first corresponding array of spaced-apart rollers being
uniformly distributed along an outer surface of a single ring member
concentrically mounted to said second shaft, while said rollers of said second
corresponding array of spaced-apart rollers being uniformly distributed along an
inner surface of said ring member, said first and second corresponding arrays of
spaced-apart rollers being shifted in phase by an angle which is equal to the
quotient of 360 degrees by the number of rollers.
10. A transmission
mechanism as defined in claim 9, wherein said ring member geometrically
corresponds to a segment of a sphere.
11. A transmission mechanism as
defined in claim 9, wherein said ring member extends from a periphery of a disc
member secured to said second shaft.
12. A transmission mechanism for
transmitting motion between a rotating shaft and a linearly translating member
having nonparallel axes, comprising a set of cams disposed at axially
spaced-apart locations on said rotating shaft for rotation therewith about said
axis of said rotating shaft, and corresponding arrays of spaced-apart rollers
connected to said linearly translating member for movement therewith, each said
cam being in a common plane with one of said corresponding arrays of
spaced-apart rollers placed in axially extending rows on said linearly
translating member, said axially extending rows being parallel to a direction of
motion of said linearly translating member, and wherein each said cams has a
fully convex contoured cam surface generated by a vector r.sub.c which is
defined as follows: ##EQU35##
wherein:
.psi.: is an angle of
rotation of the rotating shaft;
.alpha..sub.1 : is the angle between the
rotating shaft and the linearly translating member;
a.sub.1 : is a
distance between the axis of the rotating shaft and a reference line parallel to
the direction of motion of said linearly translating member;
a.sub.3 :
is a distance between the reference line and an axis passing through the center
of the rollers;
a.sub.4 : is a radius of the rollers;
k.sub.i :
are temporary variables, where i=1,2,3 . . .
.lambda.: is a real number
defining one specific point along the contoured cam surface
1/N: where N
is an integer denoting the number of rollers on each side of the linearly
translating member that enter in contact with corresponding cams upon a 360
degrees rotation of cam shaft;
and in which a.sub.3 satisfies the
following condition:
a.sub.3 <a.sub.1 /(2/N+1).
13. A
transmission mechanism as defined in claim 12, wherein said set of cams includes
first and second cams adapted to engage corresponding first and second arrays of
spaced-apart rollers laterally disposed on opposed longitudinal sides of said
linearly translating member.
14. A transmission mechanism as defined in
claim 13, wherein said first and second cams are shifted in phase by an angle of
180.degree..
Description
BACKGROUND OF THE INVENTION
1. Field of the Invention
The present invention relates to a transmission device and, more
particularly, to a transmission device having rolling contact surfaces for
transmitting motion between first and second moveable elements.
2.
Description of the Prior Art
The effective operation of any transmission
device relies on an accurate meshing of the parts in contact. In gear trains,
accurate meshing is relaxed in favor of assemblability, thereby giving rise to
backlash and its well-known drawbacks. While backlash in gear trains intended
for power transmission at a constant rpm can be tolerated, in automation,
machinery, such as robots, the speed reverses its sign continuously and, in
sensor-based systems, unpredictably. In automation machinery, therefore,
backlash hampers seriously the machine performance.
Robotic applications
have called for a revision of alternative transmissions. In this vein, it has
been proposed to replace gear trains by transmission devices based on the pure
rolling action of a cam with respect to a roller. The idea has been pursued in
the past, as attested by the following patents:
U.S. Pat. No. 3,170,333
by Emil Umbricht discloses a rotary positioning device specifically aimed at the
coupling of parallel shafts for the production of indexing motion. The device
comprises a plurality of rollers meshing with conjugate cams. The patent
discloses two conjugate cams of an asymmetric shape, with high curvature
changes.
U.S. Pat. No. 3,525,268 by Ronald H. Kenny discloses a parallel
shaft cam drive intended for the production of either indexing motion or speed
reduction. As its name states, this device pertains only to the coupling of
parallel shafts. It comprises also a plurality of rollers meshing with two
conjugate cams which, as in the Umbricht patent, bear asymmetric shapes with
high curvature changes.
U.S. Pat. No. 5,176,036 by William O. Harris
discloses a parallel shaft drive and indexing machine which, as its name
indicates, is limited to both parallel shafts and indexing motion. The
morphology of the device is similar to those of the Umbricht and Kenny patents.
U.S. Pat. No. 5,247,847 by Inhoy Gu discloses a cam gear assembly that
is intended as a speed-reducing device. The device consists of a
periodically-convex plate, with a shape proper of gears, meshing with a
plurality of rollers. The patent is mostly intended for the coupling of parallel
shafts, but it also contemplates the coupling of intersecting shafts.
The patents recalled above comprise invariably a cam element with a
shape exhibiting pronounced changes of curvature, which hampers (a) the accurate
machining of the profile and (b) the strength of the cam.
SUMMARY OF THE
INVENTION
It is therefore an aim of the present invention to provide a
new transmission device which is adapted to produce uniform speed transmission
between two moveable elements.
It is a further aim of the present
invention to provide a transmission device which is adapted to transmit power
from an input element to an output element through a transmission mechanism
having rolling-contact, thereby generating low friction resistance and low power
losses.
It is an aim of the present invention to provide such a
transmission device comprising conjugate cams with fully convex cam profiles.
It is a further aim of the present invention to provide a transmission
device which produces low backlash and which is thus particularly suitable in
specific areas that call for high accuracy and smooth operations.
It is
an aim of the present invention to provide a transmission device which operates
at low noise level.
It is a further aim of the present invention to
provide a transmission device which can be designed to couple parallel or
intersecting shafts at angles varying from 0 to 180 degrees.
It is a
still further aim of the present invention to provide a transmission device
which is adapted to link a revolving shaft to a translating rack.
Therefore, in accordance with the present invention, there is provided a
transmission mechanism for producing uniform speed transmission between first
and second moveable elements, comprising a set of conjugate cams adapted to
rotate with said first moveable element about a first axis, and corresponding
arrays of spaced-apart rollers connected to said second moveable element for
movement therewith, said set of conjugate cams being adapted to alternately
cooperate with said spaced-apart rollers of said corresponding arrays of
spaced-apart rollers to communicate continuous motion to one of said first and
second moveable elements in response to a driving action of the other of said
first and second moveable elements, wherein each said cam has a fully convex cam
profile.
BRIEF DESCRIPTION OF THE DRAWINGS
Having thus generally
described the nature of the present invention, reference will now be made to the
accompanying drawings, showing by way of illustration a preferred embodiment
thereof and in which:
FIG. 1 is a perspective view partly in
cross-section of a transmission device in accordance with the present invention;
FIG. 2 is a perspective view partly in cross-section showing a housing
of the device of FIG. 1;
FIG. 3 is a perspective view partly in
cross-section showing an arrangement of conjugate cams rigidly mounted on an
input shaft of the device of FIG. 1;
FIG. 4 is a perspective view partly
in cross-section showing two sets of rollers revolvably disposed about the
periphery of a circular carrier member which is in turn fixedly mounted to an
output shaft of the device of FIG. 1;
FIG. 5 is a front elevational view
of a cam member of the present invention;
FIG. 6 is a perspective view
partly in cross-section of a second preferred embodiment in accordance with the
present invention;
FIG. 7 is a perspective view partly in cross-section
showing a housing of the device of FIG. 6;
FIG. 8 is a perspective view
partly in cross-section showing an arrangement of conjugate cams rigidly mounted
on an input shaft of the device of FIG. 6;
FIG. 9 is a perspective view
partly in cross-section showing two sets of rollers revolvably disposed about
the periphery of a carrier member which is in turn fixedly mounted to an output
shaft of the device of FIG. 6;
FIG. 10 is a schematic perspective view
partly in cross-section of a transmission device in accordance with a third
embodiment of the present invention for transforming a rotary motion into a
linear motion or vice versa;
FIG. 11 is a schematic side elevational
view of the transmission device of FIG. 10;
FIG. 12 is an example of a
planar cam plate having a fully convex cam profile;
FIG. 13 is a graph
of the pressure angle distribution with .vertline.a.sub.3.vertline./a.sub.1 =0.8
of the third embodiment;
FIG. 14 illustrates an internal layout of the
first embodiment but with fully-convex cam profile;
FIG. 15 illustrates
an internal layout of the second embodiment but with fully-convex cam profile;
and
FIG. 16 is a perspective view partly in cross-section of a fourth
embodiment in accordance with the present invention.
DESCRIPTION OF THE
PREFERRED EMBODIMENTS
Now referring to the drawings, and in particular
to FIG. 1, a transmission device or speed reducer embodying the elements of the
present invention and generally designated by numeral 10 will be described.
The transmission device 10 generalrises a housing 12, an input
shaft 14 and an output shaft 16 rotatably mounted within the housing 12 and
connected to each other through a transmission mechanism having a constant speed
reduction ratio.
More specifically, as shown in FIG. 2, the housing 12
includes two side plates 18 and 20 which are spaced-apart by a front plate 22
and a rear spacer 24 secured thereto by means of any suitable fasteners 25 such
as screws or the like. A groove 26 following a curved path is defined on the
interior side of each side plate 18 and 20 for receiving the side edges of a
cover shell 28. The output shaft 16 is rotatably mounted to the housing 12
through a pair of bearings 30 which are respectively disposed within a bearing
aperture defined in the side plates 18 and 20.
The housing 12 also
includes two adjustable side plates 32 and 34 each capable of undergoing
submillimetric translations along a horizontal channel 36 and a vertical channel
38 which mate respectively with a horizontal protrusion 40 and a vertical
protrusion 42 extending from a front portion of each side plate 18 and 20. The
main purpose of this arrangement is to allow a preloading of the rollers in
contact with the cam, in order to ensure a force transmission without backlash.
Accordingly, the adjustable side plate 32 can be slidably engaged onto the front
portion of the side plate 18 while the adjustable side plate 34 can be slidably
engaged onto the front portion of the side plate 20. Each adjustable side plate
32 and 34 is provided with a bearing 43 for rotatably supporting the input shaft
14. As illustrated in FIG. 2, a nut 44 engages a screw 46 which extends through
the left corner of the front plate 22 and through the adjustable side plate 32
so as to slidably displace the adjustable side plate 32 relative to the front
portion of the side plate 18. In a similar manner, the adjustable side plate 34
may be displaced relative to the front portion of the side plate 20. Therefore,
translational misalignments of the input shaft 14 relative to the output shaft
16 may be compensated.
The housing 12 is also provided with a base
portion in the form of two L-shape plates 48 and 50 which are respectively
secured to the side plates 18 and 20 by way of fasteners 52.
Referring
now to FIG. 3, it can be seen that two cam plates 54 and 56 are mounted on the
input shaft 14 with a predetermined phase difference by means of a square key 58
which cooperates with a key way defined in both cam plates 54 and 56 to fixedly
secure the same onto the input shaft 14, as it is well known in the art.
Alternatively, and in order to allow a stiffer device, the two cam plates 54 and
56 and the shaft 14 thereof can be cut from a single blank, thereby eliminating
the necessity of having a key way and square key assembly which require precise
machining in order to avoid backlash.
More particularly, in the present
embodiment, the two cam plates 54 and 56 are symmetrically installed at an angle
of 180 degrees corresponding to each other, thereby forming a conjugate
arrangement of cams.
A separating bushing 60 having a predetermined
length is fitted on the input shaft 14 between the cam plates 54 and 56 to set
the relative axial position thereof. Furthermore, an aligning bushing 62 is
provided to set the axial position of the conjugate cams arrangement onto the
input shaft 14 when the latter is installed into the housing 12.
A lock
nut 64 and an external retaining ring 66 are mounted to the opposed ends of the
input shaft 14 to restrict the axial displacement of the input shaft 14 within
the housing 12.
As shown in FIG. 4, a carrier member 68 supporting a
first row of rollers 70 and a second row of rollers 72 is fixedly mounted on the
output shaft 16 through a coupling bushing 74 defining a key way (not shown) for
receiving a square key 76 extending on the outer periphery of the output shaft
16.
The carrier member 68 includes a front disc 78, a rear disc 80 and a
middle disc 82 with the relative axial position thereof being dictated by the
coupling bushing 74 and by both rows of rollers 70 and 72 which are respectively
disposed on either side of the middle disc 82 between the front disc 78 and the
rear disc 80. The front and the rear discs 78 and 80 are provided with cutout
portions 84 with the twofold purpose of reducing weight and easing the assembly
of the rollers. Circular holes 86, or holes of alternative shapes, are also
defined in the front disc 78, the rear disc 80 and the middle disc 82 for
further reducing the weight of the carrier member 68. The front and the rear
discs 78 and 80 are secured to the coupling bushing 74 by fasteners 88 and by
the dowel pin 87.
Each roller 85 of the first row of rollers 70 is
freely mounted on a roller pin 90 which extends through holes defined in the
middle disc 82 and the front disc 78. In a similar manner, each roller 85 of the
second row of rollers 72 is freely mounted on a roller pin 90 which extends
through holes defined in the middle disc 82 and the rear disc 80. Therefore, the
front disc 78, the middle disc 82 and the rear disc 80 rotate about the axis of
the output shaft 16 as a single unit.
As clearly seen from FIG. 4, the
first and the second rows of rollers 70 and 72 are shifted in phase by a
predetermined angle which is a function of the number of rollers 85. For
example, if the first and second rows of rollers 70 and 72 each includes eight
rollers 85 uniformly distributed around the periphery of the carrier member 68,
as in the preferred embodiment illustrated in FIGS. 1 to 5, the angle between
two adjacent rollers 85 of the same row is 360.degree./8=45.degree. and the
phase difference between the first row and the second row of rollers 70 and 72
is equal to 45.degree./2=22.5.degree..
A lock nut 92 and an external
retaining ring 94 are mounted on the opposed ends of the output shaft 16 to
restrict the axial displacement of the output shaft 16 within the housing 12.
When the input shaft 14 and the output shaft 16 are assembled to the
housing 12 as shown in FIG. 1, the cam plates 54 and 56 are respectively aligned
with the first and second rows of rollers 70 and 72 such that rotation of the
input shaft 14 will cause the cam plates 54 and 56 to alternately push on a
roller 85 of the corresponding row of rollers 70 and 72 to thus transmit a
torque from the input shaft 14 to the output shaft 16. In the contact condition
shown in FIG. 1, that is at a particular instant of operation, the torque
transmission is essentially effected through the cam plate 56 because the input
shaft 14 rotates in the direction indicated by arrow 96. It is noted that even
though, at that moment of operation, the cam plate 54 does not contribute to the
torque transmission, it does not interfere with any other parts, since it is in
relatively pure-rolling contact with the associated rollers 85 at all times.
It is noted that when it is desired that the rotation of the output
shaft 16 be in the same direction to that of the input shaft 14, the cam members
54 and 56 may be disposed within the circle (the follower circle) passing
through the centers of the rollers 70 and 72 (see FIG. 14) instead of outside as
in the above described embodiment. This internal layout of FIG. 14 corresponds
to an angle between shafts at 180.degree., while the external layout of FIG. 1
corresponds to an angle of 0.degree.. The internal layout has the advantage of
being compact.
As shown in FIG. 14, the carrier member 68' can
advantageously consist of a single ring member 69' having roller pins 90'
extending from opposed sides thereof to hold two series of rollers 70' and 72'.
It is the unique profile of both cam plates 54 and 56 that allows the
maintenance of relatively pure-rolling between the contact surface of the cam
plates 54 and 56 and the associated first and second row of rollers 70 and 72,
thereby providing a transmission device having superior efficiency and
durability.
Moreover, the profile of the cam plates 54 and 56 ensures
that a constant speed reduction ratio is obtained. As shown in FIG. 5, each cam
plate 54 and 56 consists of a planar body having an essentially convex contact
surface with the only concave portion thereof being at a dead point of this
particular cam-follower arrangement, i.e. at a point where no torque
transmission occurs, thereby preventing stress concentrations from affecting the
service life of the transmission device. It is to be noted that the dead points
of each cam-follower arrangement are out of phase by a maximum angle of
180.degree.; thus, when one of the cams operates at a dead point, its conjugate
(or conjugates) takes up the load, thus allowing for a continuous torque
transmission. More particularly, the profile of the cam plates 54 and 56 is
generated by a vector r.sub.c which is defined as follows: ##EQU1##
wherein:
1/N: speed reduction ratio, with N being an integer
equal to the number of rollers on each side of the second element;
.psi.: angle of rotation of the input shaft with respect to the housing;
.phi.: angle of rotation of the output shaft with respect to the
housing;
a.sub.1 : distance between output and input shafts;
a.sub.3 : distance between output shaft and roller centers;
a.sub.4 : radius of the rollers;
k.sub.i : temporary variables,
where i=1,2,3 . . .
.lambda.: real number defining one specific point
along the contact line. It varies continuously between .lambda..sub.min and
.lambda..sub.max.
The vector r.sub.c defines the position of a point of
the cam surface and thus, it may be used to generate a complete cam profile
which will enable to transmit a motion with a uniform velocity from an input
shaft to and output shaft having parallel axes. In other words, this equation
allows the construction of a cam profile necessary to obtain a desired speed
reduction ratio 1/N between two parallel shafts.
Applicants have found
that fully convex profiles are to be favored in the design of precision cams,
which are at the core of cam-based transmissions for speed reduction in
automation machinery. However, the above expressions are too cumbersome to allow
a terse analysis of the curvature of the cam profile. Accordingly, another
approach had to be found to set the conditions that must be satisfied to ensure
fully convex cam profiles.
If we denote with x and y the Cartesian
coordinates of a planar curve, its curvature is given by ##EQU2##
where
x' and x" denote the first and the second derivative, respectively, of x with
respect to the angle .psi. of rotation of the cam, which is assumed to turn
counterclockwise, y' and y" being defined likewise.
Now, the radius of
curvature of the cam profile is the sum of that of the pitch curve, defined as
the curve traced by the center of the roller as it moves with respect to the
cam, and the radius of the roller. The curvature being the reciprocal of the
radius of curvature, the relation between the curvatures of the pitch curve and
the cam profile follows immediately. As a consequence of the above relation, a
convex pitch curve guarantees a convex cam profile.
From the above
expression for the curvature it is apparent that the derivatives of the cam
profile with respect to .psi. will be needed. As mentioned hereinbefore, the
expressions for the x and y coordinates of the cam profile are too cumbersome to
allow for a terse analysis of the curvature. On the contrary, the Cartesian
coordinates of the pitch curve are much simpler for the purpose at hand. The
expressions for the Cartesian coordinates of the pitch curve are ##EQU3##
The relation between .phi. above and .phi. is
.phi.=.+-..phi.
where the positive sign is used for external layouts and the negative
sign for their internal counterparts. After differentiation of the above
expressions with respect to .psi., and with the definition r=a.sub.3 /a.sub.1,
we derive an expression for the curvature of the pitch curve in nondimensional
form, namely, ##EQU4##
whence it is apparent that the sign of the
curvature is that of its numerator, which will be henceforth denoted by
f(.phi.;r) to stress that the numerator is a function of the angle of rotation
of the follower, with r as a parameter, i.e.,
f(.phi.;r)=r.sup.2
(1+.phi.').sup.3 +r[(1+.phi.')(2+.phi.') cos .phi.+.phi." sin .phi.]+1
Notice that, by virtue of the linear relation between .phi. and .psi.,
.phi.'=.+-.1/N, i.e., .phi.' is a constant, while .phi."=0. The positive sign is
used for external layouts and the negative sign for their internal counterparts.
The condition on r for a convex profile is now readily derived by imposing that
f(.phi.;r) remain positive for any value of .phi.. We do this by imposing the
condition that the equation f(.phi.;r)=0 do not, admit a real root .phi., which
thus leads, for external layouts, to ##EQU5##
For internal layouts, the
same relation holds, except for the sign of 1/N, which would be negative.
Apparently, the above expression yields a complex root .phi. whenever the
fraction displayed above, which is apparently positive, is greater than unity,
i.e., ##EQU6##
The above inequality holds for ##EQU7##
The range
of values of r that satisfy this interval yield a cam profile with a curvature
that does not change its sign in the external layout. However, these conditions
are necessary, but not sufficient to obtain a realizable fully-convex cam.
Sufficient conditions on r that guarantee a fully-convex, feasible cam profile
are derived upon imposing the condition to avoid undercutting. Undercutting
occurs when the follower or the cam cannot produce the desired path. This
phenomena happens when the radius of the roller is greater than or equal to the
minimum absolute value of the radius of curvature of the pitch curve. Therefore,
to avoid undercutting, the radius of the roller must be greater than the maximum
radius of curvature of the pitch curve, and hence, ##EQU8##
which thus
rules out the first interval found above, so that ##EQU9##
We have thus
found the condition on r=a.sub.3 /a.sub.1 to guarantee a fully convex cam
profile for external layouts. Shown in FIG. 12 is an instance of the cam plate
of the first embodiment with a convex cam of minimum curvature equal to zero and
N=5, with r=0.6944 and a.sub.1 =75 mm.
For internal layouts ##EQU10##
gives the interval to ensure an acceptable fully-convex cam profile.
In operation, the rotation of the input shaft 14 directly drives the cam
plates 54 and 56 which will in turn act on the rollers 85 to cause the carrier
member 68 rotating the output shaft 16 in the opposite direction with a reduced
rotational speed according to the profile of the cam plates 54 and 56. Indeed,
the motion of the carrier member 68 and thus of the output shaft 16 depends upon
the shape of the cam plates 54 and 56.
FIG. 6 shows a second possible
embodiment of the present invention wherein the longitudinal axis of the input
shaft 204 intersects the longitudinal axis of the output shaft 206 at right
angles, but the device can accommodate other angles between shafts. As shown in
FIG. 7, the housing 202 of the second embodiment is essentially the same as the
one described above except that the input shaft 204 is supported by a pair of
bearings 205 disposed in a bearing housing 207 secured to a front plate 208. It
is also noted that the housing 202 lacks any adjustable side plates. The
remaining features of the housing 202 are similar to those of the embodiment
shown in FIGS. 1 to 5, and thus their duplicate description will be omitted.
Referring to FIG. 8, it can be seen that two cams 212 and 214 are
mounted apart from each other onto the input shaft 204 with a predetermined
phase difference (180.degree. in this embodiment) by means of two square keys
216 and 218 which respectively cooperate with a key way defined in both cams 212
and 214 to fixedly secure the same onto the input shaft 204, as it is well known
in the art.
A lock nut 220 is provided at each end of the input shaft to
restrict the axial displacement of the cams 212 and 214 and of the input shaft
itself within the housing 202.
Referring now to FIG. 9, it can be seen
that a carrier member 222 supporting an internal row of rollers 224 and an
external row of rollers 226 is mounted to the output shaft 206 for rotation
therewith. The carrier member 222 is provided with a key way (not shown) which
is adapted to slidably engage a square key 228 extending along a portion of the
length of the output shaft 206. The axial positioning of the carrier member 222
to the output shaft 206 is ensured by an aligning bushing 230.
The
carrier member 222 includes a disc 232 and an integral ring 234 extending at an
angle from the periphery of the disc 232 for supporting the internal and
external sets of rollers 224 and 226. Geometrically, the integral ring 234
corresponds to a segment of a sphere, i.e. a portion of a sphere contained
between two parallel planes both intersecting the sphere. The integral ring 234
has holes 236 regularly distributed along the surface thereof for roller pins
238 to pass through.
As easily seen from FIG. 9, each roller pin 238 has
a head 240 and a longitudinal body having a threaded portion which is adapted to
cooperate with a bolt 242 to retain the roller pin 238 on the integral ring 234
and to restrict the axial displacement of a roller 244 mounted onto the
longitudinal body of the roller pin 238. The roller pins 238 are alternately
assembled to the integral ring 234 with the longitudinal body thereof extending
inwardly and outwardly of the integral ring 234 such that adjacent roller pins
238 extend in opposite direction with respect to each other, whereby the
internal and external row of rollers 224 and 226 are shifted in phase by an
angle which is equal to 360.degree./number of roller pins.
The rollers
244 have a frusto-conical shape. As best seen in FIG. 9, the rollers 244
disposed outwardly of the integral ring 234 are mounted to the roller pins 238
with the smallest radius section thereof facing the head 240 of the roller pins
238, whereas the rollers 244 disposed inwardly of the integral ring 234 are
mounted on the roller pins 238 with the greatest radius section thereof facing
the head 240 of the roller pins 238.
As for the first embodiment, a lock
nut 246 and an external retaining ring 248 are mounted on the opposed ends of
the output shaft 206 to restrict the axial displacement of the output shaft 206
within the housing 202.
When the input shaft 204 and the output shaft
206 are assembled to the housing 202 as shown in FIG. 6, the cams 212 and 214
are respectively in rolling contact with the internal and external rows of
rollers 224 and 226 such that rotation of the input shaft 204 will cause the
cams 212 and 214 to alternately push on a roller 244 of the corresponding row of
rollers 224 and 226 to thus transmit a torque from the input shaft 204 to the
output shaft 206.
As for the first embodiment, the profile of the cams
212 and 214 is the key element to ensure that a constant speed reduction ratio
is obtained. Accordingly, the profile of the cams 212 and 214 is generated by a
position vector r.sub.c which is expressed as follows: ##EQU11##
wherein:
.alpha..sub.1 : angle between output and input shafts;
.alpha..sub.3 : angle between output shaft and the axis of rotation of
the rollers;
.alpha..sub.4 : angle of the roller cone;
The
external layout illustrated in FIG. 6 is characterized by sin .alpha..sub.1
>sin .alpha..sub.3, whereas the internal layout (FIG. 15), i.e. wherein the
common axis of rotation of the cams 212' and 214' is located inside the circular
conic surface (follower cone) passing through all the axes of the rollers 244',
is characterized by sin .alpha..sub.1 <sin .alpha..sub.3. An internal layout
leads to an output angular-velocity vector with a component along the input
angular velocity vector in the direction of the latter, what we call
sense-preservation of the transmitted angular velocity. Likewise, an external
layout leads to such a component pointing in the opposite direction of the input
angular-velocity vector, what we call sense-reversal. Hence, an internal layout
becomes indispensable when applications call for sense-preservation. If the
sense of the output velocity is not an issue, then, still in the presence of
input and output shafts intersecting at an angle other than 90 degrees, an
internal layout is attractive, because it allows for a more compact transmission
device.
In the second embodiment, the pitch curve and the cam profile of
the previous one become curves on the unit sphere. The contact surface of the
cam is a conic surface produced by a ray stemming from the center of the sphere,
as it traverses the generatrix of the cam profile. As in the case of the first
embodiment, the algebraic relations defining the generatrix of the cam profile
are much more elaborate than those pertaining to the pitch surface. Likewise,
the convexity of the latter guarantees that of the cam contact surface. We will
thus work with the position vector e of a point on the generatrix of the pitch
surface. Moreover, the roller becomes a frustoconic surface of cone angle
.alpha..sub.3. While .alpha..sub.1 can attain any real value, we limit ourselves
here to the specific case in which .alpha..sub.1 =90.degree., which is
henceforth termed an orthogonal embodiment. In this embodiment, the generatrix
of the pitch surface is given by the unit vector e, defined as ##EQU12##
The geodetic curvature of the generatrix of the pitch surface is given
by ##EQU13##
with e'=de/d.psi. and e"=d.sup.2 e/d.psi..sup.2, and hence,
the condition for a convex contact cam surface is that the numerator of the
above expression do not change its positive sign, i.e.,
F(.psi.;
.alpha..sub.3)=e.multidot.(e'.times.e")>0
Expressions for the
foregoing derivatives and the mixed product were obtained with the aid of Maple
6 (a computer-algebra software package produced by Waterloo Maple Inc., of
Waterloo, Ontario, Canada). In our second embodiment, we have used a value of
.alpha..sub.3 given by ##EQU14##
and if we recall that .phi.=.psi./N,
the expressions for e and its derivatives lead to a highly simplified
expression, as obtained with the aid of Maple 6, namely, ##EQU15##
Now,
the condition sought is found by requiring that the roots .psi. of F(.psi.; N)
be complex. Maple 6 finds that, of the three roots cos (.psi./N) of F(.psi.; N),
only one is real, namely, ##EQU16##
with A and B given below: ##EQU17##
where C is defined as ##EQU18##
Although the above expression
for cos (.psi./N) appears complex because of the negative radical, it is in fact
real, because the two complex numbers A-B in the numerator and B in the
denominator turn out to be real multiples of each other, as calculated by Maple
6. This real root was evaluated for values of N between 2 and 20, the results
being shown in Table 1. For completeness, an entry with N=1 is also included. As
this table attests, all the values of cos (.psi./N) that verify F(.psi.; N)=0,
except for that corresponding to N=1, are greater than unity. As a consequence,
all orthogonal embodiments with .alpha..sub.3 =arctan (1/N) have a fully-convex
cam contact surface. N cos (V) N cos (V)
1 1.0 11 9.602584814
2 1.139944858 12 10.59398794
3 1.895708196 13 11.58672400
4 2.789219940 14 12.58050459
5 3.728836123 15 13.57511911
6 4.689626710 16 14.57041010
7 5.662012338 17 15.56625743
8 6.641474230 18 16.56256790
9 7.625585479 19 17.55926804
10 8.612920501 20 18.55629913
Table 1: Values of cos .psi., for various values of N, that
produce a zero geodetic curvature of the spherical pitch surface.
The
embodiment shown in FIG. 6 exhibits a convex profile, with N=8 and a3=arctan
(1/8)=0.1243549945 rad=7.125016344'. in this case, ##EQU19##
where
j=-1+L . Nevertheless, the foregoing relation yields an imaginary value of
.psi., namely, .psi.=j2.580764586.
FIG. 10 shows a third embodiment of
the present invention which may act as a substitute for a conventional rack and
pinion transmission to communicate a revolution of a first element into a linear
motion of another element or vice versa. More specifically, the transmission
device 300 comprises a rotary shaft 304 on which a pair of spaced-apart cam
plates 306 and 308 are mounted with a predetermined phase difference
(180.degree. in this embodiment) for respectively engaging first and second rows
of rollers 310 and 312 distributed on opposed longitudinal sides of an elongated
member 314.
It is noted that the cams 306 and 308 may be cut with the
rotary shaft 304 from a unique blank, in one single piece to add stiffness to
the transmission device 300. This concept is correspondingly applicable to the
first two embodiments, i.e., to the transmission devices 10 and 200.
Unlike the first two embodiments, the transmission device 300 does not
require a housing, since one of the moveable elements of the transmission
mechanism, i.e. the elongated member 314, has a translational motion, and hence,
the length of the stroke thereof is limited only by each application, the device
300 providing for an unlimited stroke length. However, the rotary shaft 304 is
mounted on a supporting frame (not shown) , which plays the role of a housing,
by means of bearings 316 provided at opposed ends thereof. The elongated member
314 or rack is supported by way of rollers 318. Each roller 318 is journalled at
opposed ends thereof within bearings 320 having respective external rings fixed
to the supporting frame (not shown). The rollers 318 are thus constrained to a
pure rotation about their revolving axes, without translating.
The
rollers 310 and 312 are mounted on pins 322 extending at right angles from both
sides of the elongated member 314. The pins 322 are uniformly distributed on
each side of the elongated member 314. As shown in FIG. 10, the first and second
row of rollers 310 and 312 are shifted in phase by a predetermined distance
which is a function of the number of rollers.
The transmission device
300 is characterized by a speed transmission factor or pitch p defined as the
quotient between the linear velocity of the elongated member 314 (m/s) by the
angular velocity of the rotating shaft 304 (rad/sec).
As for the first
and second embodiments, the profile of the cam members 306 and 308 is the key
element to ensure that a constant speed transmission is obtained. Accordingly,
the profile of the cam members 306 and 308 is generated by a position vector
r.sub.c, which is expressed as follows: ##EQU20##
wherein:
1/N:
where N is an integer denoting the number of rollers on each side of the
linearly translating member that enter in contact with corresponding cams upon a
full turn of the cam shaft;
.psi.: angle of rotation of the rotating
shaft with respect to the supporting frame;
.alpha..sub.1 : angle
between the rotating shaft and the longitudinal axis of the elongated member;
a.sub.1 : distance between the revolving axis of the rotating shaft and
a reference line parallel to a longitudinal axis of the elongated member;
a.sub.3 : distance between the reference line and roller centers;
a.sub.4 : radius of rollers;
As seen in FIG. 11, the center of
the rollers 310 and 312 are disposed along a common line 324 which is spaced by
a distance a.sub.3 from a reference line 326 parallel to a longitudinal axis of
the elongated member 314. The variable a.sub.1 is also established with
reference to the line 324.
As in the first two embodiments, in this
embodiment the algebraic expressions for the cam profile are much more
cumbersome than those for the pitch curve.
For this reason, we set the
conditions for a fully-convex profile based on the pitch curve, the Cartesian
coordinates x and y of any of its points being given below: ##EQU21##
In
our case .alpha..sub.1 =-.pi./2, the above expressions thus becoming ##EQU22##
Now, the curvature of the pitch curve is derived from the same formula
used for the first embodiment, which thus yields ##EQU23##
with A(.psi.)
and B(.psi.) given by ##EQU24##
Moreover, the translation z.sub.3 of the
slider 314 of FIG. 10 in terms of the angle of rotation .psi. of the cam is
given by ##EQU25##
where N=8 and a.sub.3 =-73.92 mm in the embodiment
shown in the abovementioned figure. We thus have: ##EQU26##
Substituting
z.sub.3 and z'.sub.3 into the expressions for the Cartesian coordinates of an
arbitrary point of the pitch curve, and its subsequent derivatives, yields, for
the numerator of K, ##EQU27##
The necessary condition for a convex
profile is thus A(.psi.)>0, for .psi..epsilon.[0, 2.pi.]. Moreover, for
convenience, we introduce the ratio R=(a.sub.1
+.vertline.a.sub.3.vertline.)/.vertline.a.sub.3.vertline., where the absolute
value is used because a.sub.3 can attain negative values, as it indeed does in
the embodiment disclosed in FIG. 10. The above expression for A(.psi.) then
becomes ##EQU28##
It is thus apparent that A(.psi.; r) does not change
sign in [0, 2.pi.] if the above quadratic equation in R has no real roots, for
any value of .psi.. This occurs if the discriminant of the above equation is
nonpositive, i.e., if ##EQU29##
The above inequality yields two ranges
of values of R, namely, ##EQU30##
and hence,
To avoid
undercutting, moreover, we must have .vertline.a.sub.3.vertline.<a.sub.1
/(2/N+1), which thus leads to the range
a.sub.3 <a.sub.1 /(2/N+1)
for a convex pitch curve, and hence, for a convex cam profile.
For a fixed value of a.sub.1, the smaller a.sub.3, the smaller the
overall pressure angle. To obtain a convex cam with a relatively small pressure
angle, we set the .vertline.a.sub.3.vertline./a.sub.1 ratio as 0.8. The pressure
angle is plotted in FIG. 13.
The third embodiment of FIGS. 10 and 11 can
be modified to allow for a convex cam profile, the new design parameters being
a.sub.1 =92.4 mm, a.sub.3 =-73.92 mm, with N=8.
In operation, the rotary
shaft 304 may be driven to cause the cam members 306 and 308 to alternately act
on the corresponding row of rollers 310 and 312 to translate the elongated
member 314 in a direction parallel to the longitudinal axis thereof.
Alternatively, the elongated member 314 may be driven such as to successively
push a roller of the first and second rows 310 and 312 against the cam members
306 and 308, respectively, to cause the rotary shaft 304 to rotate.
It
is understood that although the longitudinal axis of the elongated member 314 is
at right angles with the axis of the shaft 304 in the embodiment illustrated in
FIG. 10, the transmission device 300 can accommodate other angles.
FIG.
16 illustrates a modified version of the transmission device of FIG. 1. The
transmission device of FIG. 16 differs from the one in FIG. 1 in that it is
provided with fully convex cam plates 54" and 56". Furthermore, counterweights
55" and 57" are provided on the input shaft 14" for balancing same. The shaft
14", the cams 54" and 56", and the counterweights 55' and 57" are preferably
integral. Also four coil springs, represented by their axis 23", have been added
between the front plate 22" and a plate 21" defining four spring seats 27"
arranged at the corner of an imaginary square. The springs provide for
adjustment to compensate for submillimetric distance variations between the cam
shaft 14" and the roller carrying shaft 16". Moreover, the carrier member 68"
consists of a single disc and the number of rollers 70" and 72" on each side of
the carrier member 68" has been limited to five in view of the fully convex
profile of the cams 54" and 56" and in order to obtain a suitable contact
pressure between the rollers 70" and 72" and the cams 54" and 56".
One
advantage of the four above-described embodiments and variations thereof resides
in the fact that they allow for a reversal of both the direction of the input
speed and the roles of the input and output elements.
The present
invention is not limited to the above-described embodiments. For example, double
trains or multiple trains (i.e. multistage transmission devices) can be provided
and linked to obtain a higher speed ratio of the transmission. The present
invention also includes in its scope a construction in which the three above
described embodiments are used in combination.
* * * * *