• Relatively little flexibility in the
design of the geneva mechanism.
One factor alone (the number of slots
in the output member) determines the
characteristics of the motion. As a
result, the ratio of the time of motion
to the time of dwell cannot exceed
one-half, the output motion cannot be
uniform for any finite portion of the
indexing cycle, and it is always oppo-
site in sense to the sense of input
rotation. The output shaft, moreover,
must always be offset from the input
shaft.
Many modifications of the standard
external geneva have been proposed,
97
ODD SHAPES IN PLANETARY GIVE
SMOOTH STOP AND GO
This intermittent-motion mechanism for automatic
processing machinery combines gears with lobes;
some pitch curves are circular and some are noncircular.
This intermittent-motion mechanism
combines circular gears with noncircular
gears in a planetary arrangement, as
shown in the drawing.
The mechanism was developed by
Ferdinand Freudenstein, a professor of
mechanical engineering at Columbia
University. Continuous rotation applied
to the input shaft produces a smooth,
stop-and-go unidirectional rotation in the
output shaft, even at high speeds.
This jar-free intermittent motion is
sought in machines designed for packag-
ing, production, automatic transfer, and
processing.
Varying differential. The basis for
Freudenstein’s invention is the varying
differential motion obtained between two
sets of gears. One set has lobular pitch
circles whose curves are partly circular
and partly noncircular.
The circular portions of the pitch
curves cooperate with the remainder of
the mechanism to provide a dwell time or
stationary phase, or phases, for the out-
put member. The non-circular portions
act with the remainder of the mechanism
to provide a motion phase, or phases, for
the output member.
Competing genevas. The main com-
petitors to Freudenstein’s “pulsating
planetary” mechanism are external
genevas and starwheels. These devices
have a number of limitations that
include:
• Need for a means, separate from the
driving pin, for locking the output
member during the dwell phase of
the motion. Moreover, accurate man-
ufacture and careful design are
required to make a smooth transition
from rest to motion and vice versa.
• Kinematic characteristics in the
geneva that are not favorable for
high-speed operation, except when
the number of stations (i.e., the num-
ber of slots in the output member) is
large. For example, there is a sudden
change of acceleration of the output
member at the beginning and end of
each indexing operation.
At heart of new planetary (in front view, circular set stacked behind noncircular set), two sets
of gears when assembled (side view) resemble conventional unit (schematic).
including multiple and unequally spaced
driving pins, double rollers, and separate
entrance and exit slots. These proposals
have, however, been only partly success-
ful in overcoming these limitations.
Differential motion. In deriving the
operating principle of his mechanism,
Freudenstein first considered a conven-
tional epicyclic (planetary) drive in
which the input to the cage or arm
causes a planet set with gears
2 and 3 to
rotate the output “sun,” gear
4, while
another sun, gear
1, is kept fixed (see
drawing).
Letting
r
1
, r
2
, r
3
, r
4
, equal the pitch
radii of the circular
1, 2, 3, 4, then the
output ratio, defined as:
is equal to:
Now, if r
1
= r
4
and r
2
= r
3
, there is no
“differential motion” and the output
remains stationary. Thus if one gear pair,
say
3 and 4, is made partly circular and
partly noncircular, then where
r
2
= r
3
and
r
1
= r
4
for the circular portion, gear 4
dwells. Where r
2
≠ r
3
and r
1
≠ r
4
for the
noncircular portion, gear
4 has motion.
The magnitude of this motion depends
Sclater Chapter 4 5/3/01 10:44 AM Page 97
on the difference in radii, in accordance
with the previous equation. In this man-
ner, gear
4 undergoes an intermittent
motion (see graph).
Advantages. The pulsating planetary
approach demonstrates some highly use-
ful characteristics for intermittent-
motion machines:
• The gear teeth serve to lock the out-
put member during the dwell as well
as to drive that member during
motion.
• Superior high-speed characteristics
are obtainable. The profiles of the
pitch curves of the noncircular gears
can be tailored to a wide variety of
desired kinematic and dynamic char-
acteristics. There need be no sudden
terminal acceleration change of the
driven member, so the transition from
dwell to motion, and vice versa, will
be smooth, with no jarring of
machine or payload.
• The ratio of motion to dwell time is
adjustable within wide limits. It can
even exceed unity, if desired. The
number of indexing operations per
revolution of the input member also
can exceed unity.
• The direction of rotation of the out-
put member can be in the same or
opposite sense relative to that of the
input member, according to whether
the pitch axis
P
34
for the noncircular
portions of gears
3 and 4 lies wholly
outside or wholly inside the pitch
surface of the planetary sun gear
1.
• Rotation of the output member is
coaxial with the rotation of the input
member.
• The velocity variation during motion
is adjustable within wide limits.
Uniform output velocity for part of
the indexing cycle is obtainable; by
varying the number and shape of the
lobes, a variety of other desirable
motion characteristics can be
obtained.
• The mechanism is compact and has
relatively few moving parts, which
can be readily dynamically balanced.
Design hints. The design techniques
work out surprisingly simply, said
Freudenstein. First the designer must
select the number of lobes
L
3
and L
4
on
the gears
3 and 4. In the drawings, L
3
= 2
and
L
4
= 3. Any two lobes on the two
gears (i.e., any two lobes of which one is
on one gear and the other on the other
gear) that are to mesh together must have
the same arc length. Thus, every lobe on
gear
3 must mesh with every lobe on gear
4, and T
3
/T
4
= L
3
/L
4
= 2/3, where T
3
and
T
4
are the numbers of teeth on gears 3
and 4. T
1
and T
2
will denote the numbers
of teeth on gears
1 and 2.
Next, select the ratio
S of the time of
motion of gear
4 to its dwell time, assum-
ing a uniform rotation of the arm
5. For the
gears shown,
S = 1. From the geometry,
(
θ
30
+ ∆
θ
30
)L
3
= 360º
and
S = ∆
θ
3
/
θ
30
Hence
θ
30
(1 + S)L
3
= 360º
For
S = 1 and L
3
+ 2,
θ
30
= 90º
and
∆
θ
3
= 90º
Now select a convenient profile for
the noncircular portion of gear
3. One
profile (see the profile drawing) that
Freudenstein found to have favorable
high-speed characteristics for stop-and-
go mechanisms is
r
3
= R
3
The profile defined by this equation
has, among other properties, the charac-
teristic that, at transition from rest to
motion and vice versa, gear
4 will have
zero acceleration for the uniform rotation
of arm
5.
In the above equation,
λ is the quan-
tity which, when multiplied by
R
3
, gives
the maximum or peak value of
r
3
– R
3
,
differing by an amount
h′ from the radius
R
3
of the circular portions of the gear.
The noncircular portions of each lobe
are, moreover, symmetrical about their
midpoints, the midpoints of these por-
tions being indicated by
m.
1
2
1
2
330
3
+−
−
λπθθ
θ
cos
()
∆
98
Output motion (upper curve) has long dwell periods; velocity curve (center) has smooth tran-
sition from zero to peak; acceleration at transition is zero (bottom).
Sclater Chapter 4 5/3/01 10:44 AM Page 98
To evaluate the quantity λ,
Freudenstein worked out the equation:
where
R
3
λ = height of lobe
To evaluate the equation, select a suit-
able value for
µ that is a reasonably sim-
ple rational fraction, i.e., a fraction such
as
3
⁄
8
whose numerator and denominator
are reasonably small integral numbers.
Thus, without a computer or lengthy
trial-and-error procedures, the designer
can select the configuration that will
achieve his objective of smooth intermit-
tent motion.
µ
α
== +
=++
R
A
RR R
SSLL
3
33 4
34
1
()
()
λ
µ
µ
ααµα αµ
ααµ
=
−
×
+−+ −−+
−+
1
11
1
2
[ ( )][ ( )]
[( )]
SS
A metering pump for liquid or gas has an
adjustable ring gear that meshes with a
special-size planet gear to provide an
infinitely variable stroke in the pump.
The stroke can be set manually or auto-
matically when driven by a servomotor.
Flow control from 180 to 1200 liter/hr.
(48 to 317 gal./hr.) is possible while the
pump is at a standstill or running.
Straight-line motion is key. The
mechanism makes use of a planet gear
whose diameter is half that of the ring
gear. As the planet is rotated to roll on the
inside of the ring, a point on the pitch
diameter of the planet will describe a
straight line (instead of the usual hypocy-
cloid curve). This line is a diameter of the
ring gear. The left end of the connecting
rod is pinned to the planet at this point.
The ring gear can be shifted if a sec-
ond set of gear teeth is machined in its
outer surface. This set can then be
meshed with a worm gear for control.
Shifting the ring gear alters the slope of
the straight-line path. The two extreme
positions are shown in the diagram. In
the position of the mechanism shown, the
pin will reciprocate vertically to produce
the minimum stroke for the piston.
Rotating the ring gear 90º will cause the
pin to reciprocate horizontally to produce
the maximum piston stroke.
The second diagram illustrates
another version that has a yoke instead of
a connecting rod. This permits the length
of the stroke to be reduced to zero. Also,
the length of the pump can be substan-
tially reduced.
99
Profiles for noncircular gears are circular
arcs blended to special cam curves.
CYCLOID GEAR MECHANISM
CONTROLS STROKE OF PUMP
An adjustable ring gear meshes with a planet gear having half of its diameter to provide an
infinitely variable stroke in a pump. The adjustment in the ring gear is made by engaging other
teeth. In the design below, a yoke replaces the connecting rod.
Sclater Chapter 4 5/3/01 10:44 AM Page 99
CONVERTING ROTARY-TO-LINEAR MOTION
A compact gear system that provides lin-
ear motion from a rotating shaft was
designed by Allen G. Ford of The Jet
Propulsion Laboratory in California. It
has a planetary gear system so that the
end of an arm attached to the planet gear
always moves in a linear path (drawing).
The gear system is set in motion by a
motor attached to the base plate. Gear
A,
attached to the motor shaft, turns the case
assembly, causing Gear
C to rotate along
Gear
B, which is fixed. The arm is the
same length as the center distance
between Gears
B and C. Lines between
the centers of Gear
C, the end of the arm,
and the case axle form an isosceles trian-
gle, the base of which is always along the
plane through the center of rotation. So
the output motion of the arm attached to
Gear
C will be in a straight line.
When the end of travel is reached, a
switch causes the motor to reverse,
returning the arm to its original position.
100
The end of arm moves in a straight line because of the triangle effect (right).
NEW STAR WHEELS CHALLENGE
GENEVA DRIVES FOR INDEXING
Star wheels with circular-arc slots can be analyzed
mathematically and manufactured easily.
Star Wheels vary in shape, depending on the degree of indexing that must be done during one input revolution.
Sclater Chapter 4 5/3/01 10:44 AM Page 100
A family of star wheels with circular
instead of the usual epicyclic slots (see
drawings) can produce fast start-and-stop
indexing with relatively low acceleration
forces.
This rapid, jar-free cycling is impor-
tant in a wide variety of production
machines and automatic assembly lines
that move parts from one station to
another for drilling, cutting, milling, and
other processes.
The circular-slot star wheels were
invented by Martin Zugel of Cleveland,
Ohio.
The motion of older star wheels with
epicyclic slots is difficult to analyze and
predict, and the wheels are hard to make.
The star wheels with their circular-arc
slots are easy to fabricate, and because
the slots are true circular arcs, they can
be visualized for mathematical analysis
as four-bar linkages during the entire
period of pin-slot engagement.
Strong points. With this approach,
changes in the radius of the slot can be
analyzed and the acceleration curve var-
ied to provide inertia loads below those
of the genevas for any practical design
requirement.
Another advantage of the star wheels
is that they can index a full 360º in a rel-
atively short period (180º). Such one-
stop operation is not possible with
genevas. In fact, genevas cannot do two-
stop operations, and they have difficulty
producing three stops per index. Most
two-stop indexing devices available are
cam-operated, which means they require
greater input angles for indexing.
101
The one-stop index motion of the unit can be designed to take longer to complete its
indexing, thus reducing its index velocity.
Geared star sector indexes smoothly a full 360º during a 180º rotation of the
wheel, then it pauses during the other 180º to allow the wheel to catch up.
An accelerating pin brings the output wheel up to speed. Gear sectors mesh to keep the output rotating beyond 180º.
Sclater Chapter 4 5/3/01 10:44 AM Page 101
Operating sequence. In operation, the
input wheel rotates continuously. A
sequence starts (see drawing) when the
accelerating pin engages the curved slot
to start indexing the output wheel clock-
wise. Simultaneously, the locking sur-
face clears the right side of the output
wheel to permit the indexing.
Pin C in the drawings continues to
accelerate the output wheel past the mid-
point, where a geneva wheel would start
deceleration. Not until the pins are sym-
metrical (see drawing) does the accelera-
tion end and the deceleration begin. Pin
D then takes the brunt of the deceleration
force.
Adaptable. The angular velocity of the
output wheel, at this stage of exit of the
acceleration roller from Slot 1, can be
varied to suit design requirements. At
this point, for example, it is possible
either to engage the deceleration roller as
described or to start the engagement of a
constant-velocity portion of the cycle.
Many more degrees of output index can
be obtained by interposing gear-element
segments between the acceleration and
deceleration rollers.
The star wheel at left will stop and
start four times in making one revolution,
while the input turns four times in the
same period. In the starting position, the
output link has zero angular velocity,
which is a prerequisite condition for any
star wheel intended to work at speeds
above a near standstill.
In the disengaged position, the angu-
lar velocity ratio between the output and
input shafts (the “gear” ratio) is entirely
dependent upon the design angles
α
and
β and independent of the slot radius, r.
Design comparisons. The slot radius,
however, plays an important role in the
mode of the acceleration forces. A four-
stop geneva provides a good basis for
comparison with a four-stage “Cyclo-
Index” system.
Assume, for example, that
α = β =
22.5º. Application of trigonometry
yields:
which yields
R = 0.541A. The only
restriction on
r is that it be large enough
to allow the wheel to pass through its
mid-position. This is satisfied if:
There is no upper limit on
r, so that
slot can be straight.
r
RA
ARA
A>
−
−−
≈
( cos )
cos
.
1
2
01
α
α
RA=
+
sin
sin( )
β
αβ
102
The accelerating force of star wheels (curves A, B, C) varies with input rota-
tion. With an optimum slot (curve C), it is lower than for a four-stop geneva.
This internal star wheel has a radius difference to
cushion the indexing shock.
Star-wheel action is improved with curved slots over the radius r, centered on the initial-
contact line OP. The units then act as four-bar linkages, 00
1
PQ.
Sclater Chapter 4 5/3/01 10:44 AM Page 102
GENEVA MECHANISMS
103
The driving follower on the rotating
input crank of this geneva enters a slot
and rapidly indexes the output. In this
version, the roller of the locking-arm
(shown leaving the slot) enters the slot to
prevent the geneva from shifting when it
is not indexing.
The output link remains stationary
while the input gear drives the planet
gear with single tooth on the locking
disk. The disk is part of the planet gear,
and it meshes with the ring-gear geneva
to index the output link one position.
The driven member of the first geneva acts as the driver for the second
geneva. This produces a wide variety of output motions including very
long dwells between rapid indexes.
When a geneva is driven by
a roller rotating at a constant
speed, it tends to have very
high acceleration and decelera-
tion characteristics. In this
modification, the input link,
which contains the driving
roller, can move radially while
being rotated by the groove
cam. Thus, as the driving roller
enters the geneva slot, it moves
radially inward. This action
reduces the geneva accelera-
tion force.
One pin locks and unlocks the geneva; the second pin rotates the
geneva during the unlocked phase. In the position shown, the drive pin is
about to enter the slot to index the geneva. Simultaneously, the locking pin
is just clearing the slot.
A four-bar geneva produces a long-dwell motion from
an oscillating output. The rotation of the input wheel
causes a driving roller to reciprocate in and out of the slot
of the output link. The two disk surfaces keep the output in
the position shown during the dwell period.
Sclater Chapter 4 5/3/01 10:44 AM Page 103
The key consideration in the design of genevas
is to have the input roller enter and leave the geneva
slots tangentially (as the crank rapidly indexes the
output). This is accomplished in the novel mecha-
nism shown with two tracks. The roller enters one
track, indexes the geneva 90º (in a four-stage
geneva), and then automatically follows the exit
slot to leave the geneva.
The associated linkage mechanism locks the
geneva when it is not indexing. In the position
shown, the locking roller is just about to exit from
the geneva.
This geneva arrangement has a chain with an extended
pin in combination with a standard geneva. This permits a
long dwell between each 90º shift in the position of the
geneva. The spacing between the sprockets determines the
length of dwell. Some of the links have special extensions
to lock the geneva in place between stations.
104
The coupler point at the extension of
the connecting link of the four-bar mech-
anism describes a curve with two
approximately straight lines, 90º apart.
This provides a favorable entry situation
because there is no motion in the geneva
while the driving pin moves deeply into
the slot. Then there is an extremely rapid
index. A locking cam, which prevents the
geneva from shifting when it is not
indexing, is connected to the input shaft
through gears.
The input link of a normal geneva
drive rotates at constant velocity, which
restricts flexibility in design. That is, for
given dimensions and number of sta-
tions, the dwell period is determined by
the speed of the input shaft. Elliptical
gears produce a varying crank rotation
that permits either extending or reducing
the dwell period.
This arrangement permits the roller to exit and enter the driving
slots tangentially. In the position shown, the driving roller has just
completed indexing the geneva, and it is about to coast for 90º as it
goes around the curve. (During this time, a separate locking device
might be necessary to prevent an external torque from reversing
the geneva.)
Sclater Chapter 4 5/3/01 10:44 AM Page 104
The output in this simple mechanism is prevented from turning in either
direction—unless it is actuated by the input motion. In operation, the drive
lever indexes the output disk by bearing on the pin. The escapement is
cammed out of the way during indexing because the slot in the input disk is
positioned to permit the escapement tip to enter it. But as the lever leaves
the pin, the input disk forces the escapement tip out of its slot and into the
notch. That locks the output in both directions.
A crank attached to the planet gear can make point
P
describe the double loop curve illustrated. The slotted
output crank oscillates briefly at the vertical positions.
105
This reciprocator transforms rotary motion into a
reciprocating motion in which the oscillating output
member is in the same plane as the input shaft. The out-
put member has two arms with rollers which contact the
surface of the truncated sphere. The rotation of the
sphere causes the output to oscillate.
The input crank contains two planet gears. The center
sun gear is fixed. By making the three gears equal in
diameter and having gear
2 serve as an idler, any member
fixed to gear
3 will remain parallel to its previous posi-
tions throughout the rotation of the input ring crank.
The high-volume 2500-ton press is designed to shape
such parts as connecting rods, tractor track links, and
wheel hubs. A simple automatic-feed mechanism makes
it possible to produce 2400 forgings per hour.
Sclater Chapter 4 5/3/01 10:44 AM Page 105
MODIFIED GENEVA DRIVES
Most of the mechanisms shown here add a varying velocity
component to conventional geneva motion.
Fig. 1 With a conventional external geneva drive, a constant-
velocity input produces an output consisting of a varying velocity
period plus a dwell. The motion period of the modified geneva shown
has a constant-velocity interval which can be varied within limits.
When spring-loaded driving roller a enters the fixed cam b, the out-
put-shaft velocity is zero. As the roller travels along the cam path, the
output velocity rises to some constant value, which is less than the
maximum output of an unmodified geneva with the same number of
slots. The duration of constant-velocity output is arbitrary within limits.
When the roller leaves the cam, the output velocity is zero. Then the
output shaft dwells until the roller re-enters the cam. The spring pro-
duces a variable radial distance of the driving roller from the input
shaft, which accounts for the described motions. The locus of the
roller’s path during the constant-velocity output is based on the veloc-
ity-ratio desired.
Fig. 2 This design incorporates a planet gear in the drive mecha-
nism. The motion period of the output shaft is decreased, and the
maximum angular velocity is increased over that of an unmodified
geneva with the same number of slots. Crank wheel a drives the unit
composed of planet gear b and driving roller c. The axis of the driving
roller coincides with a point on the pitch circle of the planet gear.
Because the planet gear rolls around the fixed sun gear d, the axis of
roller c describes a cardioid e. To prevent the roller from interfering
with the locking disk f, the clearance arc g must be larger than is
required for unmodified genevas.
106
Fig. 3 A motion curve similar to that of Fig. 2 can be derived by driv-
ing a geneva wheel with a two-crank linkage. Input crank a drives
crank b through link c. The variable angular velocity of driving roller d,
mounted on b, depends on the center distance L, and on the radii M
and N of the crank arms. This velocity is about equivalent to what
would be produced if the input shaft were driven by elliptical gears.
Sclater Chapter 4 5/3/01 10:44 AM Page 106
Không có nhận xét nào:
Đăng nhận xét