Introduction to Mechanisms
Yi Zhang
with
Susan Finger
Stephannie Behrens
A wheel provided with suitably shaped teeth, receiving an intermittent
circular motion from an oscillating or reciprocating member, is called
a ratchet wheel. A simple form of ratchet mechanism is shown
in Figure 8-1.
Figure 8-1 Ratchet
A is the ratchet wheel, and B is an oscillating
lever carrying the driving pawl, C. A supplementary
pawl at D prevents backward motion of the wheel.
When arm B moves counterclockwise, pawl C will force the
wheel through a fractional part of a revolution dependent upon the
motion of B. When the arm moves back (clockwise), pawl C
will slide over the points of the teeth while the wheel remains at
rest because of fixed pawl D, and will be ready to push the
wheel on its forward (counterclockwise) motion as before.
The amount of backward motion possible varies with the pitch of the
teeth. This motion could be reduced by using small teeth, and the
expedient is sometimes used by placing several pawls side by side on
the same axis, the pawls being of different lengths.
The contact surfaces of wheel and pawl should be inclined so that they
will not tend to disengage under pressure. This means that the common
normal at N should pass between the pawl and the ratchet-wheel
centers. If this common normal should pass outside these limits, the
pawl would be forced out of contact under load unless held by
friction. In many ratchet mechanisms the pawl is held against the
wheel during motion by the action of a spring.
The usual form of the teeth of a ratchet wheel is that shown in the
above Figure, but in feed mechanisms such as used on many machine
tools it is necessary to modify the tooth shape for a reversible pawl
so that the drive can be in either direction.
The following SimDesign example of a ratchet also includes a four bar linkage.
If you try this mechanism, you may turn the crank of the link mechanism. The rocker will drive the driving pawl to drive the ratchet wheel. The corresponding SimDesign data
file is:
/afs/andrew.cmu.edu/cit/ce/rapidproto/simdesign/ratchet.sim
A special form of a ratchet is the
overrunning clutch. Have you ever thought about what kind of
mechanism drives the rear axle of bicycle? It is a free-wheel
mechanism which is an overrunning clutch. Figure 8-2 illustrates a
simplified model. As the driver delivers torque to the driven member,
the rollers or balls are wedged into the tapered recesses. This is
what gives the positive drive. Should the driven member attempt to
drive the driver in the directions shown, the rollers or balls become
free and no torque is transmitted.
Figure 8-2 Overrunning clutch
A pair of rotating members may be designed so that, for continuous
rotation of the driver, the follower will alternately roll with the
driver and remain stationary. This type of arrangement is know by the
general term intermittent gearing. This type of gearing occurs
in some counting mechanisms, motion-picture machines, feed mechanisms,
as well as others.
Figure 8-3 Intermittent gearing
The simplest form of intermittent gearing, as illustrated in Figure 8-3
has the same kind of teeth as ordinary gears designed for
continuous rotation. This example is a pair
of 18-tooth gears modified to meet the requirement that the follower
advance one-ninth of a turn for each turn of the driver. The interval
of action is the two-pitch angle (indicated on both gears). The single
tooth on the driver engages with each space on the follower to
produce the required motion of a one-ninth turn of the follower. During
the remainder of a driver turn, the follower is locked against
rotation in the manner shown in the figure.
To vary the relative movements of the driver and follower, the meshing
teeth can be arranged in various ways to suit requirements. For
example, the driver may have more than one tooth, and the periods of
rest of the follower may be uniform or may vary considerably. Counting
mechanisms are often equipped with gearing of this type.
An interesting example of intermittent
gearing is the Geneva Wheel shown in Figure 8-4. In this
case the driven wheel, B, makes one fourth of a turn for
one turn of the driver, A, the pin, a,
working in the slots, b, causing the motion of B.
The circular portion of the driver, coming in contact with the
corresponding hollow circular parts of the driven wheel, retains it in
position when the pin or tooth a is out of action. The wheel
A is cut away near the pin a as shown, to provide
clearance for wheel B in its motion.
Figure 8-4 Geneva wheel
If one of the slots is closed, A can only move through part of
the revolution in either direction before pin a strikes the
closed slot and thus stops the motion. The device in this modified
form was used in watches, music boxes, etc., to prevent
overwinding. From this application it received the name Geneva
stop. Arranged as a stop, wheel A is secured to the spring
shaft, and B turns on the axis of the spring barrel. The
number of slots or interval units in B depends upon the desired
number of turns for the spring shaft.
An example of this mechanism has been made in SimDesign, as in the following picture.
The corresponding SimDesign data file is:
/afs/andrew.cmu.edu/cit/ce/rapidproto/simdesign/geneva.sim
The engine of an automobile is usually located in front part. How does
it connect to the rear axle of the automobile? In this case,
universal joints are used to transmit the motion.
Figure 8-5 Universal joint
The universal joint as shown in Figure 8-5 is also known in the
older literature as Hooke's coupling. Regardless of how it is
constructed or proportioned, for practical use it has essentially the
form shown in Figure 8-6, consisting of two semicircular forks 2
and 4, pin-jointed to a right -angle cross 3.
Figure 8-6 General form for a universal joint
The driver 2 and the follower 4 make the complete revolution at the
same time, but the velocity ratio is not constant throughout the
revolution. The following analysis will show how complete information
as to the relative motions of driver and follower may be obtained for
any phase of the motion.
 
Figure 8-7 Analysis of a universal joint
If the plane of projection is taken perpendicular to the axis of 2,
the path of a and b will be a circle AKBL as
shown in Figure 8-7.
If the angle between the shafts is  , the path of c and
d will be a circle that is projected as the ellipse
ACBD, in which
OC = OD = OKcos =
OAcos
(8-1)
If one of the arms of the driver is at A, an arm of the
follower will be at C. If the driver arm moves through the
angle  to P, the
follower arm will move to Q. OQ will be perpendicular
to OP; hence: angle COQ = . But angle COQ is the
projection of the real angle describes by the follower. Qn is
the real component of the motion of the follower in a direction
parallel to AB, and line AB is the intersection of the
planes of the driver's and the follower's planes. The true angle  described by the follower, while
the driver describes the angle , can be found by revolving
OQ about AB as an axis into the plane of the circle
AKBL. Then OR = the true length of OQ, and
ROK = = the true
angle that is projected as angle COQ = .
Now
tan
= Rm/Om
and
tan
= Qn/On
But
Qn = Rm
Hence
Therefore
tan
=
costan
The ratio of the angular motion of the follower to that of the driver
is found as follower, by differentiating above equation, remembering
that is constant
Eliminating
:
Similarly, can be eliminated:
According to the above equations, when the driver has a uniform
angular velocity, the ratio of angular velocities varies between
extremes of cos and
1/cos. These
variations in velocity give rise to inertia forces, torques, noise,
and vibration which would not be present if the velocity ratio were
constant.
By using a double joint shown on the right in
Figure 8-7, the variation of angular motion between driver and
follower can be entirely avoided. This compensating arrangement is to
place an intermediate shaft 3 between the driver and follower
shafts. The two forks of this intermediate shaft must lie in the same
plane, and the angle between the first shaft and the intermediate
shaft must exactly be the same with that between the intermediate
shaft and the last shaft. If the first shaft rotates uniformly, the
angular motion of the intermediate shaft will vary according to the
result deduced above. This variation is exactly the same as if the
last shaft rotated uniformly, driving the intermediate
shaft. Therefore, the variable motion transmitted to the intermediate
shaft by the uniform rotation of the first shaft is exactly
compensated for by the motion transmitted from the intermediate to the
last shaft, the uniform motion of either of these shafts will impart,
through the intermediate shaft, uniform motion to the other.
Universal joints, particularly in pairs, are used in many
machines. One common application is in the drive shaft which connects
the engine of an automobiles to the axle.
Complete Table of Contents
- 1 Physical Principles
- 2 Mechanisms and Simple Machines
- 3 More on Machines and Mechanisms
- 4 Basic Kinematics of Constrained Rigid Bodies
- 5 Planar Linkages
- 6 Cams
- 7 Gears
- 8 Other Mechanisms
- 8.1 Ratchet Mechanisms
- 8.2 Overrunning Clutch
- 8.3 Intermittent Gearing
- 8.4 The Geneva Wheel
- 8.5 The Universal Joint
- 8.5.1 Analysis of a Universal Joint
- 8.5.2 Double Universal Joint
- Index
- References
sfinger@ri.cmu.edu
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