# Introduction to Mechanisms

Yi Zhang
with
Susan Finger
Stephannie Behrens

## Chapter 8. Other Mechanisms

### 8.1 Ratchet Mechanisms

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

### 8.2 Overrunning Clutch

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.

### 8.3 Intermittent Gearing

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.

### 8.4 The Geneva Wheel

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

### 8.5 The Universal Joint

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.

### 8.5.1 Analysis of a Universal Joint

#### 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.

### 8.5.2 Double Universal Joint

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.

1 Physical Principles
2 Mechanisms and Simple Machines
3 More on Machines and Mechanisms
4 Basic Kinematics of Constrained Rigid Bodies
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