5 Planar Linkages
5.1 Introduction
5.1.1 What are Linkage Mechanisms?
Have you ever wondered what kind of mechanism causes the wind shield wiper on the front
widow of car to oscillate ( Figure 5-1a)? The mechanism, shown in Figure 5-1b, transforms the
rotary motion of the motor into an oscillating motion of the windshield wiper.
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Figure 5-1 Windshield wiper
Let's make a simple mechanism with similar behavior. Take some cardboard and make four
strips as shown in Figure 5-2a.
Take 4 pins and assemble them as shown in Figure 5-2b.
Now, hold the 6in. strip so it can't move and turn the 3in. strip. You will see that the 4in. strip
oscillates.
Figure 5-2 Do-it-yourself four bar linkage mechanism
The four bar linkage is the simplest and often times, the most useful mechanism. As we
mentioned before, a mechanism composed of rigid bodies and lower pairs is called a linkage
(Hunt 78). In planar mechanisms, there are only two kinds of lower pairs --- revolute pairs and
prismatic pairs.
The simplest closed-loop linkage is the four bar linkage which has four members, three moving
links, one fixed link and four pin joints. A linkage that has at least one fixed link is a mechanism.
The following example of a four bar linkage was created in SimDesign in
simdesign/fourbar.sim
Figure 5-3 Four bar linkage in SimDesign
This mechanism has three moving links. Two of the links are pinned to the frame which is not
shown in this picture. In SimDesign, links can be nailed to the background thereby making them
into the frame.
How many DOF does this mechanism have? If we want it to have just one, we can impose one
constraint on the linkage and it will have a definite motion. The four bar linkage is the simplest
and the most useful mechanism.
Reminder: A mechanism is composed of rigid bodies and lower pairs called linkages (Hunt 78).
In planar mechanisms there are only two kinds of lower pairs: turning pairs and prismatic pairs.
5.1.2 Functions of Linkages
The function of a link mechanism is to produce rotating, oscillating, or reciprocating motion
from the rotation of a crank or vice versa (Ham et al. 58). Stated more specifically linkages may
be used to convert:
1. Continuous rotation into continuous rotation, with a constant or variable angular velocity
ratio.
2. Continuous rotation into oscillation or reciprocation (or the reverse), with a constant or
variable velocity ratio.
3. Oscillation into oscillation, or reciprocation into reciprocation, with a constant or variable
velocity ratio.
Linkages have many different functions, which can be classified according on the primary goal
of the mechanism:
Function generation: the relative motion between the links connected to the frame,
Path generation: the path of a tracer point, or
Motion generation: the motion of the coupler link.
5.2 Four Link Mechanisms
One of the simplest examples of a constrained linkage is the four-link mechanism. A variety of
useful mechanisms can be formed from a four-link mechanism through slight variations, such as
changing the character of the pairs, proportions of links, etc. Furthermore, many complex link
mechanisms are combinations of two or more such mechanisms. The majority of four-link
mechanisms fall into one of the following two classes:
1. the four-bar linkage mechanism, and
2. the slider-crank mechanism.
5.2.1 Examples
Parallelogram Mechanism
In a parallelogram four-bar linkage, the orientation of the coupler does not change during the
motion. The figure illustrates a loader. Obvioulsy the behavior of maintaining parallelism is
important in a loader. The bucket should not rotate as it is raised and lowered. The corresponding
SimDesign file is simdesign/loader.sim.
Figure 5-4 Front loader mechanism
Slider-Crank Mechanism
The four-bar mechanism has some special configurations created by making one or more links
infinite in length. The slider-crank (or crank and slider) mechanism shown below is a four-bar
linkage with the slider replacing an infinitely long output link. The corresponding SimDesign file
is simdesign/slider.crank.sim.
Figure 5-5 Crank and Slider Mechanism
This configuration translates a rotational motion into a translational one. Most mechanisms are
driven by motors, and slider-cranks are often used to transform rotary motion into linear motion.
Crank and Piston
You can also use the slider as the input link and the crank as the output link. In this case, the
mechanism transfers translational motion into rotary motion. The pistons and crank in an internal
combustion engine are an example of this type of mechanism. The corresponding SimDesign file
is simdesign/combustion.sim.
Figure 5-6 Crank and Piston
You might wonder why there is another slider and a link on the left. This mechanism has two
dead points. The slider and link on the left help the mechanism to overcome these dead points.
Block Feeder
One interesting application of slider-crank is the block feeder. The SimDesign file can be found
in simdesign/block-feeder.sim
Figure 5-7 Block Feeder
5.2.2 Definitions
In the range of planar mechanisms, the simplest group of lower pair mechanisms are four bar
linkages. A four bar linkage comprises four bar-shaped links and four turning pairs as shown in
Figure 5-8.
Figure 5-8 Four bar linkage
The link opposite the frame is called the coupler link, and the links whick are hinged to the
frame are called side links. A link which is free to rotate through 360 degree with respect to a
second link will be said to revolve relative to the second link (not necessarily a frame). If it is
possible for all four bars to become simultaneously aligned, such a state is called a change point.
Some important concepts in link mechanisms are:
1. Crank: A side link which revolves relative to the frame is called a crank.
2. Rocker: Any link which does not revolve is called a rocker.
3. Crank-rocker mechanism: In a four bar linkage, if the shorter side link revolves and the
other one rocks (i.e., oscillates), it is called a crank-rocker mechanism.
4. Double-crank mechanism: In a four bar linkage, if both of the side links revolve, it is
called a double-crank mechanism.
5. Double-rocker mechanism: In a four bar linkage, if both of the side links rock, it is
called a double-rocker mechanism.
5.2.3 Classification
Before classifying four-bar linkages, we need to introduce some basic nomenclature.
In a four-bar linkage, we refer to the line segment between hinges on a given link as a bar where:
s = length of shortest bar
l = length of longest bar
p, q = lengths of intermediate bar
Grashof's theorem states that a four-bar mechanism has at least one revolving link if
s + l <= p + q
(5-1)
and all three mobile links will rock if
s+l>p+q
(5-2)
The inequality 5-1 is Grashof's criterion.
All four-bar mechanisms fall into one of the four categories listed in Table 5-1:
Case l + s vers. p + q Shortest Bar Type
1 < Frame Double-crank
2 < Side Rocker-crank
3 < Coupler Doubl rocker
4 = Any Change point
5 > Any Double-rocker
Table 5-1 Classification of Four-Bar Mechanisms
From Table 5-1 we can see that for a mechanism to have a crank, the sum of the length of its
shortest and longest links must be less than or equal to the sum of the length of the other two
links. However, this condition is necessary but not sufficient. Mechanisms satisfying this
condition fall into the following three categories:
1. When the shortest link is a side link, the mechanism is a crank-rocker mechanism. The
shortest link is the crank in the mechanism.
2. When the shortest link is the frame of the mechanism, the mechanism is a double-crank
mechanism.
3. When the shortest link is the coupler link, the mechanism is a double-rocker mechanism.
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5.2.4 Transmission Angle
In Figure 5-11, if AB is the input link, the force applied to the output link, CD, is transmitted
through the coupler link BC. (That is, pushing on the link CD imposes a force on the link AB,
which is transmitted through the link BC.) For sufficiently slow motions (negligible inertia
forces), the force in the coupler link is pure tension or compression (negligible bending action)
and is directed along BC. For a given force in the coupler link, the torque transmitted to the
output bar (about point D) is maximum when the angle between coupler bar BC and output bar
CD is /2. Therefore, angle BCD is called transmission angle.
(5-3)
Figure 5-11 Transmission angle
When the transmission angle deviates significantly from /2, the torque on the output bar
decreases and may not be sufficient to overcome the friction in the system. For this reason, the
deviation angle =| /2- | should not be too great. In practice, there is no definite upper limit for
, because the existence of the inertia forces may eliminate the undesirable force relationships
that is present under static conditions. Nevertheless, the following criterion can be followed.
5.2.5 Dead Point
When a side link such as AB in Figure 5-10, becomes aligned with the coupler link BC, it can
only be compressed or extended by the coupler. In this configuration, a torque applied to the link
on the other side, CD, cannot induce rotation in link AB. This link is therefore said to be at a
dead point (sometimes called a toggle point).
Figure 5-10 Dead point
In Figure 5-11, if AB is a crank, it can become aligned with BC in full extension along the line
AB1C1 or in flexion with AB2 folded over B2C2. We denote the angle ADC by and the angle DAB
by . We use the subscript 1 to denote the extended state and 2 to denote the flexed state of links
AB and BC. In the extended state, link CD cannot rotate clockwise without stretching or
compressing the theoretically rigid line AC1. Therefore, link CD cannot move into the forbidden
zone below C1D, and must be at one of its two extreme positions; in other words, link CD is at
an extremum. A second extremum of link CD occurs with = 1.
Note that the extreme positions of a side link occur simultaneously with the dead points of the
opposite link.
In some cases, the dead point can be useful for tasks such as work fixturing (Figure 5-11).
Figure 5-11 Work fixturing
In other cases, dead point should be and can be overcome with the moment of inertia of links or
with the asymmetrical deployment of the mechanism (Figure 5-12).
Figure 5-12 Overcoming the dead point by asymmetrical deployment (V engine)
5.2.6 Slider-Crank Mechanism
The slider-crank mechanism, which has a well-known application in engines, is a special case of
the crank-rocker mechanism. Notice that if rocker 3 in Figure 5-13a is very long, it can be
replaced by a block sliding in a curved slot or guide as shown. If the length of the rocker is
infinite, the guide and block are no longer curved. Rather, they are apparently straight, as shown
in Figure 5-13b, and the linkage takes the form of the ordinary slider-crank mechanism.
Figure 5-13 Slider-Crank mechanism
5.2.7 Inversion of the Slider-Crank Mechanism
Inversion is a term used in kinematics for a reversal or interchange of form or function as
applied to kinematic chains and mechanisms. For example, taking a different link as the fixed
link, the slider-crank mechanism shown in Figure 5-14a can be inverted into the mechanisms
shown in Figure 5-14b, c, and d. Different examples can be found in the application of these
mechanisms. For example, the mechanism of the pump device in Figure 5-15 is the same as that
in Figure 5-14b.
Figure 5-14 Inversions of the crank-slide mechanism
Figure 5-15 A pump device
Keep in mind that the inversion of a mechanism does not change the motions of its links relative
to each other but does change their absolute motions.
