Introduction to 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.
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
Take 4 pins and assemble them as shown in Figure
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
In planar mechanisms, there are only two kinds of
lower pairs ---
revolute pairs and
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
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:
- Continuous rotation into continuous rotation, with a constant or
variable angular velocity ratio.
- Continuous rotation into oscillation or reciprocation (or the
reverse), with a constant or variable velocity ratio.
- 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.
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:
- the four-bar linkage mechanism, and
- the slider-crank 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
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.
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
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:
- Crank: A side link which revolves relative to the frame is
called a crank.
- Rocker: Any link which does not revolve is called a rocker.
- 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.
- Double-crank mechanism: In a four bar linkage, if both of the
side links revolve, it is called a double-crank mechanism.
- Double-rocker mechanism: In a four bar linkage, if both of the
side links rock, it is called a double-rocker mechanism.
Before classifying four-bar linkages, we need to introduce some
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
and all three mobile links will rock if
s + l > p + q
The inequality 5-1 is Grashof's criterion.
All four-bar mechanisms fall into one of the four categories listed in
Table 5-1 Classification of Four-Bar Mechanisms
|| l + s vers. p + q
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:
- When the shortest link is a side link,
the mechanism is a crank-rocker mechanism. The shortest
link is the crank in the mechanism.
- When the shortest link is the frame of the
mechanism, the mechanism is a double-crank mechanism.
- When the shortest link
is the coupler link, the mechanism is a double-rocker mechanism.
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.
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.
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)
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
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
- 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
- 5.1 Introduction
- 5.1.1 What are Linkage Mechanisms?
- 5.1.2 Functions of Linkages
- 5.2 Four Link Mechanisms
- 5.2.1 Examples
- 5.2.2 Definitions
- 5.2.3 Classification
- 5.2.4 Transmission Angle
- 5.2.5 Dead Point
- 5.2.6 Slider-Crank Mechanism
- 5.2.7 Inversion of the Slider-Crank Mechanism
- 6 Cams
- 7 Gears
- 8 Other Mechanisms