A Brief Introduction
A mechanism transfers motion and force from a source to an output. Mechanisms include gear
trains, linear to rotary converters, clockworks and more. Mechanisms are everywhere you look
from the drive train of your car to the lead screw in your CD player. It is likely that you will use a
mechanism in your robot.
A linkage is a simple mechanism containing rigid bars (the links) connected through pin and
prismatic (linear) joints. Linkages are simple in structure but powerful in function, providing
elegant solutions to motion control problems.
For example, here are two versions of the slider crank, a common linkage,. Images are taken
from http://www.flying-pig.com/mechanisms/, a wonderful site for learning about basic mechanisms
that you should explore.
At the right is a 4-bar linkage that has three moving links and one ground
link. By varying the lengths of the moving links and the location of anchor
joints on the ground link, a surprising range of tip motion can be
You may end up using a linkage in your robot, perhaps the rotary-to-
linear slider crank or a 4-bar. It is likely you will design or specify linkages
in your future career as a professional mechanical engineer. In ME3222
Design and Manufacturing II you will learn how to design mechanisms,
including linkages and will use analysis software to synthesize a linkage
to produce a desired output motion. In this activity you will build three
basic linkages with simple materials. The construction methods, precision
and utility of what you build will not be memorable, but you should walk
away with a new appreciation for mechanisms and many ideas for how to
use linkages in your machines, including your robot.
Deliverables: At home and before the lab, build and run all 3
mechanisms. Along with constructing and measuring the mechanisms,
there are some homework problems which should be completed. Some
of the homework questions may be difficult, but make a good faith effort even through you will not
be graded on whether the equation is correct and your derivations will only be looked at by your
group members . Bring the mechanisms, ruler, completed homework problem and a calculator
with you to lab.
Supplies you need: Sheet of link cutouts, 12 brass fasteners for the joints, office tape to cover
brass heads and anchor ground links. (ME2011 students will get the sheet and the fasteners in
lecture.) Tools you need: scissors and a ruler with a millimeter scale.
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The boxed sections labeled “Lab Exercise” will be done in lab with results entered on a supplied
worksheet. The questions labeled “Homework” are what you should do in advance to prepare for
the lab, but will not be turned in.
The mechanisms work reasonably well if you are careful in your construction methods. Link
members are cut from the sheet. No need to be careful in the outline cuts but do be careful to get
the holes exactly at the marked locations. To make holes for joints, prick with a sharp object or
fold the link in half at the hole and nick with scissors. Make as small a hole as you can for tight
tolerances. To connect links, align the holes and gently push a brass fastener through. Think
about the assembly order so that links can rotate all the way. Spin the fastener which will enlarge
the hole resulting in a smoothly operating joint with no backlash. Bend the legs of the fastener flat
against the paper. After the mechanism is complete, flip it onto its back with the heads up and
tape over any brass fastener heads that tend to catch links as the mechanism operates. When
ready to operate, lightly tape the ground link to the table.
The slider-crank is a basic rotary to linear mechanism. It is handy for generating reciprocating
motion because the motor does not have to reverse direction. A cartoon of a slider crank is on the
front page of this document and an abstracted slider-crank is shown below. Link a, the crank,
travels round in a circle through angle theta. Link c, the slider is constrained by bearings to travel
along the dotted line.
Homework Question #1: Using your high school trigonometry, write the equation, x = f(theta) for
the displacement x of the tip of link c as a function of crank angle theta. Let x = 0 for when theta =
0 with positive x going from right to left as shown in the picture above. Leave link lengths a and b
as variables in the equation. You might want to look up the law of sines and the law of cosines to
help you in deriving the equation.
Homework Question #2: Write a second equation where the link lengths are replaced with the
lengths in mm of the slider crank you will build, a = 40 mm, b = 80 mm.
Homework Question #3 (optional): Imagine pushing against link c with a force F in the direction
shown by the arrow marked x in the figure. Now imagine that there is a motor driving link a and
that the motor is producing a torque T acting in the clockwise direction that results in a force
through link c that perfectly counteracts force F. In other words, the motor is acting through the
slider crank linkage to resist your pushing force so that link c does not move. You can see that
the magnitude of torque T to balance your pushing force F will depend on angle theta. Your task
is to write the equation T = f(theta, F) for the torque at balances pushing force F with the linkage
at angle theta.
LINKAGE #1: Slider crank
Construct the slider crank mechanism shown below which has four links and three joints. The link
proportions are b = 2*a. Anchor the ground link g to the table. Form a U-shaped channel with an
index card to use as a guide so that link c is constrained to travel in a straight line. Tape the
channel to the table. Go back and read the construction techniques if the system does not run
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index card channel
Run the crank (link a) by hand (normally the crank would be anchored to the motor shaft) and
admire your work.
LAB EXERCISE: Set your constructed linkage drive crank to angles shown in the table on the
work sheet. Start with theta = 0 and mark the x = 0 position of the tip of link c on the index card
guide. Then move the crank to the next angle and mark the location of the tip. After all angles are
marked, use the mm scale of a ruler to measure the distance between each mark and the x = 0
mark. Record your data on the worksheet table. Explain any differences between the theoretical
and measured positions. (Warning: Do not “adjust” your measurements to fit the theory as that
would be scientific and engineering fraud.)
In your slider crank, the center of rotation of the crank is in line with the line of action of the
piston. Is it possible to have a slider crank where the crank rotates about a point off the piston
line? Try it.
The piston requires a linear bearing, in this case the index card guide, to constrain the motion of
link c to a straight line. Can you name an easy to make or easy to purchase linear motion
devices? For example, a drawer slide is a linear motion device. Work with your group to come up
with at least 10 linear motion devices.
A 4-bar linkage has four links and four joints. One of the links is grounded and two of the joints
are fixed joints which means what you actually see moving are three links. A generic 4-bar is
Link a is the driver crank and is attached to the motor. Link b is the coupler and link c is the output
crank. The tip of link c traces the desired output. The fourth link d is the ground link which does
not move and is represented by the distance d between the two ground joints. By changing the
lengths of a, b, c and the distance between the ground joints, an infinite set of motions for the
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driven tip can be achieved. A course in kinematics will teach you how to design 4-bars to achieve
a given trace function. In this activity, you will build one 4-bar that traces an approximate straight
line and another that acts like an insect leg. One advantage of doing straight line motion with a 4-
bar rather than a slider-crank is that the 4-bar has all rotary and no sliding joints.
LINKAGE #2: Straight line motion with a 4-bar
This is a remarkable 4-bar linkage where the tip of tracer bar b follows a straight line path over
much of the curve as crank a rotates
For straight line motion, the link lengths are in the proportions a = 2x, b = 10x, c = 5x, d = 4x and
the joint between links b and c is located at the midpoint of link b. Confirm that the link lengths on
your cut out sheet are in these proportions.
Construct this 4-bar. Before assembling link b, cut a small hole, about the size of a marker tip at
the place marked “output”.
LAB EXERCISE: Anchor ground link d to the table and tape a piece of paper under tracer link b.
Operate by rotating link a which would normally be attached to a motor. Every few degrees make
a pen mark at the output of link b as link a rotates 360 degrees. Connect the dots Or, you might
be able to let the link drag a marker across the paper to make a continuous line. You should see
that a good part of the trace approximates a straight line.
Measurement: With a ruler, draw a best-fit straight line through the trace over the part where the
trace is closest to being straight. Mark where the actual trace deviates from the straight line by
more than 5 mm and mark these limits. Measure in mm the length of the straight line trace
between those two limits and enter on the worksheet. Which person in your group was able to get
the straightest line and why?
Challenge: Come up with six robot ideas for which this mechanism would be useful.
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LINKAGE #3: Insect leg
(Adapted from the book “Mechanical Devices” by B. Rorabaugh)
Here is an interesting mechanism that lets your rotary motor drive a leg in somewhat of a walking
motion. Link a is the input crank that is connected to a gear motor. The insect foot is at the tip of
link c. Link d is the ground link which is the robot frame.
For this 4-bar, the proportions are a = 2x b = 5x, c = 10.5x, and the joint between link c and link b
is 5.5x away from the ground pivot on link c. The horizontal distance between the two ground
pivots is 5.0x, and vertical distance between the two pivots is 3.5x.
Build the mechanism. Tape link d to the edge of a notebook at approximately the angle shown in
the drawing where the notebook represents the robot frame. Hold the notebook so that leg c
dangles straight down. Operate the crank and admire your insect. One key feature of this
mechanism is that it converts continuous rotation of a motor to reciprocating motion of a leg.
Challenge: Do you think that a platform with 4 of these legs placed strategically could be one way
to implement a robotic walking machine?
LAB EXERCISE: Run your insect leg in lab.
Challenge: Come up with six robot ideas for which this mechanism would be useful and share
those ideas with your group.
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