Four bars joined at pivots turn a crank into a rocking arm, if their lengths allow it. Students find a crank that jams, learn the rule that predicts whether it can turn all the way (shortest + longest ≤ the other two), and design a windscreen wiper with a big enough swing.
A windscreen wiper swings back and forth, but its motor turns round and round. Four bars do it. Why can't they be any lengths you like?
A linkage is bars joined at pivots. In a four-bar linkage, one bar is the frame (the ground: it doesn't move, and holds two pivots), the crank turns on one pivot, the rocker swings on the other, and the coupler joins their ends.
Four-bar linkages are everywhere: windscreen wipers, bike pedals and legs, oil pumps, folding chairs, excavator arms.
Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.
Make your prediction first. Help opens once you've locked it in.
PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
A longer crank
TryTry it on the board and compare the result with your prediction.
Try: find the jam
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Turn the crank once. Watch the rocker.
Use the board. This ticks when you get there.
Make the crank 55 mm and turn it.
Use the board. This ticks when you get there.
Try crank lengths between 30 and 55 mm. What is the longest crank that still turns all the way (in mm)?
InvestigateWork out why it happens, one step at a time.
Worked example: the rule
Worked exampleBefore each step, say what you think comes next.
sorted: 30,60,70,80List the four lengths, shortest to longest.
30+80=110Add the shortest and the longest.
60+70=130Add the other two.
110≤130✓No more than the other two: one bar can go all the way round. The crank is the shortest, so it is the crank.
55+80=135>60+70=130×With a 55 mm crank: more than the other two, so no bar can go all the way round. It jams.
Ava thinks her crank turns all the way. Click the line where her working first goes wrong.
ModifyChange the problem on purpose and see what happens.
Vary: a wiper
Need a hint?
A longer crank swings the rocker farther.
Another hint
Keep shortest + longest no more than the other two.
Another hint
Try 40 mm.
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A wiper must swing at least 100°, and its crank must turn all the way. Change the crank and turn it.
Use the board. This ticks when you get there.
PractiseA few on your own, to check you have it.
The rule
Coupler 90, rocker 70 and ground 80 mm. What is the longest crank (shorter than all of them) that still turns all the way, in mm?
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
A wiper to a range
Need a hint?
Start by making the crank longer: that grows the swing.
Another hint
If it jams, check the rule: shortest + longest against the other two.
Another hint
A shorter rocker also swings farther.
Try it yourselfYour edits are saved in this browser.
Design a linkage whose crank turns all the way and whose rocker swings between 110° and 130°. Change any bars and turn it.
Use the board. This ticks when you get there.
For teachers
Students turn a four-bar linkage, find that a longer crank jams, learn the rule for whether the crank can turn fully (shortest + longest ≤ the other two, with the crank shortest), design a wiper with a swing of at least 100°, and search for a linkage whose swing falls in a range. About 45 minutes.
I can name the four bars of a four-bar linkage and what each one does.
I can use the rule to say whether a crank can turn all the way round.
I can change bar lengths to meet a swing criterion.
Before class
Card strips and split pins make linkages students can feel jam. Meccano strips work too.
Simplifications: rigid bars, no friction, pivots with no play, and the linkage is assembled one way up (the board keeps the coupler and rocker meeting above the frame). The rule (Grashof's condition) is about whether some bar can turn fully; which one depends on which bar is shortest.
Curriculum hooks: Design and Technologies, engineering principles and systems (primary); Maths, inequalities and measurement (Years 9–10).
Running the lesson
Predict: most students choose a bigger swing. Keep their reason for after the jam.
Try: have students find the exact longest crank, then check it with the sum.
Worked example: write the four lengths sorted on the board every time.
Vary: the shortest-frame question opens double cranks; a picture of a locomotive's coupling rods helps.
The hard part (on purpose)
Students add the wrong pair, or forget that the crank must be the shortest bar. Have them sort the four lengths before anything else.
When students get stuck
A student adds the wrong two bars. Ask: Which is the shortest and which is the longest? Fix: Sort first, then add the ends of the list.
A student expects a longer crank always to give a bigger swing. Ask: Did it turn all the way? Fix: Only up to the limit the rule sets.
A student is lost after changing several bars. Ask: Change one bar at a time. What did that one change do? Fix: One change, one test.
Stretch and support
Stretch: Find a linkage where the rule holds exactly (shortest + longest = the other two). What does it do at the moment it lines up, and why do engineers avoid it?
Support: Give a table: four lengths, sorted, shortest + longest, other two, turns fully? Fill the first row together.
Where it leads
Next: the Capstone projects (Physical computing) include a lift and a sorter where these mechanisms meet motors.
Real examples: windscreen wipers, bike and leg, oil pump jacks, excavator arms, folding chairs, Theo Jansen's Strandbeest.
a longer crank: A longer crank looks like it simply gives a bigger swing. Committing first makes the jam surprising: past a certain length the bars cannot reach, and the crank cannot turn all the way.
Try: find the jam: Students lengthen the crank in steps and find the last length that still turns: the rule comes after they have met it.
Worked example: the rule: The rule (Grashof's): add the shortest and longest bars; if that is no more than the other two added, one bar can turn all the way, and it is the crank if the crank is the shortest. Ava adds the wrong pair, planted after the method.
Vary: a wiper: A longer crank gives a bigger swing, up to the limit the rule sets: students meet a swing criterion within the rule.
the rule: Apply the rule to new lengths, find a limit, and name what a long crank gives.
a wiper to a range: Several bars may change now: students search a design space with the rule as a guide and the board as the test.
Now you can
Name the four bars of a four-bar linkage and what each one does.
Use the rule to say whether a crank can turn all the way round.