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.
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Students will be able to
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.
Timing
Predict and Try 10 min · Worked example 10 min · Vary 10 min · Practise and Make 15 min
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
What you will see
A question to ask
The fix
A student adds the wrong two bars.
Which is the shortest and which is the longest?
Sort first, then add the ends of the list.
A student expects a longer crank always to give a bigger swing.
Did it turn all the way?
Only up to the limit the rule sets.
A student is lost after changing several bars.
Change one bar at a time. What did that one change do?
One change, one test.
Answers to the checks
a longer crank: The crank is 30 mm and turns all the way round. Make it 55 mm (the other bars stay the same). What happens? It jams. With a 55 mm crank, at some angles the coupler and rocker together can't reach the crank's end, so the linkage locks.
Turn it: Turn the crank once. Watch the rocker.
The crank turns all the way; the rocker swings back and forth.
Too long: Make the crank 55 mm and turn it.
It jams: the bars can't reach at some angles.
The longest crank: Try crank lengths between 30 and 55 mm. What is the longest crank that still turns all the way (in mm)? 50 50 mm. 50 + 80 = 130, exactly the other two (60 + 70); any longer and it jams.
Check 30, 80, 60, 70, then 55, 80, 60, 70: Bars 40, 90, 60 and 70 mm (crank 40). Can the crank turn all the way? Yes: 40 + 90 = 130 is no more than 60 + 70 = 130 Shortest + longest = 40 + 90 = 130, the other two = 130. Not more, and the crank is the shortest, so it turns (only just).
Find the mistake: Ava thinks her crank turns all the way. Click the line where her working first goes wrong. line 2 Line 2: add the shortest and the longest: 55 + 80 = 135. That is more than 60 + 70 = 130, so the crank jams.
A wiper: A wiper must swing at least 100°, and its crank must turn all the way. Change the crank and turn it.
A 40 mm crank swings the rocker 100°, and 40 + 80 ≤ 60 + 70.
Shortest frame: If the shortest bar is the frame (the ground), and the rule holds, what happens? Both the crank and the rocker can turn all the way round With the frame shortest, both bars on fixed pivots can turn fully: a double crank, used in some train wheels and drag-link mechanisms.
Yes or no: Bars 20, 50, 40 and 45 mm (crank 20). Can the crank turn all the way? Yes 20 + 50 = 70, no more than 40 + 45 = 85.
Again: Bars 35, 100, 50 and 70 mm (crank 35). Can the crank turn all the way? No 35 + 100 = 135, more than 50 + 70 = 120: it jams.
A limit: 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? 60 crank + 90 ≤ 70 + 80 = 150, so the crank can be up to 150 − 90 = 60 mm.
Into range: Design a linkage whose crank turns all the way and whose rocker swings between 110° and 130°. Change any bars and turn it.
For example crank 40, coupler 70, rocker 50, ground 70 (110°), or crank 45, coupler 80, rocker 60, ground 70 (118°).
Solutions
One possible solution for each task. Other correct programs are fine.
Vary: a wiper › A wiper
Open task: any program that does what the step asks.
Vary: a wiper › Shortest frame
Open task: any program that does what the step asks.
a wiper to a range › Into range
Open task: any program that does what the step asks.
a wiper to a range › Evaluate
Open task: any program that does what the step asks.
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.
Exit ticket
Bars 25, 75, 55 and 60 mm, with the 25 mm bar as the crank. Can the crank turn all the way? Show the sums.
25 + 75 = 100, 55 + 60 = 115; 100 ≤ 115 and the crank is shortest, so yes.
What each step is for
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.
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.