Linkages and four-bar mechanisms for QCE Engineering Unit 4
“Explain how linkages and the four-bar mechanism transmit and change motion, identify common linkage types, and analyse the motion they produce from their geometry”
A linkage is a chain of rigid links connected by pivoted joints that transmits and transforms motion. The four-bar linkage, the most important, has four links (one of them fixed, the frame) joined in a loop; by choosing the link lengths it can act as a crank-rocker (continuous rotation in, oscillation out), a double-crank or a double-rocker. The crank-slider converts rotary motion into reciprocating linear motion (engine pistons, pumps). Simpler linkages include the reverse-motion linkage, which reverses direction, the bell-crank, which changes the direction of a force through an angle, and the parallel-motion linkage, which keeps a part level. The output motion is set entirely by the geometry: the link lengths and pivot positions.
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What this dot point is asking
QCAA wants you to explain linkages: assemblies of rigid bars (links) joined by pivots that transmit force and transform one motion into another. You need to recognise the common arrangements, especially the four-bar mechanism and the crank-slider, and work out the motion they produce from their link lengths. Linkages sit alongside cams and gears as a core way mechanisms reshape motion.
The answer
What a linkage does
A linkage is a set of rigid bars, called links, joined at pivots (pin joints). One link is usually fixed to the machine frame; the others move. Because the links are rigid and the joints constrain how they move relative to each other, a linkage transmits force and transforms an input motion at one link into a specific, repeatable output motion at another. Linkages are valued for being simple, strong and reliable, with no teeth to wear or belts to slip.
The four-bar linkage
The four-bar linkage is the fundamental closed-loop linkage: four links joined in a loop, with one link fixed as the frame. The behaviour depends entirely on the relative link lengths:
- Crank-rocker: one link rotates fully (the crank) while another swings back and forth (the rocker). This converts continuous rotation into oscillation, as in a windscreen-wiper drive.
- Double-crank (drag-link): both the input and output links rotate fully.
- Double-rocker: neither input nor output link makes a full rotation; both oscillate.
Grashof's idea captures which type you get: if the sum of the shortest and longest links is no greater than the sum of the other two, at least one link can rotate fully.
The crank-slider
The crank-slider replaces one pivoted link with a sliding block. A rotating crank drives a connecting rod, which pushes a slider back and forth in a straight track. This converts rotary motion into reciprocating linear motion (and the reverse), and it is the mechanism inside every piston engine, pump and compressor.
Simpler linkages
- Reverse-motion linkage: a single link pivoted in the middle so the output moves opposite to the input.
- Bell-crank linkage: an L-shaped link that changes the direction of a force or motion through an angle, as in a bicycle brake.
- Parallel-motion (parallelogram) linkage: keeps a component at a constant angle while it moves, used in drawing boards and toolboxes.
Find the stroke of a crank-slider
A crank-slider has a crank of length rotating about a fixed pivot, connected by a rod to a slider that moves along a straight line through the pivot. Find the total stroke (the distance the slider travels between its two extreme positions).
In a crank-slider whose slider line passes through the crank pivot, the slider is furthest out when the crank and rod are in line on the far side, and closest in when the crank points back toward the slider. The difference between these two positions is twice the crank radius:
So the slider reciprocates over a stroke for each full turn of the crank. This is why engine designers speak of stroke: it is set by the crank throw, here , giving a travel. Doubling the crank length would double the stroke.
Why this matters for machines and mechanisms
Linkages are how engineers produce a precise path or convert motion using only bars and pins, with no slipping or meshing parts. The four-bar and crank-slider appear in engines, pumps, robot arms, suspensions and countless tools. Choosing the link lengths to get the required output motion, and checking which links can rotate fully, is a direct application of geometry to design in the Unit 4 engineered solution.
- Forgetting one link is the fixed frame
- A four-bar linkage has four links, but one is the stationary frame. Counting only the three moving bars misreads the mechanism.
- Assuming any four-bar gives full rotation
- Whether a link can rotate fully depends on the link lengths (the Grashof condition). Some four-bar linkages only rock.
- Confusing crank-slider stroke with crank length
- The stroke is twice the crank radius, not the crank radius itself, when the slider line passes through the pivot.
- Treating a linkage like a gear for ratios
- Linkage output speed varies through the cycle and depends on geometry and angle; it is not a single fixed ratio like a gear pair.
Exam-style questions
Questions in the style of QCAA exam questions on this dot point, each with a worked answer. They are written by ExamExplained unless tagged "Past paper"; the year shows the paper a question is modelled on.
Practice6 marksIn a crank-slider mechanism the crank has a radius of . Determine the maximum linear displacement (stroke) of the slider, and explain how the stroke would change if the crank radius were doubled.Show worked answer →
A 6 mark determine-and-explain question rewards the stroke geometry then the proportional reasoning.
In a crank-slider, the slider stroke equals twice the crank radius, because the slider moves from one extreme (crank aligned outward) to the other (crank aligned inward): .
If the crank radius doubled to , the stroke would also double to , because stroke is directly proportional to crank radius.
Markers reward the relationship , the value of , and the correct conclusion that doubling the radius doubles the stroke.
Practice4 marksExplain how a four-bar linkage in a crank-rocker arrangement converts continuous rotation of one link into oscillation of another, and identify what fixes this behaviour.Show worked answer →
A 4 mark explain answer needs the input-output motion and the role of link lengths.
In a crank-rocker four-bar linkage the shortest link (the crank) can rotate fully while the opposite link (the rocker) swings back and forth between two limits, so continuous rotary input becomes oscillating output. The coupler transmits motion between them. Whether a four-bar acts as a crank-rocker is fixed by the relative link lengths (the Grashof condition), so the geometry, not the speed, determines the motion produced.
Markers reward describing rotary-in to oscillating-out and identifying the link-length geometry as what determines the behaviour.