A crank-rocker mechanism is a four-bar linkage in which the shortest link rotates fully as the input crank while one of the adjacent links oscillates back and forth as the rocker. Franz Reuleaux formalised the kinematic analysis of these linkages in his 1875 work Theoretische Kinematik, which still underpins how we classify them today. The crank converts continuous shaft rotation into a bounded swing at the rocker, giving designers a cheap, reliable way to drive oscillating tools, beams, or wipers from a single motor. You see it on every windscreen wiper linkage and on millions of oil-field pumpjacks worldwide.
Crank-rocker Mechanism Interactive Calculator
Vary the four link lengths to check the Grashof condition, estimated rocker swing, transmission angle, and linkage risk.
Equation Used
The calculator sorts the four link lengths to test the Grashof condition. A positive margin means s + l is less than or equal to p + q. For a crank-rocker, the crank should also be the shortest link. The swing estimate uses the two dead-centre distances a + b and |b - a|.
- Planar four-bar with ground, crank, coupler, and rocker in order.
- The input crank is intended to be the shortest link.
- Links are rigid and pin clearances are neglected.
- Rocker swing is estimated from the two dead-centre positions.
Inside the Crank-rocker Mechanism
A crank-rocker is one of four Grashof-class four-bar linkages. You have four rigid links — ground, crank, coupler, and rocker — joined by four revolute pivots. For the crank to make full 360° rotations while the rocker only swings, the link lengths must satisfy the Grashof condition: s + l ≤ p + q, where s is the shortest link, l the longest, and p and q the other two. The shortest link must also be the crank. Get this wrong and you end up with a double-rocker or a drag-link, and the input shaft binds before completing a turn.
The crank drives the coupler, which pushes and pulls the rocker through a swing angle that depends purely on geometry. Two extreme positions — the dead-centre positions — define the swing limits. Between these extremes the rocker accelerates, decelerates, stops, reverses, and repeats. Because the forward and return strokes do not take the same crank angle, you get a time ratio (Q) greater than 1 — the return is faster than the working stroke, which is exactly what you want on a shaper, a pumpjack, or any cutting tool with a load-bearing forward pass.
Where it goes wrong: if the transmission angle (the angle between the coupler and the rocker) drops below about 40°, the linkage loses mechanical advantage and the pins start to take radial loads they were not designed for. You will hear knocking, see oval pin holes after a few hundred hours, and eventually the bushings will pound out. Pivot clearance also matters — slop above 0.1 mm at any joint stacks up across four pivots and shows as visible chatter at the rocker tip.
Key Components
- Ground link (frame): The fixed reference between the crank pivot and the rocker pivot. Its length sets the geometry of the entire linkage. On a steel weldment the centre-distance tolerance should be held to ±0.1 mm or you will throw off the swing angle by more than 1°.
- Crank: The shortest link, driven in continuous 360° rotation by a motor or gearbox. Typical input speeds run 30 to 300 RPM in industrial use. The crank pin must run on a bushing or needle bearing rated for full reversed loading every revolution.
- Coupler (connecting rod): Transmits force from the crank pin to the rocker pin. It carries pure tension and compression in the ideal case, but any pivot misalignment forces it into bending. Coupler stiffness sets the high-frequency behaviour — a flexible coupler shows up as lag at the rocker tip above ~120 RPM.
- Rocker: The output link, swinging through a bounded angle typically 60° to 120° depending on link ratios. The rocker pivot bearing sees the highest cyclic load in the system because of the lever arm to the working tool.
- Revolute joints (4 pivots): Each pin joint must have radial clearance under 0.05 mm for precision builds. Use hardened pins (60 HRC) running in oil-impregnated bronze or needle bearings. A single sloppy joint will dominate end-effector error.
Industries That Rely on the Crank-rocker Mechanism
Crank-rockers show up anywhere you need cheap, repeatable oscillation from a rotating shaft. They handle high cyclic loads when sized correctly, run for decades with minimal maintenance, and need only one motor to drive a complex motion. The geometry also gives you a built-in quick-return action for free, which is why machine-tool designers reached for it long before servos and ball-screws existed. Pivot wear is the main long-term failure mode — once the pin holes elongate, the time ratio drifts and the swing angle grows, which is why you grease the joints and inspect for ovality.
- Oil & gas: Lufkin Mark II pumpjacks use a crank-rocker geometry to convert gearbox rotation into the up-and-down stroke of the polished rod down the well casing.
- Automotive: Bosch windscreen wiper linkages on the VW Golf Mk7 use a crank-rocker pair to swing each wiper arm through roughly 90° from a single motor.
- Machine tools: The Cincinnati shaper Model 24 uses a crank-rocker quick-return linkage to drive the cutting ram, giving a slow forward cut and fast return.
- Textile machinery: Sulzer projectile weaving looms use crank-rockers to drive the heddle frames up and down for shed formation at speeds above 300 picks per minute.
- Agriculture: John Deere pull-type sickle mowers use a crank-rocker to oscillate the cutter bar back and forth at around 1000 strokes per minute.
- Toys and animatronics: Disney audio-animatronic figures from the 1960s used crank-rocker linkages driven by cam-shafts to nod, wave, and bow with repeatable choreography.
The Formula Behind the Crank-rocker Mechanism
The swing angle of the rocker — call it Δθ — is what you actually feel at the output. It depends only on link lengths, not on speed or load, which is why crank-rockers are so predictable. At the low end of typical link-ratio range (rocker length close to ground length) you get a tight 30-50° swing suitable for valve actuators and small wipers. At the nominal mid-range you land around 60-90°, which covers most pumpjacks and shapers. Push the geometry to the high end and you can hit 120° swings, but transmission angle drops below 40° at the extremes and the linkage starts knocking. The formula below uses the law of cosines on the two dead-centre triangles to give you the swing angle directly.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| Δθ | Total rocker swing angle between dead-centre positions | degrees or radians | degrees |
| r1 | Crank length (shortest link) | mm | in |
| r2 | Ground link length (fixed frame) | mm | in |
| r3 | Coupler length | mm | in |
| r4 | Rocker length | mm | in |
Worked Example: Crank-rocker Mechanism in a Glassware Annealing Lehr Conveyor Pusher
A glass-bottle plant in Veracruz is retrofitting the entry pusher on a Pennekamp annealing lehr. The pusher arm needs to oscillate back and forth to nudge hot bottles onto the mesh belt at roughly 80 cycles per minute. Geometry: crank r1 = 60 mm, ground r2 = 200 mm, coupler r3 = 220 mm, rocker r4 = 180 mm. The Grashof check passes (60 + 220 = 280 ≤ 200 + 180 = 380), so the crank rotates fully and the rocker oscillates. Find the rocker swing angle and the resulting pusher tip travel for a 250 mm pusher arm.
Given
- r1 = 60 mm
- r2 = 200 mm
- r3 = 220 mm
- r4 = 180 mm
- Ltip = 250 mm
Solution
Step 1 — compute the rocker angle at the extended dead-centre position (crank and coupler in line, lengths add):
Step 2 — compute the rocker angle at the folded dead-centre position (coupler and crank overlap, lengths subtract):
Step 3 — nominal swing angle is the difference, and tip travel comes from the chord across that arc at the 250 mm tip radius:
At the low end of the typical pusher range — say you shorten the crank to r1 = 40 mm — Δθ drops to roughly 29° and the tip travel falls to about 125 mm. That is too short to clear a 110 mm bottle reliably; the pusher will graze the next bottle in line. At the high end, lengthening the crank to r1 = 80 mm pushes Δθ up to about 63° and tip travel to roughly 263 mm, but the minimum transmission angle drops to around 32°, which is below the 40° rule of thumb. The linkage will knock at the dead-centre and you will see pin-hole ovality within a few hundred shifts. The 60 mm crank is the sweet spot.
Result
Nominal rocker swing is 45. 3° giving a pusher tip travel of 193 mm at 250 mm radius — enough to push a bottle clear of the next one in line with margin. The 40 mm crank case gives only 125 mm of tip travel which is too tight, while the 80 mm case delivers 263 mm but at a transmission angle that hammers the joints, so 60 mm is the right call. If your built linkage measures less than the predicted 193 mm, suspect three things first: (1) frame weldment distortion shifting r2 beyond ±0.5 mm, which directly biases both dead-centre angles, (2) coupler bending under load if you used a thin-section bar instead of a 12 mm wall tube, and (3) crank-pin bushing wear opening up the effective r1 by 0.3-0.5 mm, which broadens the swing in an uncontrolled way and shows up as a growing time ratio drift over months.
When to Use a Crank-rocker Mechanism and When Not To
A crank-rocker is not the only way to get oscillating motion from a motor. The two main alternatives are a Scotch yoke — which gives pure sinusoidal output from a slot and pin — and a cam-and-follower with a return spring. Each has a different set of strengths on the dimensions that actually matter when you are sizing a machine.
| Property | Crank-rocker | Scotch yoke | Cam-and-follower |
|---|---|---|---|
| Typical operating speed | 30-500 RPM | 0-1500 RPM | 0-3000 RPM |
| Output motion profile | Asymmetric quick-return swing | Pure sinusoidal linear | Arbitrary, designed to suit |
| Cost (relative) | Low — 4 pins, 4 links | Medium — slotted yoke needs ground finish | High — cam grinding is precision work |
| Lifespan at rated load | 20,000-100,000 hours with greased bushings | 5,000-15,000 hours (slot wear) | 10,000-50,000 hours (cam pitting) |
| Load capacity | High — pure pin loading | Medium — slot side loads limit force | High but follower-spring dependent |
| Geometric complexity | Low — algebraic link sizing | Very low | High — needs cam-profile computation |
| Best application fit | Pumpjacks, shapers, wipers | Compressor pistons, vibration tables | Engine valves, indexing |
Frequently Asked Questions About Crank-rocker Mechanism
Run two checks. First, the Grashof inequality: s + l ≤ p + q, where s is the shortest link and l the longest. If that fails you have a non-Grashof linkage that cannot complete a full crank rotation. Second — and this is the one people miss — the shortest link must be adjacent to the ground, and it must be the link you intend to drive. If the shortest link is the coupler, you get a double-rocker; if it is the ground link, you get a drag-link with two full rotators. Sketch the four links to scale on paper and rotate the crank with a pencil through every 30° before you commit to a weldment.
Time ratio is set by the angle between the two dead-centre positions of the crank, and that geometry is sensitive to effective link lengths. As the crank-pin and rocker-pin bushings wear, the effective r1 and r4 increase by the radial clearance — typically 0.2-0.5 mm per joint after heavy service. The dead-centre angles drift, the forward stroke takes a slightly different fraction of the cycle, and Q creeps up. If you have a critical time-ratio application, gauge the pin holes annually and replace the bushings once ovality exceeds 0.15 mm.
If your output motion needs to be sinusoidal — say you are driving a vibration test or a piston that benefits from harmonic motion — go Scotch yoke. If you want quick-return (slow working stroke, fast return) or you need to carry serious force at the output, go crank-rocker. The crank-rocker also wins on lifespan because all four joints are pure revolute pairs running on bushings, while the Scotch yoke's sliding pin-in-slot pair wears out 3-5× faster at the same load.
Transmission angle is the angle between the coupler and the rocker at any crank position. It controls how much of the coupler force goes into useful torque on the rocker versus radial pin load. Below 40° the cosine term drops fast and the radial component dominates — pins start hammering, and you get audible knocking. The 40° rule is conservative; for low-speed light-load builds you can run down to 30°, but for anything above 100 RPM or carrying real force, hold ≥ 45° at both extremes. Compute it at the two dead-centres because that is where it is worst.
Almost always pivot slop. Each of the four pin joints contributes a small angular error proportional to its radial clearance divided by the link length to the next pin. Stack four sloppy joints with 0.2 mm clearance each on short links and you can easily see 3-5° of extra swing at the rocker — beyond what the geometry should give. Check by indicating each pin while loading the rocker by hand. The other suspect is a mis-machined ground link: if r2 is short by even 1-2 mm, the dead-centre angles shift and the swing widens.
You can, but the rocker position is a non-linear function of crank angle, so equal stepper increments do not give equal rocker increments. Near the dead-centres the rocker barely moves for tens of crank degrees; mid-stroke it sweeps fast. If you need linear indexing at the rocker, build a lookup table from the position equations, or pick a different mechanism entirely — a rack-and-pinion or a ball-screw will give you proportional motion without the dead-centre stall zones.
References & Further Reading
- Wikipedia contributors. Four-bar linkage. Wikipedia
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