A Grashof linkage is a four-bar linkage whose link lengths satisfy the Grashof condition — the sum of the shortest and longest links is less than or equal to the sum of the other two — which guarantees at least one link can rotate fully relative to another. The rule predicts continuous-rotation behaviour to within link-length tolerance of about ±0.1 mm on a 100 mm crank. Engineers use it to decide whether a four-bar will run as a crank-rocker, drag-link, or double-rocker. You see it in oil-field pumpjacks, the Theo Jansen Strandbeest leg, and Bosch windscreen-wiper linkages.
Grashof Linkage Interactive Calculator
Vary the four link lengths and see whether the Grashof condition allows continuous rotation.
Equation Used
The Grashof test compares the shortest-plus-longest link sum with the sum of the remaining two links. If s + l is less than or equal to p + q, at least one link can rotate continuously depending on which link is grounded. The margin is calculated as (p + q) - (s + l).
- Inputs are entered in Grashof label order: s is shortest, l is longest, and p/q are the other two links.
- A nonnegative margin means the linkage satisfies the Grashof condition.
- Kinematic inversion and transmission angle are not calculated in this compact condition check.
Operating Principle of the Grashof Linkage
The Grashof condition is the gatekeeper test for any four-bar linkage. Label the four link lengths s (shortest), l (longest), and p, q (the other two). If s + l ≤ p + q, the linkage is Grashof and at least one link can make a full 360° revolution about its neighbour. If s + l > p + q, it is non-Grashof and every link is stuck oscillating — you get a double-rocker no matter which link you ground. That single inequality decides whether your motor input can drive the linkage continuously or whether you need a reciprocating actuator.
Which link you fix to ground decides the kinematic inversion. Ground the shortest link and you get a double-crank, also called a drag-link — both side links rotate fully. Ground a link adjacent to the shortest and you get a crank-rocker — the short link cranks, the opposite link rocks. Ground the link opposite the shortest and you get a double-rocker, even though the linkage is technically Grashof. The Strandbeest leg, the windscreen wiper, and the bicycle pedal-and-chainstay loop are all the same Grashof family viewed through different inversions.
Tolerance matters more than people expect. If your link lengths drift so that s + l equals p + q exactly, you have a change-point or folding linkage — at two positions per revolution the linkage goes dead and the output direction becomes ambiguous. On a 100 mm crank, that's a manufacturing window of about ±0.1 mm before behaviour shifts. Push past it and the mechanism either binds or skips through the dead point with a clunk. The other failure mode is a poor transmission angle: when the angle between coupler and output rocker drops below roughly 40°, force transmission collapses, the bearings see large side loads, and the linkage stalls under load even though kinematically it should still rotate.
Key Components
- Ground link (frame): The fixed reference link. Its length sets the spacing between the two pivot bearings on the chassis. On a typical small mechanism this is held to ��0.05 mm centre-to-centre, because pivot spacing error directly subtracts from the Grashof margin.
- Crank (input link): The link driven by the motor or actuator. In a Grashof crank-rocker this is the shortest link and rotates a full 360°. Length tolerance of ±0.1% is normal; bore-to-bore length is what counts, not the outside dimension.
- Coupler (floating link): The link that connects crank to rocker and carries no fixed pivot. It traces a coupler curve — the path used in walker legs, film advance mechanisms, and dwell linkages. Its length and the choice of tracer point on it dominate the output trajectory.
- Rocker or output link: The link that delivers motion to the load. In a crank-rocker it oscillates between two extreme angles; in a drag-link it rotates fully. Transmission angle between coupler and rocker should stay above 40° through the cycle to avoid stall.
- Revolute joints (pin pivots): Four single-degree-of-freedom pin joints connect the links. Bushing or bearing radial play of more than about 0.05 mm on a 100 mm linkage shows up as backlash at the output and effectively shortens the Grashof margin.
Where the Grashof Linkage Is Used
Grashof linkages show up anywhere you need a motor to produce rotation, oscillation, or a specific coupler curve from one continuous input. The decision a designer makes is rarely whether to use a four-bar — it's which inversion to use and where to put the tracer point. Get the link-length ratios wrong and the mechanism either jams at a change point or stalls through a low transmission angle; get them right and the same four pin joints serve a pumpjack and a walker robot.
- Oil and gas: Lufkin conventional beam pumpjacks use a crank-rocker Grashof linkage to convert prime-mover rotation into vertical rod-string stroke at the well horsehead.
- Automotive: Bosch and Valeo windscreen wiper linkages run a crank-rocker four-bar to swing each wiper arm through a 90-110° arc from a single rotary motor.
- Kinetic art and robotics: Theo Jansen's Strandbeest uses an 11-bar linkage built from coupled Grashof four-bars to produce the characteristic walking foot-path.
- Agricultural machinery: Round-baler pickup-tine bars on John Deere and New Holland balers run drag-link Grashof inversions to keep tines vertical while the carrier rotates.
- Aerospace: Boeing 737 Krueger leading-edge flap deployment linkages use a four-bar Grashof crank-rocker to swing the flap forward and down from the wing leading edge.
- Consumer products: Sewing-machine needle-bar mechanisms on Singer and Brother domestic machines are crank-rocker Grashof linkages translating motor rotation into reciprocating needle stroke.
The Formula Behind the Grashof Linkage
The Grashof inequality is the single test that decides what motion you'll get out of a four-bar before you cut a single piece of metal. Run the numbers at the low end of your link-length tolerance band and you can predict whether the worst-case build still rotates fully; run them at the nominal lengths to see how much margin you have; run them at the high end to see how close you are to the change-point boundary. The sweet spot is a Grashof number of roughly 10-20% of the longest link — enough margin that manufacturing tolerance and pivot wear won't push you across the boundary, but not so much that transmission angle suffers at the extremes.
G = (p + q) − (s + l)
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| s | Length of the shortest link | m | in |
| l | Length of the longest link | m | in |
| p | Length of one intermediate link | m | in |
| q | Length of the other intermediate link | m | in |
| G | Grashof margin — positive means Grashof, zero means change-point, negative means non-Grashof | m | in |
Worked Example: Grashof Linkage in a paper-folding feeder on a Heidelberg Stahlfolder
A bindery in Leipzig is rebuilding the sheet-grip kicker on a Heidelberg Stahlfolder TH 56 paper-folding line. The kicker swings a rubber-faced finger through a 75° arc to flick each folded signature off the delivery belt at 180 cycles per minute. The crew has four candidate link-length sets and needs to know which give a clean crank-rocker, which sit too close to the change-point, and which won't run at all. Nominal target lengths are crank 40 mm, coupler 120 mm, rocker 100 mm, ground 140 mm.
Given
- s (crank) = 40 mm
- p (coupler) = 120 mm
- q (rocker) = 100 mm
- l (ground) = 140 mm
Solution
Step 1 — identify shortest and longest links and apply the Grashof test at nominal lengths:
p + q = 120 + 100 = 220 mm
Gnom = 220 − 180 = +40 mm
G is positive, so the linkage is Grashof. The shortest link (the 40 mm crank) is adjacent to the ground link, so grounding the 140 mm gives a crank-rocker — exactly what the kicker needs. The 40 mm margin is roughly 29% of the longest link, which is healthy headroom against tolerance and wear.
Step 2 — recheck at the low end of the tolerance band, where the shop's machinist can hold ±0.3 mm bore-to-bore on each link. Worst case for Grashof margin is shortest and longest growing while intermediates shrink:
p + q = 119.7 + 99.7 = 219.4 mm
Glow = 219.4 − 180.6 = +38.8 mm
Margin barely moves — still firmly Grashof. At ±0.3 mm machining tolerance the linkage is bulletproof. You'd have to be off by more than 19 mm on a single link to reach the change-point, which is well outside any plausible build error.
Step 3 — check what happens if the bindery tries to shorten the ground link to 200 mm to fit a tighter chassis cutout, keeping the other three lengths fixed:
p + q = 120 + 100 = 220 mm
Ghigh = 220 − 240 = −20 mm
G is now negative — the linkage is non-Grashof and the crank can no longer rotate fully. Drive it with a motor and it will jam after roughly 70° of input rotation. The crew either keeps the 140 mm ground link or extends the coupler and rocker to compensate. Anywhere between G = +10 mm and G = +50 mm is the practical sweet spot for this size of mechanism — enough margin to absorb wear, not so much that transmission angle suffers near the rocker dead-points.
Result
At nominal link lengths the linkage is Grashof with a margin of +40 mm and runs cleanly as a crank-rocker — the kicker finger swings its full 75° arc at 180 cycles per minute with comfortable transmission angle. The low-end tolerance check at +38.8 mm shows the build is robust against the machinist's ±0.3 mm bore-to-bore variation, while the high-end test with a shortened 200 mm ground link goes negative at −20 mm and would jam the motor in service. If the rebuilt kicker measures fine on the bench but stalls in production, suspect three things in this order: (1) pivot bushing radial play above 0.05 mm on the coupler-to-rocker pin, which silently eats Grashof margin and shows up as intermittent dead-point hangs; (2) coupler bore-centre distance machined long by more than 0.5 mm, which pushes transmission angle below 40° at the end of the stroke and stalls under signature load; (3) ground-link mounting holes drilled at 138 mm instead of 140 mm, a 2 mm error that is invisible to eye but trims the Grashof margin by 5%.
Choosing the Grashof Linkage: Pros and Cons
Picking a Grashof four-bar is rarely the only option. Cams, slider-cranks, and six-bar linkages all compete for the same job. The right choice depends on whether you need a specific output curve, how much rotational speed you can sustain, and how tight your packaging is.
| Property | Grashof four-bar | Slider-crank | Cam-follower |
|---|---|---|---|
| Maximum input speed | Up to ~3000 RPM with rolling-element pivots | Up to ~6000 RPM (engine connecting rods) | Limited by follower-spring resonance, typically ≤2000 RPM |
| Output motion type | Rotation, oscillation, or coupler curve | Pure linear reciprocation | Any programmed motion profile |
| Path-shape flexibility | Coupler curve fixed by link ratios; limited tuning | Sinusoidal stroke only | Arbitrary — defined by cam profile |
| Component count | 4 links, 4 pin joints | 3 links, 1 slider, 3 pin joints | Cam + follower + return spring + bearings |
| Manufacturing cost | Low — turned pins and machined links | Low — but slider needs precision bore | High — cam profile requires CNC grinding |
| Sensitivity to tolerance | ±0.1 mm on 100 mm link risks change-point shift | Slider clearance dominates; 0.02 mm typical | Cam profile error directly maps to follower error |
| Service life | 10⁸+ cycles with bronze bushings | 10⁷ cycles, slider seal limited | 10⁷-10⁸ cycles, cam wear dependent on lubrication |
Frequently Asked Questions About Grashof Linkage
You're almost certainly hitting a transmission-angle problem, not a Grashof problem. The Grashof condition only tells you whether full rotation is geometrically possible — it says nothing about force transmission. When the angle between coupler and output rocker drops below about 40°, the component of coupler force that actually drives the rocker collapses and the rest goes into the bearings as side load.
Plot the transmission angle through one full revolution. If it dips below 40° anywhere, lengthen the coupler or the rocker by 5-10% and re-run. Lufkin pumpjack designers target a minimum of 45° to keep stall margin under cold-oil starting torque.
Both are Grashof, but they ground different links. Ground the shortest link and you get a drag-link — both side links rotate fully, useful when you need rotary output that's phase-shifted or non-uniform versus the input. Ground a link adjacent to the shortest and you get a crank-rocker — input rotates, output oscillates, which is what you want for wiper arms, kickers, and pumpjack horseheads.
If your driven device wants oscillation, crank-rocker. If it wants modulated rotation (think round-baler pickup tines that need to stay vertical), drag-link. Drag-links also need a more compact ground link, which can fight your packaging.
A change-point linkage occurs when s + l exactly equals p + q. At two positions per revolution, all four links become collinear and the mechanism has a momentary second degree of freedom — it can choose either branch of motion. Real builds with bushing slop or unequal coupler inertia will pick a direction at random, which is why you see the output sometimes reversing or stuttering.
If you're close to a change-point by accident, add 2-3% to one link length to get clear of it. If you intentionally want a change-point (some folding mechanisms exploit it), you must add a guide spring or geometric stop to bias the branch selection.
Link lengths scaled, but pivot clearances did not. A 0.05 mm radial bushing clearance is 0.05% of a 100 mm link but 0.1% of a 50 mm link — relative slop doubles. Worse, the absolute Grashof margin shrinks linearly with scale while machining tolerance often stays the same in absolute terms, so a +2 mm margin on the big version becomes +1 mm on the small version, and your ±0.3 mm bore-to-bore tolerance suddenly eats a third of your margin.
When scaling down, either tighten manufacturing tolerance proportionally, switch to ground pins with H7 bores, or deliberately increase the Grashof number on the smaller version by 20-30%.
Non-Grashof linkages — double-rockers — are perfectly valid when you want bounded oscillation on both input and output and you're driving with a reciprocating actuator like a hydraulic cylinder or a lead-screw. Aircraft landing gear retract linkages, excavator bucket linkages, and many vehicle suspension links are deliberately non-Grashof because the input is itself oscillatory.
Don't use non-Grashof if your input is a rotating motor without a reversal mechanism. The motor will hit the dead-point limit and either stall or break a link. The decision is driven by your input source, not by Grashof being inherently better or worse.
Rule of thumb: keep absolute manufacturing tolerance below 10% of your Grashof margin G. If G = +20 mm, tolerance per link should be ±2 mm worst case across all four links combined — practically that's about ±0.5 mm per link. Tighter than that buys nothing; looser and you'll see units at the end of the production tolerance stack-up exhibiting different output strokes.
The hidden tolerance is pivot-pin bore-to-bore distance, not the outside link dimension. Drill jigs or single-setup CNC machining the two bores in one operation eliminates 80% of the variation seen in field-built linkages.
Every point on the coupler traces a different curve as the linkage runs. Points near the coupler-rocker pin trace something close to a circular arc; points near the coupler-crank pin trace a more complex egg-shape; points well off the coupler line can trace approximate straight lines (Watt's linkage) or figure-eights (Strandbeest foot path). The coupler curve is locked to the link ratios — you can't change it without changing the linkage.
Pick your tracer point by simulating in software like Linkage or SAM, or build a cardboard model and trace with a pen. Theo Jansen iterated his Strandbeest leg geometry over thousands of generations of evolutionary search to find the link ratios and tracer point that give the smooth flat-bottomed walking curve.
References & Further Reading
- Wikipedia contributors. Four-bar linkage. Wikipedia
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