The Four-Bar Linkage: The Simplest Machine That Draws Curves
Strip a machine down to its skeleton and you often find a four-bar linkage: four rigid links joined in a loop by four pivots. One link is the immovable frame, and the other three fight it out to turn, rock, and trace curves. It is arguably the most-studied mechanism in engineering — because the cheapest possible motion machine is also one of the most versatile.
The four members
- Ground — the fixed link. Everything is measured relative to it.
- Crank — the input link, pinned to the ground. Depending on its length it may rotate fully (a crank) or only swing back and forth (a rocker).
- Coupler — the floating link that connects crank tip to follower tip. Every point on the coupler moves on its own closed curve — the coupler curve.
- Follower — the output link, pinned back to the ground.
What the animation shows
Watch the white point on the coupler: it glides around a closed, egg-shaped path. Each different point on the same coupler traces a different curve — some oval, some kidney-shaped, some with sharp cusps. This is the four-bar’s superpower: one mechanism, infinitely many output paths, just by moving where you attach the work.
Designers exploit this by choosing a coupler curve that matches a task — a straight segment for a film-advance mechanism, a near-circular arc for a walking machine’s foot, a figure-eight for a stirring motion.
Grashof’s law: will the crank turn all the way around?
There is a beautiful shortcut for predicting a four-bar’s character. Sort the four link lengths (s = shortest, l = longest, p and q the other two):
Grashof condition: s + l ≤ p + q
- If the condition holds, at least one link can rotate fully. The shortest link becomes a crank, and you get continuous rotary input — ideal for motors.
- If it fails (s + l > p + q), no link can rotate fully: all three moving links just rock. You get a double-rocker, useful for compliant, oscillating motions.
This single inequality tells a designer in ten seconds whether the mechanism can be driven by a continuously rotating shaft.
Crank-rocker vs double-crank
With the Grashof condition satisfied, which link you choose as ground decides the behaviour:
- Crank-rocker — crank rotates, follower rocks. This is the classic foot-pump and windshield-wiper layout.
- Double-crank (drag-link) — both side links rotate. Used to couple two rotating shafts at slightly different phases.
- Double-rocker — only the coupler spins through full turns while both side links rock.
Where four-bars live around you
- Windshield wipers — motor turns a crank; two followers rock the blades.
- Bicycle rear suspension — many designs are four-bar linkages with the wheel as one link.
- Walking robots and the Klann linkage — extensions of the four-bar that turn rotation into stepping.
- Film projectors — coupler curves chosen to move film frame-by-frame, then hold still.
- Car hood latches, dump-truck tails, engine rocker covers — anywhere a small rotation must become a shaped path.
Why engineers still love it
No sliding parts, no bearings in dirty places, no lubrication drama — just four pins. A four-bar can be cut from sheet metal and assembled with four rivets, yet produce precisely engineered motion. When a task needs only a few degrees of shaped motion, a four-bar is almost always cheaper and more reliable than a cam or a servo.
Key takeaways
- Four links, four pivots, one loop — the minimum machine that transforms rotation.
- Every coupler point traces its own closed curve; pick the point, pick the path.
- Grashof’s inequality (s + l ≤ p + q) instantly predicts full rotation vs rocking.
- Ground choice decides the behaviour: crank-rocker, drag-link, or double-rocker.