Linkages and Four-Bar Mechanisms

A four-bar linkage is four rigid bars joined in a loop by hinges, with one bar fixed. Turn one link and the other two follow a motion set entirely by the lengths. Grashof's rule predicts from the lengths alone whether a link can turn all the way around.

Ground link Input crank Coupler Output rocker Coupler point position

Grashof's rule

Call the shortest link $s$, the longest $l$, and the other two $p$ and $q$. If

$$s + l \le p + q$$

then at least one link can rotate all the way around. Which motion you get depends on which link is shortest. If the shortest link is the input crank, you get a crank-rocker. If the shortest is the ground, both side links turn fully, a double crank. If the inequality fails, no link turns fully, a double rocker.

What to notice

The red curve is the path of one point on the coupler. Small changes in length make very different shapes, from ovals to figure eights, which is why linkages are used to make walking legs, windshield wipers, and grippers that move in a set path from a single motor. Make the input crank too long and the loop can no longer close at some angles, so the crank stops and reverses instead.