Four-bar linkage

Example

A planar four-bar linkage consists of four rigid bars joined in a closed loop by four revolute joints in the plane. One bar, the ground link, is fixed, and the bar opposite to it is the coupler. The two others are the crank and the rocker, each pivoted on the ground.

Mobility check: there are $N = 4$ links and $j = 4$ joints, each with $f_i = 1$. The Chebychev–Grübler–Kutzbach criterion gives $M = 3(4 - 1 - 4) + 4 = 1$. Indeed, for generic bar lengths, the angle of the crank fixes the other positions up to at most two choices. These are the two ways of closing a triangle.

Grashof condition: let $s$ and $l$ be the shortest and the longest bar lengths, and $p$, $q$ the other two. If $s + l < p + q$, then the shortest bar can make full turns relative to its two neighbours. If $s + l > p + q$, no bar makes a full turn relative to another.