Grashof condition

Definition

For a planar four-bar linkage with link lengths s (shortest), l (longest), and p, q (the other two), Grashof’s condition is:

s + l ≤ p + q

When the inequality holds strictly, at least one link can rotate fully relative to ground. Such a linkage is called Grashof.

When the inequality holds with equality, the linkage is at a change-point — a degenerate transition between sub-classes. At a change-point the four bars momentarily collinear and the mechanism can pop into either of two assembly modes, which is generally something a designer wants to avoid.

When the inequality is violated, no link can rotate fully — the linkage is non-Grashof (sometimes called a triple-rocker).

Sub-classes

If Grashof’s law is satisfied, the position of the shortest link s in the mechanism determines the sub-class:

Position of shortest link	Sub-class	Behaviour
Ground link	Crank-rocker	The link adjacent to s rotates fully (the crank); the other moving link oscillates (the rocker).
Coupler link	Double-rocker	Both moving links oscillate; the coupler rotates fully.
Side link (input or output)	Double-crank (drag-link)	Both moving links rotate fully.

The crank-rocker is the workhorse of mechanism design — input rotates continuously, output sweeps an arc. Most balance-shaft drives, four-bar suspensions, oscillating-pump linkages, and our own worked practica end up as crank-rockers when the synthesis is done right.

Why it ships first in the pipeline

Among the four classical defect filters, Grashof is the cheapest:

• One inequality.
• Evaluated on link lengths.
• Constant time.
• No simulation, no Newton-Raphson, no Jacobian assembly.

It rejects non-rotatable candidates before any expensive position-solve runs. On a practical synthesis sweep over 70k+ geometric candidates, Grashof filtering plus the branch test alone collapses the surviving set by an order of magnitude.

What Grashof does not tell you

A Grashof linkage is rotatable. That does not mean it is useful. The linkage might still:

• Have a transmission angle so close to 0° that mechanical advantage collapses near a dead-point.
• Sweep a coupler curve that misses the precision points you wanted.
• Be a crank-rocker where the output (rocker) is the side you wanted to drive — function vs structure ordering matters.
• Generate large reaction forces at the joints under load.

So Grashof is necessary but not sufficient. The other three classical filters — order, branch, transmission angle — and the dynamics validation all run downstream.

See also

• Four-bar linkage — the canonical mechanism Grashof’s law applies to.
• Transmission angle — the second filter, applied after Grashof.
• Practica case study — Grashof applied end-to-end on a worked university problem.