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 + qWhen 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 |
|---|---|---|
| Side link (input or output) | Crank-rocker | The shortest link is the crank and rotates fully; the opposite side link oscillates (the rocker). |
| Ground link | Double-crank (drag-link) | Both side links rotate fully relative to ground. |
| Coupler link | Double-rocker | Both side links oscillate; the coupler rotates 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.