# Triple Rocker

*Mechanics Corner*

A Journal of Applied Mechanics and Mathematics by DrD

July 31, 2017

**Triple Rocker**

Over at the Kinematics of Machines club, I recently ask if anyone could show me an example of a four-bar linkage that would be classed as a triple rocker. In the terminology of four-bar linkages, a link is classed as either a crank or a rocker:

Crank - can rotate in a complete circle

Rocker - cannot rotate in a complete circle]

Thus my question was for an example of a four-bar linkage where no link is able to rotate around a full circle. My request has not generated any answers, but fortunately, I stumbled onto one.

Since the definition of a rocker is a link that cannot rotate completely, it is evident that the linkage shown is in fact a Triple Rocker. None of the links is able to move through a complete revolution. If we try to rotate the input (left) link further down, it cannot happen without stretching the combination of the coupler and the output (right) links. When the input link (left side) gets to the top, again its motion is stopped by the need to stretch the coupler and output link. Thus, a figure I drew as an illustration for something else turns out to be a Triple Rocker, the item I was looking to find.

In connection with four-bar linkages, some readers will have heard of Grashof's theorem. Let

s = length of shortest link

L = length of the longest link

p, q = lengths of the two intermediate links

Grashof's theorem says that a necessary and sufficient condition for at least one link to be a crank (able to rotate entirely around), it is necessary that

s + L < p + q

This inequality is not satisfied for the four-bar that I drew by chance, so Grashof's theorem says that none of the links can be a crank. That is precisely the condition required for a Triple Rocker (a ground link plus three moving but not fully rotating links). So, there you have it. That is an example of a Triple Rocker, and we now have the criteria for identifying such as a four-bar linkage that does not satisfy Grashof's Theorem.

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