Coincidentally, the Paucellier linkage was invented by a French Naval Lieutenant by the name of Paucellier.
It was the first mechanism employing only rotational joints to convert rotary motion to true, rather than approximate straight line motion. (The crank/piston arrangement found in most engines is an example of a mechanism, which converts rotary motion of the crank to linear motion of the piston, however it involves a sliding joint at the piston.)
The mechanism remained a curiosity rather than a particularly useful one, probably because it only worked for a portion of a complete revolution.
The mechanism remained a curiosity rather than a particularly useful one, probably because it only worked for a portion of a complete revolution.
The lines of the figure will all have their lengths constrained. In addition, the location of points A and F will be constrained.
We will need to constrain the lengths in such a way that the following hold:
|AB| = |AC|
|CE| = |EB| = |BD| = |CD|
|EF| = |FA|
The crank is the line EF. The output motion is acquired at point D.
For convenience in analyzing the output, we’ll put A and F both on the y-axis:
As AF is 4, we need to set EF to 4 also.
Set AB and AC to the same value: 14 in this case, and the four smaller links to their value: 8
With all the links constrained, we can move the mechanism (if you break it, don’t worry, use the undo key, or read on…)
Did D actually move in a straight line? To find out, let’s drag D into a geometry link in eActivity:
First create a Geometry Link from the eActivity menu:
Now drag D and drop it into the link:
We can now drag point E through its range of motion and watch the coordinates update in the geometry link field:
We see that the y coordinate stays constant, and hence traces a straight line.
You’ll notice that we were very careful in only dragging E through a small arc of its possible motion. What happens if we drag it farther?
If E goes too far round, then it is impossible to find locations for C and B which match the constraints. These elements and any others which depend on them will disappear from the screen. (Don’t worry, you can get them back either with Undo, or by dragging E back.)
Another interesting situation can be created if you drag E too far in one jump. Specifically if you drag it outside of the triangle ABC:
What has happened here is that we have jerked the mechanism into another configuration: one in which B and C are superimposed. You can Undo this, or drag B back out beyond E:
