The Peaucellier Linkage
A machine that transforms radial motion into linear motion.
Mechanical Linkages had its time of fame through the mid 19th century. With the rapid growth of industry there was a great interest in the theoretical aspects of the movement machines, the result was the study of Mechanical Linkages which are abstractions of real world machines consisting only of Joints, Anchors and Rigid Bars. An important problem was to construct a machine that would transform radial motion into straight-line motion, the above applet demonstrates a solution to this problem found in the 1860s. This mechanical linkage turned to be of little use in reality due to its complicated nature and the development of oil lubricants which made simpler machines, which could only approximate linear motion more feasible.
How this machine works?
Consider a circle in the complex plane with radius 1 and center:(0,0) the inversion function has the property that it takes circles passing through the origin to straight lines. It turns out that there is a mechanical linkage that implements this function, that is if the coordinate of a Joint (Input Joint) is Z the coordinate of another Joint (output Joint) is . In fact this Linkage is the base of the above machine:
|Place the mouse over the green joint(input joint) and view the red joint(output joint).|
Now In order to construct a mechanical linkage which transforms circular movement into linear movement all is needed is to restrict the Input Joint to rotate on a circle passing through the origin, this is done by placing an anchor in distance r from the origin and connecting it to the Input Joint by a rigid Bar.
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