Mechanisms : March 2016

Peaucillier's Mechanism: Mathematicians and engineers had being searching for almost a century to find the solution to a straight line linkage but all had failed until 1864 when a French army officer Charles Nicolas Peaucellier came up with his inversor linkage. Interestingly, he did not publish his findings and proof until 1873, when Lipmann I. Lipkin, a student from University of St. Petersburg, demonstrated the same working model at the World Exhibition in Vienna. Peaucellier acknowledged Lipkin's independent findings with the publication of the details of his discovery in 1864 and the mathematical proof.

Image 14
Let's turn to a skeleton drawing of the Peaucellier-Lipkin linkage in Image 14. It is constructed in such a way that $OA = OB$ and $AC=CB=BP=PA$ . Furthermore, all the bars are free to rotate at every joint and point $O$ is a fixed pivot. Due to the symmetrical construction of the linkage, it goes without proof that points $O$ , $C$ and $P$ lie in a straight line. Construct lines $OCP$ and $AB$ and they meet at point $M$ . Since shape $APBC$ is a rhombus $AB \perp CP$ and $CM = MP$ Now, $(OA)^2 = (OM)^2 + (AM)^2$ $(AP)^2 = (PM)^2 + (AM)^2$ Therefore, $\begin{align} (OA)^2 - (AP)^2 & = (OM)^2 - (PM)^2\\ & = (OM-PM)\cdot(OM + PM)\\ & = OC \cdot OP\\ \end{align}$ Let's take a moment to look at the relation $(OA)^2 - (AP)^2 = OC \cdot OP$ . Since the length $OA$ and $AP$ are of constant length, then the product $OC \cdot OP$ is of constant value however you change the shape of this construction.

Image 15
Refer to Image 15. Let's fix the path of point $C$ such that it traces out a circle that has point $O$ on it. $QC$ is the extra link pivoted to the fixed point $Q$ with $QC=QO$ . Construct line $OQ$ that cuts the circle at point $R$ . In addition, construct line $PN$ such that $PN \perp OR$ . Since, $\angle OCR = 90^\circ$ We have $\vartriangle OCR \sim \vartriangle ONP and \frac{OC}{OR} = \frac{ON}{OP}$ . Moreover $OC \cdot OP = ON \cdot OR$ . Therefore $ON = \frac {OC \cdot OP}{OR} =$ constant, i.e. the length of $ON$ (or the x-coordinate of $P$ w.r.t $O$ ) does not change as points $C$ and $P$ move. Hence, point $P$ moves in a straight line.

Mechanisms

Tuesday, March 8, 2016

Peaucellier's Mechanism