Peaucellier–Lipkin linkage

File:Peaucellier-Lipkin Inversor.gif

Animation for Peaucellier–Lipkin linkage:

Dimensions:
Cyan Links = a
Green Links = b
Yellow Links = c

The Peaucellier–Lipkin linkage (or Peaucellier–Lipkin cell, or Peaucellier–Lipkin inversor), invented in 1864, was the first true planar straight line mechanism – the first planar linkage capable of transforming rotary motion into perfect straight-line motion, and vice versa. It is named after Charles-Nicolas Peaucellier (1832–1913), a French army officer, and Yom Tov Lipman Lipkin (1846–1876), a Lithuanian Jew and son of the famed Rabbi Israel Salanter.

Until this invention, no planar method existed of converting exact straight-line motion to circular motion, without reference guideways. In 1864, all power came from steam engines, which had a piston moving in a straight-line up and down a cylinder. This piston needed to keep a good seal with the cylinder in order to retain the driving medium, and not lose energy efficiency due to leaks. The piston does this by remaining perpendicular to the axis of the cylinder, retaining its straight-line motion. Converting the straight-line motion of the piston into circular motion was of critical importance. Most, if not all, applications of these steam engines, were rotary.

The mathematics of the Peaucellier–Lipkin linkage is directly related to the inversion of a circle.

Earlier Sarrus linkage

There is an earlier straight-line mechanism, whose history is not well known, called the Sarrus linkage. This linkage predates the Peaucellier–Lipkin linkage by 11 years and consists of a series of hinged rectangular plates, two of which remain parallel but can be moved normally to each other. Sarrus' linkage is of a three-dimensional class sometimes known as a space crank, unlike the Peaucellier–Lipkin linkage which is a planar mechanism.

Geometry

File:PeaucellierApparatus.PNG

Geometric diagram of a Peaucellier linkage

In the geometric diagram of the apparatus, six bars of fixed length can be seen: Template:Mvar, Template:Mvar, Template:Mvar, Template:Mvar, Template:Mvar, Template:Mvar. The length of Template:Mvar is equal to the length of Template:Mvar, and the lengths of Template:Mvar, Template:Mvar, Template:Mvar, and Template:Mvar are all equal forming a rhombus. Also, point Template:Mvar is fixed. Then, if point Template:Mvar is constrained to move along a circle (for example, by attaching it to a bar with a length halfway between Template:Mvar and Template:Mvar; path shown in red) which passes through Template:Mvar, then point Template:Mvar will necessarily have to move along a straight line (shown in blue). In contrast, if point Template:Mvar were constrained to move along a line (not passing through Template:Mvar), then point Template:Mvar would necessarily have to move along a circle (passing through Template:Mvar).

Many different over-all proportions of this linkage are possible. Since points Template:Mvar, Template:Mvar, Template:Mvar must be collinear at all points in the linkage's motion, and countless arm length combinations are viable, then mirror symmetry across Template:Mvar isn't necessary. With Template:Mvar staying collinear, the only requirement to achieve the intended straight-line motion of Template:Mvar are that AB = AD, that BC = DC, and for Template:Mvar to be constrained to a circular path which crosses Template:Mvar. Otherwise, there is no fixed relationship between the lengths of the sides of the Template:Mvar figure, the radius of the constraining circular path of Template:Mvar, and the lengths of Template:Mvar or Template:Mvar.

Mathematical proof of concept

Collinearity

First, it must be proven that points Template:Mvar, Template:Mvar, Template:Mvar are collinear. This may be easily seen by observing that the linkage is mirror-symmetric about line Template:Mvar, so point Template:Mvar must fall on that line.

More formally, triangles △BAD and △BCD are congruent because side Template:Mvar is congruent to itself, side Template:Mvar is congruent to side Template:Mvar , and side Template:Mvar is congruent to side Template:Mvar . Therefore, angles ∠ABD and ∠CBD are equal.

Next, triangles △OBA and △OBC are congruent, since sides Template:Mvar and Template:Mvar are congruent, side Template:Mvar is congruent to itself, and sides Template:Mvar and Template:Mvar are congruent. Therefore, angles ∠OBA and ∠OBC are equal.

Finally, because they form a complete circle, we have

$\mathrm{\angle }OBA+\mathrm{\angle }ABD+\mathrm{\angle }DBC+\mathrm{\angle }CBO=360^{\circ }$

but, due to the congruences, ∠OBA = ∠OBC and ∠DBA = ∠DBC, thus

$\begin{array}{rl}& 2\times \mathrm{\angle }OBA+2\times \mathrm{\angle }DBA=360^{\circ }\\ & \mathrm{\angle }OBA+\mathrm{\angle }DBA=180^{\circ }\end{array}$

therefore points Template:Mvar, Template:Mvar, and Template:Mvar are collinear.

Inverse points

Let point Template:Mvar be the intersection of lines Template:Mvar and Template:Mvar. Then, since Template:Mvar is a rhombus, Template:Mvar is the midpoint of both line segments Template:Mvar and Template:Mvar. Therefore, length Template:Mvar = length Template:Mvar.

Triangle △BPA is congruent to triangle △DPA, because side Template:Mvar is congruent to side Template:Mvar, side Template:Mvar is congruent to itself, and side Template:Mvar is congruent to side Template:Mvar . Therefore, angle ∠BPA = angle ∠DPA. But since ∠BPA + ∠DPA = 180°, then 2 × ∠BPA = 180°, ∠BPA = 90°, and ∠DPA = 90°.

Let:

$\begin{array}{rl}& x={\ell }_{BP}={\ell }_{PD}\\ & y={\ell }_{OB}\\ & h={\ell }_{AP}\end{array}$

Then:

${\ell }_{OB}\cdot {\ell }_{OD}=y(y+2x)=y^2+2xy$

${{\ell }_{OA}}^2=(y+x{)}^2+h^2$ (due to the Pythagorean theorem)

${{\ell }_{OA}}^2=y^2+2xy+x^2+h^2$(same expression expanded)

${{\ell }_{AD}}^2=x^2+h^2$ (Pythagorean theorem)

${{\ell }_{OA}}^2-{{\ell }_{AD}}^2=y^2+2xy={\ell }_{OB}\cdot {\ell }_{OD}$

Since Template:Mvar and Template:Mvar are both fixed lengths, then the product of Template:Mvar and Template:Mvar is a constant:

${\ell }_{OB}\cdot {\ell }_{OD}=k^2$

and since points Template:Mvar, Template:Mvar, Template:Mvar are collinear, then Template:Mvar is the inverse of Template:Mvar with respect to the circle (O,k) with center Template:Mvar and radius Template:Mvar.

Inversive geometry

Thus, by the properties of inversive geometry, since the figure traced by point Template:Mvar is the inverse of the figure traced by point Template:Mvar, if Template:Mvar traces a circle passing through the center of inversion Template:Mvar, then Template:Mvar is constrained to trace a straight line. But if Template:Mvar traces a straight line not passing through Template:Mvar, then Template:Mvar must trace an arc of a circle passing through Template:Mvar. Q.E.D.

A typical driver

File:The Peaucellier-Lipkin linkage with a rocker-slider four-bar as its driver.gif

Slider-rocker four-bar acts as the driver of the Peaucellier–Lipkin linkage

Peaucellier–Lipkin linkages (PLLs) may have several inversions. A typical example is shown in the opposite figure, in which a rocker-slider four-bar serves as the input driver. To be precise, the slider acts as the input, which in turn drives the right grounded link of the PLL, thus driving the entire PLL.

Historical notes

Sylvester (Collected Works, Vol. 3, Paper 2) writes that when he showed a model to Kelvin, he “nursed it as if it had been his own child, and when a motion was made to relieve him of it, replied ‘No! I have not had nearly enough of it—it is the most beautiful thing I have ever seen in my life.’”

Cultural references

A monumental-scale sculpture implementing the linkage in illuminated struts is on permanent exhibition in Eindhoven, Netherlands. The artwork measures , weighs , and can be operated from a control panel accessible to the general public.

See also

• Linkage (mechanical)
• Straight line mechanism

Bibliography

• — proof and discussion of Peaucellier–Lipkin linkage, mathematical and real-world mechanical models
• (and references cited therein)
• Hartenberg, R.S. & J. Denavit (1964) Kinematic synthesis of linkages, pp 181–5, New York: McGraw–Hill, weblink from Cornell University.

External links

• How to Draw a Straight Line, online video clips of linkages with interactive applets.
• How to Draw a Straight Line, historical discussion of linkage design
• Interactive Java Applet with proof.
• Java animated Peaucellier–Lipkin linkage
• Jewish Encyclopedia article on Lippman Lipkin and his father Israel Salanter
• Peaucellier Apparatus features an interactive applet
• A simulation using the Molecular Workbench software
• A related linkage called Hart's Inversor.
• Modified Peaucellier robotic arm linkage (Vex Team 1508 video)