Math and Motion: A Look at Chebyshev’s Works on Linkages

This is a fascinating subject. I stumbled into it (without knowing of Chebyshev) while playing with a "lego walker" back in 2018. Even did a couple of blog posts [0,1] (in danish, and mostly for my own benefit) exploring the geometry.

Only recently did I learn of the Chebyshev linkage, which has a "lambda variant" [2] is basically what I was looking for in the walker (except it needs to be upside down, as in Chebyshevs own "walker" or "plantigrade" [3]).

[0]: https://hack42.dk/2018/04/lego-walker/ [1]: https://hack42.dk/2018/04/geogebra-og-gaamaskiner/ [2]: https://en.wikipedia.org/wiki/Chebyshev_lambda_linkage [3]: https://en.wikipedia.org/wiki/File:Tchebyshevs_plantigrade_m...

(Edit: formatting)

For anyone interested there is a 748 page atlas of available curves, mechanically generated and published in 1951. It's out of print but there are pdfs on various sites. The pages look like this:

https://www.gutenberg.org/files/27106/27106-h/images/fig-45....

Analysis of the Four-Bar Linkage Its Application to the Synthesis of Mechanisms

by John A. Hrones and George L. Nelson

Hardcover ISBN: 9780262080033 Published: June 15, 1951

Today the pages can be software generated:

https://www.cedarville.edu/cf/engineering/kinematics/ccapdf/...

https://demonstrations.wolfram.com/CouplerCurveAtlasForTheFo...

The purpose was to allow engineers to design (on paper) mechanisms where a motor could drive a crank link connected to a rocker link via a coupler link (think of it as a plane being moved) and a certain point on the coupler could describe a complex curved or straight path. You figured out roughly the path you wanted and went looking for it in the atlas. Each atlas page had varying integer combinations of linkage lengths and the coupler link had 5 to 7 curves traced out for different points on it.

"It was used in steam engines to translate circular motion into rectilinear. The full mechanism is shown in Fig. 6, where Watt’s parallelogram is the upper part of it. The machine transforms the circular motion of the point KK through the rocking of the points EE and BB into an almost rectilinear motion of the points MM and CC, which enables a truly rectilinear motion of the piston SS, but with some side tension."

I am pretty sure it is the other way around ?

https://technicshistory.com/2021/10/10/the-steam-revolution/

Chebyshev also has a famous inequality in probability.

https://en.m.wikipedia.org/wiki/Chebyshev's_inequality

https://youtu.be/d5pnfFvggYk

If you want to go into more depth on this, the book "How Round is Your Circle" is a worthwhile read. As I recall, it was a bit dense and math-heavy, and made for tough subway reading.

This article is crying out for some interactive animations.

Something like what's on this page, for each mechanism:

https://mechanicalexpressions.com/explore/kinematics/4-bar-l...

If you want to quickly sketch and simulate the motion of linkages, I can recommend using Solvespace:

https://solvespace.com/index.pl

I've seen a couple of sites with linkages like these. Does anyone know of a source of first principles from whence one could build their own?

Is there an app/site where one can create/test their own linkages?