4 ms·

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

72 points by signa11 4y ago

japanuspus 4y ago
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/ https://hack42.dk/2018/04/lego-walker/ [1]: https://hack42.dk/2018/04/geogebra-og-gaamaskiner/ https://hack42.dk/2018/04/geogebra-og-gaamaskiner/ [2]: https://en.wikipedia.org/wiki/Chebyshev_lambda_linkage https://en.wikipedia.org/wiki/Chebyshev_lambda_linkage [3]: https://en.wikipedia.org/wiki/File:Tchebyshevs_plantigrade_machine.gif https://en.wikipedia.org/wiki/File:Tchebyshevs_plantigrade_m... (Edit: formatting)
- japanuspus 4y ago
  For anyone wondering about the domain name: Week 42 is the "autumn break" in Denmark: kids are home from school, you are still filled with travel from your summer holiday, weather is typically ok -- so in short, it is the perfect time do work on fun projects with the kids.
  - madcaptenor 4y ago
    As an American, it always surprises me that people in other countries actually use week numbers in common use. I first ran into this when I was planning a trip in Norway and read something about how the train I had hoped to take would be under repairs in, I think, week 31.
BlueTemplar 4y ago
"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/ https://technicshistory.com/2021/10/10/the-steam-revolution/
JKCalhoun 4y ago
Is there an app/site where one can create/test their own linkages?
- navane 4y ago
  only desktop apps afaik https://blog.rectorsquid.com/category/programming/linkage-program https://blog.rectorsquid.com/category/programming/linkage-pr... https://solvespace.com/index.pl https://solvespace.com/index.pl
melling 4y ago
Chebyshev also has a famous inequality in probability. https://en.m.wikipedia.org/wiki/Chebyshev's_inequality https://en.m.wikipedia.org/wiki/Chebyshev's_inequality https://youtu.be/d5pnfFvggYk https://youtu.be/d5pnfFvggYk
- Jarmsy 4y ago
  Also known for his studies of quadrilateral meshes with equal edge lengths. https://cseweb.ucsd.edu/~alchern/projects/ChebyshevNets/ https://cseweb.ucsd.edu/~alchern/projects/ChebyshevNets/ https://mathshistory.st-andrews.ac.uk/Extras/Chebyshev_nets/ https://mathshistory.st-andrews.ac.uk/Extras/Chebyshev_nets/
  - madcaptenor 4y ago
    Also known for having lots of variant spellings of his name (https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6500167 https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6500167) due to every Latin-alphabet language having different conventions for transliterating from Russian.
    - HPsquared 4y ago
      That must be a real headache when counting citations.
      - madcaptenor 4y ago
        Wikipedia (https://en.wikipedia.org/wiki/Pafnuty_Chebyshev https://en.wikipedia.org/wiki/Pafnuty_Chebyshev) says "It is one of the most well known data-retrieval nightmares in mathematical literature."
bunabhucan 4y ago
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.jpg 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/fccca.htm https://www.cedarville.edu/cf/engineering/kinematics/ccapdf/... https://demonstrations.wolfram.com/CouplerCurveAtlasForTheFourBarLinkage/ 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.
mauvehaus 4y ago
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.
HPsquared 4y ago
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-linkage.html https://mechanicalexpressions.com/explore/kinematics/4-bar-l...
phkahler 4y ago
If you want to quickly sketch and simulate the motion of linkages, I can recommend using Solvespace: https://solvespace.com/index.pl https://solvespace.com/index.pl
navane 4y ago
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?

72 points by signa11 4y ago

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/ https://hack42.dk/2018/04/lego-walker/ [1]: https://hack42.dk/2018/04/geogebra-og-gaamaskiner/ https://hack42.dk/2018/04/geogebra-og-gaamaskiner/ [2]: https://en.wikipedia.org/wiki/Chebyshev_lambda_linkage https://en.wikipedia.org/wiki/Chebyshev_lambda_linkage [3]: https://en.wikipedia.org/wiki/File:Tchebyshevs_plantigrade_machine.gif https://en.wikipedia.org/wiki/File:Tchebyshevs_plantigrade_m... (Edit: formatting)

japanuspus 4y ago
For anyone wondering about the domain name: Week 42 is the "autumn break" in Denmark: kids are home from school, you are still filled with travel from your summer holiday, weather is typically ok -- so in short, it is the perfect time do work on fun projects with the kids.
- madcaptenor 4y ago
  As an American, it always surprises me that people in other countries actually use week numbers in common use. I first ran into this when I was planning a trip in Norway and read something about how the train I had hoped to take would be under repairs in, I think, week 31.

BlueTemplar 4y ago

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

JKCalhoun 4y ago

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

navane 4y ago
only desktop apps afaik https://blog.rectorsquid.com/category/programming/linkage-program https://blog.rectorsquid.com/category/programming/linkage-pr... https://solvespace.com/index.pl https://solvespace.com/index.pl

melling 4y ago

Chebyshev also has a famous inequality in probability. https://en.m.wikipedia.org/wiki/Chebyshev's_inequality https://en.m.wikipedia.org/wiki/Chebyshev's_inequality https://youtu.be/d5pnfFvggYk https://youtu.be/d5pnfFvggYk

Jarmsy 4y ago
Also known for his studies of quadrilateral meshes with equal edge lengths. https://cseweb.ucsd.edu/~alchern/projects/ChebyshevNets/ https://cseweb.ucsd.edu/~alchern/projects/ChebyshevNets/ https://mathshistory.st-andrews.ac.uk/Extras/Chebyshev_nets/ https://mathshistory.st-andrews.ac.uk/Extras/Chebyshev_nets/
- madcaptenor 4y ago
  Also known for having lots of variant spellings of his name (https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6500167 https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6500167) due to every Latin-alphabet language having different conventions for transliterating from Russian.
  - HPsquared 4y ago
    That must be a real headache when counting citations.
    - madcaptenor 4y ago
      Wikipedia (https://en.wikipedia.org/wiki/Pafnuty_Chebyshev https://en.wikipedia.org/wiki/Pafnuty_Chebyshev) says "It is one of the most well known data-retrieval nightmares in mathematical literature."

bunabhucan 4y ago

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.jpg 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/fccca.htm https://www.cedarville.edu/cf/engineering/kinematics/ccapdf/... https://demonstrations.wolfram.com/CouplerCurveAtlasForTheFourBarLinkage/ 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.

mauvehaus 4y ago

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.

HPsquared 4y ago

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

phkahler 4y ago

If you want to quickly sketch and simulate the motion of linkages, I can recommend using Solvespace: https://solvespace.com/index.pl https://solvespace.com/index.pl

navane 4y ago

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?