
A pendulum theorem - mgdo
http://fermatslibrary.com/s/a-pendulum-theorem
======
Jun8
Hmm, this reminded me of an empirical discovery I made: When walking with an
open coffee cup at hand oscillating your hand vertically in random fashion
greatly reduces the coffee resonance and the sloshing effect.

This problem has been investigated before
([http://www.sciencemag.org/news/2012/05/physics-spilled-
coffe...](http://www.sciencemag.org/news/2012/05/physics-spilled-coffee)) but
AFAIK the above solution has not been proposed. I wonder if the theorem given
can help motivate this finding?

~~~
adrianN
Walk without rythm and we won't spill the coffee.

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selimthegrim
This isn't original to Acheson - the single pendulum case is widely known as
the Kapitza pendulum.

[https://en.wikipedia.org/wiki/Kapitza%27s_pendulum](https://en.wikipedia.org/wiki/Kapitza%27s_pendulum)

If you speak Russian the following video with V.I. Arnold is nice

[http://www.youtube.com/watch?v=ZBSLK3Jbd2I](http://www.youtube.com/watch?v=ZBSLK3Jbd2I)

~~~
jacobolus
The paper says in its first sentence “Stephenson (1908a, b) showed that it is
possible ... He confirmed his theoretical predictions by a practical
demonstration ...”

~~~
selimthegrim
His derivation isn't nearly as complete as Kapitza's, though.

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Animats
When that was discovered it was a shock to the controls community. Inverted
pendulum stabilization was long thought to require active control.

Here's a video demonstrating passive stabilization.[1] It's so simple it's
embarrassing.

[1]
[https://www.youtube.com/watch?v=rwGAzy0noU0](https://www.youtube.com/watch?v=rwGAzy0noU0)

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idlewords
The last line of the paper mentions seeing a triple inverted pendulum recover
from even very severe perturbations and go back to vertical. I would love to
see a video of this stuff!

~~~
dools
This is a different control mechanism but it's still a cool video :)

[https://www.youtube.com/watch?v=cyN-
CRNrb3E](https://www.youtube.com/watch?v=cyN-CRNrb3E)

~~~
curiousgal
Came across this video after watching that, somehow related but equally
amazing. [https://www.youtube.com/watch?v=Lt-
KLtkDlh8](https://www.youtube.com/watch?v=Lt-KLtkDlh8)

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1jFDS
Sane link:

[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.205...](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.205.5342&rep=rep1&type=pdf)

I don't understand how people put up with a wobbling "Click here to see more"
on fermatslibrary.com.

What is next? Back to marquee?

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akssri
Mark Levi has written about these things on SIAM and has a nice video up on,

[http://www.personal.psu.edu/mxl48/Inverted_Pendulum.html](http://www.personal.psu.edu/mxl48/Inverted_Pendulum.html)

I think they've also shown that this is robustly stable for a particular
family.

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elcritch
Makes me think about drones in particular. If you can control an inverted
n-ary pendulum then controlling inverted thrust should be a straightforward
extension.

