February 2012


      

      

soliton is a kind of solitary, stable and localised wave which acts in many ways like a particle. They are useful in describing a diverse range of physical phenomena, and their mathematics is a large and active topic of research.

One way of demonstrating the idea of solitons is the coupled pendulum model: Imagine a series of pendulums attached at even spacings along a horizontal rod, free to rotate only in the plane perpendicular to the rod, and connected to their neighbouring pendulums by torsion springs. If the pendulum at one end is pulled over the top of the rod it introduces a twist into the system. The weight of the pendulums makes them want to hang straight down, and along most of the rod they do, but at this twist they cannot. Because of the pendulums pulling down either side of the twist, it remains localised and cannot spread out. It can move along the rod like a wave, but unlike a normal wave, it cannot simply dissipate and disappear (except by leaving one end of the rod). These twists can be in either direction (sometimes called kink and anti-kink solitons), and interestingly they can pass through each other and emerge unchanged. Curious about this behaviour, I had a go at simulating it in Kangaroo :

Another model for solitons can be made with playing cards :

(I think I first encountered this model in the book ‘The New Ambidextrous Universe’ by Martin Gardner)

A form of soliton even occurs in traffic jams, and here is yet another model of soliton like behaviour – the Jacob’s ladder toy.

I see a link here with the the idea of bistable structures. A normal elastic object has a single unstressed configuration, and if you bend it away from this shape(provided you do not bend it too far) it will always try and spring back to that one rest configuration. A bistable object on the other hand has 2 different states in which it is at rest, and in between a more highly stressed state. An example of a bistable structure that some readers may remember are the ‘snap bracelets‘ popular during the early 90′s.

These were very similar to metal measuring tape – a strip of thin metal with a slight curvature across its short direction when straight along its length. A normal metal tape measure is not actually bistable, because it is still stressed when in its rolled state, but it is much less stressed in the rolled or straight configurations than in the in between state.

This allows a sharp bend to move along the tape while staying localized, a bit like the pendulum soliton.

I’ve recently been exploring the idea of whether it is possible to create something similar, but on a surface, not just a linear element.

Here I am also drawing on the idea of auxetic materials – defined as having a negative Poisson’s ratio. This means that unlike most materials which get thinner in cross section as you stretch them along their length, auxetics surprisingly actually get fatter. This can be achieved by small scale structures within the material behaving as linkages.

Many of the origami textured surfaces I have explored in previous posts actually have auxetic properties. Something I am recently trying to do is combine auxetic and bistable properties in a single sheet material through 3d printing. I’ll update when the test print arrives, but I’m hoping that the transition region between the two different contracted states of the material will behave something like a 2 dimensional version of the unstable region of the tape measure, causing some interesting out of plane buckling.

I’ve barely scratched the surface of the real study of solitons, and what I’ve mentioned here are only really toy models of a highly complex subject. For some more related reading :

Wikipedia article on the Sine-Gordon equation

Many Faces of Solitons

A real version of the pendulum model, and another

and there are also all sorts of exotic Soliton related objects, such as SkyrmionsTwistors Instantons and Hopfions

For more on auxetics, check out Rod Lakes’ page

and for more on bistable structures try Simon Guest’s page

Finally – sorry for such a long hiatus, it’s good to be writing again, more posts coming soon.

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