# How does this tensegrity table work?

I have assembled below a desk toy which seems to defy laws of physics at first glance (objects can be placed on top of it up to a certain limit, since it is already under strain).

The toy is in fact an example of a tensegrity sculpture, where a system of components is held under static equilibrium by some combination of tension forces. In this case, there is a very tightly stretched elastic band and four metal chains.

In this design the two beams are at an angle, so I guess what actually happens in terms of force diagrams is that the tension force in the elastic band creates a torque which tries to rotate the top table clockwise, but it is constrained by tension forces in the stretched chains which hold it in place.

• I'm thinkin the weight up top is held up by the elastic in the middle. This would induce a torque moving the top side to side, but the chains keep that from happening. So the elastic pulls up then chains pull down and to the side. Commented Jun 12 at 20:27
• OK yes that makes sense.
– Tom
Commented Jun 12 at 20:28
• Not sure what your question is. What holds the table up/still. You seem to have answered that. Commented Jun 12 at 20:28
• Re, "two beams are at an angle [and therefore] "the elastic band creates a torque..." The way I see it, the circular top and the beam that supports it constitute a single rigid object. Likewise, the circular base and its attached beam. If you extend the line of the elastic band, it looks to me as if it passes right through the centers of those two objects and, what's maybe more important, the extended line is equidistant from the attachment points of the four chains. Commented Jun 12 at 21:07
• Commented Jun 13 at 0:23

Stable mechanical equilibrium broadly means that any movement would incur a net energy penalty.

This is intuitive when we see a ball at rest in a dip, for example; it’s clear that any rolling would incur an increase in gravitational potential energy.

Even a ball hanging from a spring, for example, is straightforwardly analyzed; the string stretches until the benefit from dropping in a gravity field no longer pays for the increased strain energy in the string material.

How much additional strain energy? For objects being stretched, a stiffness $$k$$ corresponds to a restoring force of magnitude $$F=k(\delta x)$$ and a strain energy increase of $$k(\delta x)^2/2$$ for a small stretch $$\delta x$$. (Another way to look at this is that the effective stiffness at equilibrium corresponds to the curvature or second derivative of the energy landscape for small perturbations. A deeper minimum corresponds to greater stability.) What's more, if the object (now idealized as Hookean, i.e., linear elastic) is already preloaded by substantial stretching $$x_\mathrm{preload}$$, the strain energy increase from additional $$\delta x$$ is boosted to $$k\frac{(x_\mathrm{preload}+\delta x)^2}{2}-k\frac{x_\mathrm{preload}^2}{2}\approx kx_\mathrm{preload}\delta x\gg k\frac{(\delta x)^2}{2},$$ so the energy penalty to shifting away from equilibrium in the direction of preload can be made much more severe. (To complete the analysis for the other side of the energy curve, we'd consider the associated stiffness for a perturbation in that direction.) This is relevant to the discussion that follows.

So-called tensegrity structures can be visually appealing because it’s not immediately clear what’s incurring the energy penalty; thus, objects seem to levitate—counterintuitively. Further, a mode of easy movement may seem obvious, and it’s interesting if that mode doesn’t activate.

In the picture, ignoring the chains for a moment, the white "strap" is perhaps loose and looks unstable—we'd expect the top to immediately rotate down and to the right. Ignoring the "strap," we know that chains have no compression strength—they can only pull, and so the table again seems destined to collapse. It emerges that the strap is actually an elastomer under large preloaded tension, not limited to the top's weight. The chains are actually preventing rising, twisting, and rotation, as any motion would stretch at least one of them and incur a large strain energy penalty. The larger the preloading, the stiffer the assembly.

(I am grateful to the commenters for identifying key aspects of this design.)

• You can also see that that the chains are attached to swivels and allowed to twist, so they are free to move up and down on the hoops and also to twist.
– Tom
Commented Jun 12 at 21:28
• In the picture above, the table top is held up by the white elastic cable at the bottom; Because of mass distribution differences to the right, there is easily a net torque rotating (picking pivot @ elastic-wood interface) to fall to the right. Thus the two chains on the left must be pulling a little bit more than just weight of chains. In any case, chains, like strings, can only pull, and only pull in the direction of the chains themselves, so the elastic band is necessarily pulling the table top up with rather tremendous force. Commented Jun 13 at 5:26
• "It emerges that what the chains are actually doing is preventing this twisting, as any motion would stretch them and incur too much of a strain energy." The primary role of the chains is to keep the elastic band in tension. If you watch videos about these (Steve Mould has a good one on Youtube) you can twist these with ease. But twisting requires the top and bottom to come closer together increasing the tension on the band. As soon as the external twisting force is released, it will return to its stable configuration. Commented Jun 13 at 15:31

What's confusing here is probably that the lifting is dose via the tensile force of the rubber band, which due to the placement on the arms pulls down the bottom and up the top part.

Mechanically, this is equivalent to the following sketch:

The red spring is in compression (so pushes the blue end borad apart) and counteracted by the black chains.

• This answer is correct, but I think it might not be obvious how a rubber band in tension acts like a spring in compression. That's the whole 'trick' to these designs. Commented Jun 13 at 15:38

Less scientifically speaking, the chains prevent the upper piece from going higher up than their length (they resist the whole structure "stretching"), but do nothing against it falling lower, while the rubber band resists "squishing" of the structure, and actually tries to "stretch it" by pulling on the ends together.

This creates an equilibrium where nothing moves as the forces from the band and from the chains oppose each other.

If it got any taller, the chains would have to snap, and if it got squished, the band would eventually snap, in any case, the chains would go slack.

The upper piece cannot tilt in any way, either, as there are three chains, so any tilt would have to snap at least one of them as that part of the table would go further away than the chain's length.

This table works very simply once you see or know it:

The white band in the middle supports the upper part against gravity.

The metal chains make it so the upper part does not fall over.

If you were very, very, very careful and patient and in a room with absolutely no air movement, you could in theory only use the white band and skip the chains. But the design being so top-heavy, it would be virtually (but not physically) impossible to have it stable for any amount of time.

"Tensegrity" is a fancy word and just means that key parts of the design are under tension (i.e., wobbly parts like chains and such). It only seems "magic" because we are not used to it. Mechanical engineers use this all the time - anytime you see a steel cable on a bridge, the cable is under tension (obviously) and helps keep the bridge up. Those "spider nets" on children's playgrounds are tensegrity designs. Actual spider nets are. And so on and forth.

The most obvious is our own body: there is not a single bone that is physically fixed to another, or rotating within another using form fit (like a screw or piston connection in mechanical engineering). Every single bone is fixed to other bones via parts under tension (i.e., ligaments, muscles).

We know that chains and elastic bands can support tension forces, but are useless against compression forces. The wooden parts on the other hand can be considered as a rigid structure.

The key observation is that the bottom of the elastic band is connected to the top of the table and the top of the elastic band is connected to the bottom of the table. So downward force on the table top creates tension, not compression in the elastic band.

However, if we only had the elastic band then the top of the table would "fall over". because it's point of support would be below it's centre of mass.

The chains stop this from happening. Any rotation of the table top would require either elongating one of the chains, or increasing the length of the elastic band.

To increase rigidity the system likely incorporates preload tension. That is the tension in the elastic band is more than is needed to hold up the top of the table, with the remaining force being taken up by the chains.

Others have given more scientific answers, but to me its principle becomes obvious when I picture what happens if you remove the rubber band and squeeze the hook parts together using your thumb and index finger.