How does this tensegrity table work?

Question

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.

Answer 1

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.)

Answer 2

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.

Answer 3

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.

Answer 4

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).

Answer 5

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.

Answer 6

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.