Toughening tensegrities

Needle Tower by Kenneth Snelson (1968):
"these structures, pure tensegrity that is, are quite elastic and flexible; too much so for use as antennas with dishes mounted on top.
Swaying in the wind might be a disadvantage." Snelson FAQ.

Standing tough

Structures like the needle tower are spectacular, and very light in construction. Perhaps the earliest structure like this is the kite framework

which cannot change shape unless a string lengthens or a cross-piece shortens or bends. (Proving this needs some algebra, but every kite-maker's hands know it directly.) Humans aerodynamics has thus used such "tensegrities" — Buckminster Fuller's name for them — without rigid joints, and with rods joined only via tight cords, for a long time. Their lightness and (sometimes) stiffness has made them interesting for outer space applications since at least 1992.
Tensegrities have equally 'obvious' uses in practical architecture, but not all designs are as tough as the kite. Save for a few pedestrian bridges, few large structures have been built: there are no large big-load-bearing ones.

Snelson, the creator of the tower, expects none.



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Making a needle tower larger, and stiffening it to support its own weight (let alone external loads) requires higher and higher tensions in the cables, and thus higher compression forces in the bars: supporting these needs heavier bars and cables; added weight requires greater stiffness; this needs higher tensions; and so on, until the lightness and attraction is gone, and we do better with a classic truss framework.

Snelson is content with sculpture: he sees such deformability of "these structures, pure tensegrity that is" as inherent. In a quoted mail he wrote that they "are also very flexible and I know of no instance where they've been put to use for any practical purpose".
Engineers, though, do like to find practical applications.
For most applications, it is impractical to admit easy, large, deformation.

Many engineering researchers also see large flexing as inevitable. Rhode-Barbarigos et al. state flatly that "Tensegrity action involves large displacements", Whittier that "they are not conventionally rigid". Gómez remarks that "Elasticity multiplication is inherent to them: When separating two struts by a certain distance, the stretching of the tendons is much less than this amount", so the system yields far more than the individual tendons (cables) do. There is dispute about what truly qualifies as a tensegrity, but even for Snelson's narrow criteria such easy deformation is not inevitable. It holds for most tensegrities in the literature, but firmer tensegrities do exist, starting with the kite (as a planar structure).

Soft and hard structures

     
     (a)                                        (b)
[replace by videos of identical-material versions]
Here are two three-bar forms: a classic Snelson nine-cable, and one with ten cables. (It is not enough to add a cable: the bar geometry is also different.) The cable tensions are similar.
Press gently on the top of (a): it goes slightly downward, but twists a long way. Press firmly on the top of (b), and it hardly moves. Put equal weights on top of them. The one on (a) wobbles dramatically, twisting around the vertical; the one on (b) is steady.
Design (b) is for many authors 'infinitesimally stable', or 'infinitesimally rigid'. That term suggests 'just a tiny bit' stable or rigid — far less impressive than for instance 'super-stable' — but in making real structures with tough cable, this test is far more useful. There is a legitimate logic of infinitesimal quantities, very useful in real science, but this is a twisted use of the word, amounting to something like 'not admitting even infinitely small changes in shape, even when we neglect changes in cable length that are infinitesimal compared to the bar end movements'. This sounds stronger, but not very clear, and confusing when we think about structures made with real cables, which all admit some finite deformation under a finite force. Let us save 6 out of 7 syllables, 10 out of 15 letters, by calling it fully stable.

To the optimiser, the glass is too big.

Half-full is a waste of glass: better put the drink in a glass that it fills to the brim, and slops out if you drink un-cautiously. Sure.

Least energy

Why are so many tensegrities so wobbly?
Be careful what you optimise for. You may get it.
Whatever we call the fully ('infinitesimally') stable test, a tensegrity with rubber cables (the material on most how-to pages) that fails that test is often easy to make, and as robust as any other rubbery shape. It deforms easily, but that comes with the material. Rubber cable systems often settle to shapes that are stable but not fully stable when re-imagined with perfectly inextensible cables. As an unavoidable consequence, these shapes deform uncomfortably easily when made with real, nearly inextensible cable such as Kevlar. They are not easy to make.
Elastically-found tensegrity forms dominated the early decades. Burkhardt says of the literature available in 1980 "The basic design technique for complex structures seemed to be to build it with rubber bands or something very elastic to start with and then use those average tendon lengths to build with less elastic tendons." Snelson says "Building simple structures with basic materials such as chopsticks or soda straws and string or rubber bands is the best way to understand what the push and pull forces are doing." It may be the worst way: the designs that come of it have dominated the field for six decades. Burkhardt "can't imagine someone building a structural tensegrity using elastic tendons", but his Practical Guide gives a mathematical design principle with results similar to the energy-minimising of an elastic-cable structure: typically, results that are not fully stable in the inextensible limit.

Fewest strings

Another common 'optimisation' (perhaps motivated by elastic experiments) is the idea that a tensegrity should use as few cables as possible. How many is that?
Clearly each end p of each bar in a 3D tensegrity should have at least three cables attached, like the 9-cable form above: the tensions in just two cables can only balance the force along the bar if all three are in one plane through p, but then there is nothing to hold p in that plane. Some design approaches take that clear minimum as a target: "only tensegrity structures with three cables originating from each vertex were used" by Paul et al., because "it was of interest to study whether the evolutionary algorithm could design tensegrities using the minimum number of cables required." Thus, for
R bars with 2R bar ends, they used exactly 3(2R ) = 6R ends, and hence
c = 3R
cables. But full stability is impossible for an unattached n-bar, c-cable tensegrity unless
c 5R – 5.
(We prove this in the full, mathematical presentation this page is placeholding for.) This means >3R cables if R is over 2 — which for a non-planar tensegrity, it must be. For their 6-bar structure

3R = 18 falls short of 5R – 5 = 25 by seven. With no more detail than that, we already know it will be severely soft in at least seven independent ways.
Less drastically, the concluding section in Williams et al. assumes the structure's total count #E of 'edges' (bars + cables) and the count #N of ends satisfy
#E = 3#N – 6,
giving
c + R = 3(2R ) – 6, and thus c = 5R – 6, "so that it is truely optimal": for us, one too few. Williams does not there explain 'optimal', but elsewhere he says that anchored structures with the corresponding property "are minimal in the sense that they do not have unnecessary edges", though without defining 'necessary'. The condition #E = 3#N – 6 certainly does not give a smallest possible number (the3R cables of Paul et al. do that), and fails to reach the 5R – 5 needed for full stability. A structure may have cables that it could lose and remain stable — we could add two such to the 10-cable structure (b) above and cut them — but that structure itself is floppy if we omit anything, so each of its joins is necessary, by any definition of 'stable' or 'rigid'. Williams' remark that #E = 3#N – 6 "describes all of the structures built by Snelson" implies that Snelson never built a fully stable tensegrity like (b), whose c = 10 cables and n = 3 bars do satisfy with '=' the necessary (but not sufficient) condition c 5R – 5. This is a pity.
The 'marching' of Williams and Micheletti is creation of a path along the `rank-deficiency manifold' which excludes full stability by definition: start from a beginning point that fails full stability, and find others. In this discussion, we want neither the one, nor the others.
The 10-cable structure (b) is also a counter-example to Whittier's assertion that tensegrities "always exhibit an infinitesimal flex", which precisely means that they are never fully stable: (b) does not exhibit an infinitesimal flex. There are many weaker stability/stiffness criteria in the literature, which ensure that (if carefully made) a structure does not actually fall apart. We will not list these criteria in detail.

It is possible to do a great deal better.


This is a first step to a Web write-up of work by Tim Poston, with a lot of help from a lot of students:
Notably Badal Yadav, Ajay Harish, Siddharth Singh, Sumegha Mantri and Akshata Krishnamurthy,
though many others have added interesting work and insights [add list].
We are working on a bells-and-whistles version which will include all the necessary mathematics at an undergraduate-accessible level, scores of figures, videos, etc., and some larger fully-stable structures. But for now, that is still
pregnancy image, words 'under construction'