0.2 Tensegrity

Tensegrity and Trusses

Introduction

The term “tensegrity”, coined by Buckminster Fuller in the 1960s, is a combination of the words “tension” and “integrity”. In a tensegrity system, the compression members do not touch each other, and are stabilized by the tension members. Skelton and Oliveira, in their paper [link] define a class-k tensegrity system as “a tensegrity system such that no more than k rigid bodies make connections at a given node (with frictionless ball joints).”

Due to the high strength and low weight of tensegrity structures, Fuller imagined building “unprecedentedly large” structures [link] , perhaps even large enough to cover a city. Kenneth Snelson, a former student of Fuller's, is renowned for his tensegrity sculptures, which are on display around the world. In addition to manufactured structures, tensegrity systems are abundant in nature. The elbow is a class 3 tensegrity joint, while the foot and shoulder are class 2 tensegrity joints. In a red blood cell, the protofilament is the compression member and the spectrin dimer are the tensile member [link] .

This module further examines properties of tensegrity systems by looking at the paper “Optimal tensegrity structures in bending: The discrete Michell truss” by Robert E. Skelton and Mauricio C. de Oliveira. The following analysis and diagrams come largely from their paper, with changes made by the 2013 Michell group to help clarify their ideas for readers at the undergraduate level.

Michell spirals

Consider a set of line segments, connected as shown below. In the diagram, $\phi =\frac{\pi }{16}$ and $\beta =\frac{\pi }{6}$ .

Definition 1 Let $r_{\ell }$ define a set of radii from a common origin, $\mathbf{0}$ , for $\ell =0,1,2,...,q$ . Let $p_{\ell }$ , $\ell =0,1,2,..,q-1$ define the lengths of lines beginning at points with radius $r_{\ell }$ and terminating at points with radius $r_{\ell +1}$ . Then a Michell spiral of order $q$ is defined by the connections of lines of length $p_{\ell }$ , satisfying

where $a,c>0$ .

By a visual inspection, one can observe that

and

where $\phi$ is the angle between each radius and $\beta$ is the angle between the beam and the radius.

Therefore, using some algebra and trigonometry, one can get that

In order for the spirals to converge to the origin, it must be that $a<1$ , which corresponds to $\phi +2\beta <\pi$ .

Michell topology

The magnitude of a node is determined by the radius on which it lies. That is,

for all $i+k=m+n$ .

Notice from the diagram above that nodes with the same radius are related by a phase-shift of $2m\phi$ , where $m$ is an integer. Using complex notation, this can be expressed as

The line segments comprising the spiral are described by defining the vector connecting nodes ${\mathbf{n}}_{ik}$ and ${\mathbf{n}}_{i,k+1},$

where the vector ${\mathbf{n}}_{ik}$ has magnitude and phase given by

such that $r_k$ satisfies the conditions in the definition for some specified $r_0$ . The mirror image of all lines reflected about the horizontal axis are obtained by computing the conjugate of the vectors ${\mathbf{m}}_{ik}$ , and shown in [link] below.

Note that, by a visual inspection, ${\mathbf{m}}_{ik}=p_{i+k}{e}^{j[\beta +(i-k\left)\phi \right]}$ , where $p_m,$ $m=0,1,2,...,q,$ satisfy [link] .