A Practical Guide to Tensegrity Design

Table of Contents

6.2.2 Dome Step 2: Implement the truncation

6.2.3 Dome Step 3: Adjust the base points

Table 6.20 shows the unadjusted
coordinate values for the base points. The mathematical programming
problem for the dome treats these three points as fixed.
As mentioned in Section 6.1,
according to some definitions of tensegrity, once these points are fixed
the structure is no longer a tensegrity since it is no longer
self-supporting.^{5}

Practically speaking, this seems a useful approach to developing a dome. So, the self-support requirement some definitions make for a true tensegrity will be ignored and the constraints to fix the base points will be included with the point constraints discussed in Chapter 3. Certainly it would be possible to develop a dome which met the self-support requirement. Another truncation technique would need to be developed, but the truncation would probably be a little more ragged looking and the structural support from the fixed base points would probably be missed. The resulting dome would be more mobile though.

To facilitate construction and perhaps make the structure more aesthetically pleasing, the base points for the not-quite-a-tensegrity dome being designed here are adjusted to lie evenly spaced on a circle about the symmetry axis of the dome. This section gives the details of how that adjustment is made.

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Table 6.20: 6ν T-Octahedron Dome: Base Point Initial Raw Coordinate Values |

The symmetry axis is the line through the origin and the point $\left(1.0,1.0,1.0\right)$. It is convenient to normalize the corresponding vector so it has length 1.0, so the vector $\left(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}\right)$ is used whenever the symmetry axis is needed for computations and is called $A$.

The adjusted points lie on a circle chosen so that the points
are moved as little as possible because of the adjustment.
The radius of the circle is the average distance
of the unadjusted points from the symmetry
axis.^{6}
This value is called
${r}_{avg}$.
In addition, the adjusted points are selected so that they all
have the same value when projected onto the symmetry axis,
and it will be the average of the values
for the three unadjusted points. This common value is called
${h}_{avg}$.
The projection is computed by taking the
dot product of the point with $A$.
The component of the point
orthogonal to the axis is called
${P}_{i}^{\perp}$
and is computed using the formula
${P}_{i}^{\perp}={P}_{i}-\left({P}_{i}\cdot A\right)A$.
This data is summarized in Table 6.21.

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Table 6.21: 6ν T-Octahedron Dome: Raw Base Point Characteristics |

The adjusted value of
${P}_{22}$,
call it
${P}_{22}^{*}$,
is generated using the formula
${P}_{22}^{*}={h}_{avg}A+\frac{{r}_{avg}}{\left|{P}_{22}^{\perp}\right|}{P}_{22}^{\perp}$.
The adjusted values for the other two points are generated by
by rotating
${P}_{22}^{*}$
about the symmetry axis by
$\frac{2\mathrm{\pi}}{9}$ and
$\frac{4\mathrm{\pi}}{9}$.
Nine is chosen as the divisor for the two rotation angles
since there are nine base points when all symmetry
transformations are taken into account. The general matrix for rotating
a point about a normalized (so it has length one) vector
$\left(x,y,z\right)$
by an angle $\theta $
is:^{7}

$\left[\begin{array}{ccc}{x}^{2}+\left(1-{x}^{2}\right)cos\theta & xy\left(1-cos\theta \right)-zsin\theta & xz\left(1-cos\theta \right)+ysin\theta \\ xy\left(1-cos\theta \right)+zsin\theta & {y}^{2}+\left(1-{y}^{2}\right)cos\theta & yz\left(1-cos\theta \right)-xsin\theta \\ xz\left(1-cos\theta \right)-ysin\theta & yz\left(1-cos\theta \right)+xsin\theta & {z}^{2}+\left(1-{z}^{2}\right)cos\theta \end{array}\right]$

In the present situation, the normalized vector in question is just $A$ and the value of $\theta $ is $\frac{2\mathrm{\pi}}{9}$. Substituting these values yields the matrix:

$\left[\begin{array}{ccc}\frac{1}{3}+\frac{2}{3}cos\left(\frac{2\mathrm{\pi}}{9}\right)& \frac{\left(1-cos\left(\frac{2\mathrm{\pi}}{9}\right)\right)}{3}-\frac{sin\left(\frac{2\mathrm{\pi}}{9}\right)}{\sqrt{3}}& \frac{\left(1-cos\left(\frac{2\mathrm{\pi}}{9}\right)\right)}{3}+\frac{sin\left(\frac{2\mathrm{\pi}}{9}\right)}{\sqrt{3}}\\ \frac{\left(1-cos\left(\frac{2\mathrm{\pi}}{9}\right)\right)}{3}+\frac{sin\left(\frac{2\mathrm{\pi}}{9}\right)}{\sqrt{3}}& \frac{1}{3}+\frac{2}{3}cos\left(\frac{2\mathrm{\pi}}{9}\right)& \frac{\left(1-cos\left(\frac{2\mathrm{\pi}}{9}\right)\right)}{3}-\frac{sin\left(\frac{2\mathrm{\pi}}{9}\right)}{\sqrt{3}}\\ \frac{\left(1-cos\left(\frac{2\mathrm{\pi}}{9}\right)\right)}{3}-\frac{sin\left(\frac{2\mathrm{\pi}}{9}\right)}{\sqrt{3}}& \frac{\left(1-cos\left(\frac{2\mathrm{\pi}}{9}\right)\right)}{3}+\frac{sin\left(\frac{2\mathrm{\pi}}{9}\right)}{\sqrt{3}}& \frac{1}{3}+\frac{2}{3}cos\left(\frac{2\mathrm{\pi}}{9}\right)\end{array}\right]$

Applying this matrix once to ${P}_{22}^{*}$ yields ${P}_{23}^{*}$. Applying this matrix twice to ${P}_{22}^{*}$ yields ${P}_{24}^{*}$. The adjusted values are what appear in Table 6.17.

^{5}
See *Wang98*.
Wang goes beyond the definition by Pugh
quoted in Chapter 1 to identify the following
characteristics of a tensegrity structure:

- It is composed of compression and tension elements.
- The struts (compression elements) are discontinuous while the cables (tension elements) are continuous.
- The structure is rigidified by self-stressing.
- The structure is self-supporting.

^{6}
See Section 8.2.1 for
the formula for calculating the distance of a point from a line.

^{7}
From *Rogers76*, Chapter 3.