A Practical Guide to Tensegrity Design: 6.2.4 Dome Step 4: Add guys

]> A Practical Guide to Tensegrity Design: 6.2.4 Dome Step 4: Add guys

A Practical Guide to Tensegrity Design
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6.2.3 Dome Step 3: Adjust the base points

6.2.4 Dome Step 4: Add guys

With the truncation methodology discussed here, adding guys, and points on the ground to attach them to, is usually advisable. A valid tensegrity could be obtained without these guys, but it would be a very rickety one. Minor lateral forces applied to the structure would move it substantially. With the guys in place, the structure will resist lateral forces more robustly.

The guys are where the outer layer of tendons meets the ground. Their attachment points should be chosen so they mimic the effect of the outer-layer tendons which would have appeared in this vicinity but were discarded due to the truncation. The guy attachment points are in the same plane as the base points and will fall on a circle which is a dilatation of the base-point circle. More precisely, the attachment-point circle is chosen to be the intersection of a sphere approximating the outer layer of tendons with the ground. Call the radius of this circle ${r'}_{avg}$ . Figure 6.10 shows a cross-section of the dome and sphere with the measurements of ${r'}_{avg}$ , and $r_{avg}$ and $h_{avg}$ from Section 6.2.3, shown.

two-dimensional diagram with dome/sphere layers, base, and lengths

Figure 6.10: Double-Layer Dome: Base-Point and Guy-Attachment-Point Radii

${r'}_{avg}$ is calculated using the formula

${r'}_{avg} = \sqrt{{(\frac{Σ_{i = 1}^{\frac{n_{h}^{s}}{2}} |P_{i}'|}{\frac{n_{h}^{s}}{2}})}^{2} - h_{avg}^{2}}$

where $n_{h}^{s}$ is the number of basic points in the sphere, and $P_{i}'$ is an outer-layer basic point of the sphere. For the 6ν sphere, the value of $n_{h}^{s}$ is 18, and the value of ${r'}_{avg}$ is $\sqrt{{5.15085}^{2} - {(-2.176104)}^{2}} = 4.66860$ .

Another question is how much to rotate the guy-attachment points relative to the base points. A sensible place to start would seem to be half the angle between the base points, $\frac{π}{9}$ in this case. These can be adjusted later if that can help ease distortions of the realization of the sphere's configurations in the dome.

With this in mind, it seems reasonable to put the guy lengths in the objective function to let the computations themselves give feedback on the necessary rotation factor. The guy weights should be chosen also so as to aid the realization of the sphere's configurations in the dome as closely as possible.

Table 6.22 lists the coordinates which resulted from applying the above procedures to deriving the guy-attachment points. Table 6.23 gives the data for the one guy-attachment point which is generated using a symmetry transformation. Table 6.24 enumerates the data for the six guys which are added to the model in this step.

Point

Coordinates

x

y

z

P_{30}'

P_{31}'

P_{32}'

-4.795937	-0.711963	1.738777
-3.058334	-3.264453	2.553663
-0.477574	-4.877340	1.585790

Table 6.22: 6ν T-Octahedron Dome: Guy Attachment Point Coordinates

Point

Coordinates

x

y

z

Basic
Point

Transform
Number

P_{33}'

z_{30}'

x_{30}'

y_{30}'

P_{30}'

Table 6.23: 6ν T-Octahedron Dome: Guy-Attachment-Point Symmetry Correspondence

Member
#

End Points

Weight

Constrained
Length

Sphere
Member

175

176

177

178

179

180

$P_{30}'$	$P_{23}'$
$P_{23}'$	$P_{31}'$
$P_{31}'$	$P_{24}'$
$P_{24}'$	$P_{32}'$
$P_{32}'$	$P_{28}'$
$P_{28}'$	$P_{33}'$

0.4000

N/A

Table 6.24: 6ν T-Octahedron Dome: Guys

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6.2.5 Dome Step 5: Compute the dome