]> A Practical Guide to Tensegrity Design: 5.4 Double-Layer Tensegrities: Hexagon/Triangle Networks

5.4 Double-Layer Tensegrities: Hexagon/Triangle Networks

A second approach to designing tensegrity trusses relies on networks which have triangles alternating with hexagons and pentagons, rather than triangles alternating with triangles as with the first approach. An advantage of this approach over the approach of Section 5.3 is that it works with all symmetries; in addition to Class I and Class II breakdowns of the octahedron, geodesic breakdowns of the tetrahedron and icosahedron can be used.

Geodesic networks are used here only as a first step in the derivation of a network. In the geodesic network's triangles, attention is now also placed on the hexagons which fill up the gaps between the triangles. Thus, these triangles and the gaps between them form a system of alternating triangles and hexagons except at the vertices of the base polyhedron where a triangle, square or pentagon is substituted for a hexagon.

2v icosahedron
Figure 5.7: 2ν Icosahedron

For an example, see Figure 5.7 which illustrates a 2ν icosahedron. At this low frequency, the single triangles on each icosahedral face surround pentagonal gaps which correspond to the vertices of the base icosahedron. At higher frequencies, hexagonal gaps would appear on the edges (as in Figure 4.2) and/or the faces of the base polyhedron. At high frequencies, the hexagonal gaps dominate since the occasional pentagonal, square or triangular gaps only appear at the vertices of the base polyhedron. Hence, the final network is referred to as a "hexagon/triangle" network even though at the lowest 2ν frequency hexagons don't appear at all.

This recontextualized geodesic network is not suitable for a t-tripod-based tensegrity truss though since adjacent polylaterals share edges rather than just points. A suitable network can easily be constructed though by inscribing a smaller version of each polylateral within that polylateral by connecting the midpoints of its sides appropriately. This technique is illustrated for the 2ν icosahedron in Figure 5.8.

2v icosahedron inscribed with alternating vertex-tangent triangle/pentagon network
Figure 5.8: Hexagon/Triangle Tensegrity Network Inscribed in 2ν Icosahedron

As in Section 5.3, this network is projected on a sphere and duplicated to form an inner and outer network. The triangles on the outer network form the apexes of outwardly-pointing t-tripods (the outer convergence triangles), while the hexalaterals on the outer network form the tendons which bind them together (the outer binding hexalaterals). On the inner sphere, the hexalaterals form the apexes of inwardly-pointing t-hexapods (the inner convergence hexalaterals), and the triangles form the tendons which bind them together (the inner binding triangles). The struts and their corresponding tendons (the primary and secondary interlayer tendons) connect the triangles on the outer network with the hexalaterals on the inner network. As before, struts are placed so that the untwisting effect of the binding tendons is enhanced.

small rhombicosidodecahedron with rectangles instead of squares pierced by four axes with points labeled
Figure 5.9: 2ν Hexagon/Triangle T-Icosahedron: Coordinate System

Figure 5.9 illustrates this network as represented in Cartesian coordinates. This representation is meant to exploit the octahedral symmetries of the icosahedron as much as possible. Thus many of the symmetry points are simple signed permutations of the basic points. To capture the icosahedral symmetries however, a general transformation matrix must be introduced.

In Figure 5.9, the axis labeled P represents the five-fold symmetry axis about which the structure is transformed.12 This axis goes through a vertex of the reference unit-side-length icosahedron. The coordinates of this vertex are 12 0τ2 where tau 1+52 1.618034 is the ratio constant of the golden section. This transformation is needed to express P2 in terms of the basic point P1 .13 P2 is generated from P1 by a -72° rotation of the structure about the axis P. The matrix which achieves this transformation is:14

T 12 τ2 1 2τ -τ2 1 2τ 12 1 2τ -12 τ2

Thus each coordinate of P2 is represented as a linear combination of the coordinates of P1. In the model, this substitution could be made in all the formulas. However, it is simpler just to consider P2 as a basic point and introduce the transformation matrix as three constraints expressing the coordinates of P2 as linear combinations of the coordinates of P1.

inverted penta-pod next to tripod it shares a strut with with axes and point labels
Figure 5.10: 2ν Hexagon/Triangle T-Icosahedron: Truss Members

Figure 5.10 illustrates the basic members of the structure as well as an outline of some of the symmetry members embedded in the coordinate system which is used to analyze the structure. The low frequency of the structure means there are very few basic members to keep track of. On the other hand, the high order of symmetry of the icosahedron means that the structure as a whole will encompass about as much space as a structure based on a more complicated 4ν breakdown of the octahedron. This symmetry-induced simplicity is an important consideration in favor of icosahedral structures. Table 5.9 summarizes the member breakdown including weights for members included in the objective function and length constraints for the others.

 End Points   Weight   Constrained 
P3' P2
 N/A   3.0   Strut 
P1' P2
 2.00   N/A   Primary Interlayer Tendon 
P3' P1
 2.00   N/A   Secondary Interlayer Tendon 
P1' P3'
 N/A   1.0   Outer Convergence Tendon 
P1' P2'
 0.45   N/A   Outer Binding Tendon 
P1 P3
 1.00   N/A   Inner Binding Tendon 
P1 P2
 N/A   1.0   Inner Convergence Tendon 
Table 5.9: 2ν Hexagon/Triangle T-Icosahedron: Truss Members

The mathematical programming problem reduces to:

minimize o 2 P1' -P22 + P3' -P12 + 25 P1' -P2' 2 + P1 -P32 P1, P1', P2, P2' subject to Tendon constraints: 1 P1' -P3' 2 1 P1 -P22 Strut constraint: -9 - P3' -P22 Symmetry constraints: P2 = TP1 P2' = TP1 '

The latter "two" constraints actually represent six linear constraints in all and are the icosahedral symmetry transformations.

The next thing needed is initial coordinate values for the computation. These are derived from the coordinates the unit icosahedron vertices, in particular, the coordinates of the icosahedral triangle generated by axes permutations located in the positive octant. Table 5.10 summarizes these coordinate values. The locations of 4 V1, 4 V2 and 4 V3 are shown in Figure 5.9.

 Vertex    x     y     z  
V1   12     0     τ2  
V2   0     τ2     12  
V3   τ2     12     0  
Table 5.10: Unit Icosahedron: Selected Vertex Coordinates

Taking the midpoints of the sides of the triangle represented by these three points yields the vertices of a triangle of a half-scale version of the unit icosadodecahedron. Taking the midpoints of this second triangle and multiplying by four yields the vertices of a triangle of a unit-scale version of the the reference network for the tensegrity being analyzed here. The coordinates of the point needed are:

2+τ2 1+τ2 1+2 τ2

This serves as the initial value for P2. The initial value for P2' is computed by scaling up P2 until the strut length constraint is approximately satisfied. A value of 1.5 for the scale factor worked satisfactorily here. These initial coordinate values are summarized in Table 5.11.

 Point    x     y     z  
P1   1.809017     1.309017     2.118034  
P1'   2.713525     1.963525     3.177051  
Table 5.11: 2ν Hexagon/Triangle T-Icosahedron: Initial Basic Point Coordinates

Since P2 and P2' are being treated as control variables as well, initial values must be supplied for them. These initial values are computed by multiplying P1 and P1' by T. The coordinates of P3 expressed in terms of P1 are z1 x1 y1 . P3' has the same relationship with P1' . These last relationships fully determine the model.

The model is solved using a similar approach to that used for the 4ν t-octahedron in Section 5.3. An initial iteration is done using the penalty formulation μ= 105 in conjunction with Fletcher-Reeves. After this 10 iterations are done with the exact formulation in conjunction with Fletcher-Reeves to bring the values to convergence. The derivatives of the objective function with respect to the independent coordinate values are all less than 10-6. Member clearances are all greater than 0.15 model units.

Table 5.12 shows the values for the final lengths and relative forces;15 Table 5.13 shows the final values for the coordinates of the basic points, and Figure 5.11 shows the final design.

 Length   Relative 
1  3.000000   -11.325 
2  2.395526   4.791 
3  2.017577   4.035 
4  1.000000   4.032 
5  2.241086   1.008 
6  1.471948   1.472 
7  1.000000   5.899 
Table 5.12: 2ν Hexagon/Triangle T-Icosahedron: Final Member Lengths and Forces

 Point    x     y     z  
P1   1.635712     0.467068     1.294325  
P1'   2.427554     1.611718     1.991202  
Table 5.13: 2ν Hexagon/Triangle T-Icosahedron: Final Coordinate Values

view of final design for double-layer 2v tensegrity icosahedron
Figure 5.11: 2ν Hexagon/Triangle T-Icosahedron: Final Design

12 P stands for pentagon.

13 Cartesian coordinates allow P3 to be expressed more simply as just a permutation of P1 P3= z1 x1 y1 .

14 Derived using a formula provided in Rogers76, Chapter 3. See Section 6.2.3 for a general statement of the formula.

15 See Section 7.2 for the method of computing relative forces.