]> A Practical Guide to Tensegrity Design: 6.2 A Procedure for Designing Double-Layer Tensegrity Domes

6.2 Double-Layer Tensegrity Domes: An Example

The following steps implement the design of a double-layer dome like that described in Section 6.1:

 Step 1 Solve the tensegrity programming problem for the sphere. Step 2 Implement the topological changes required by the truncation. Step 3 Adjust the base points (the points of the truncation polylateral as they manifest themselves on the inner tendon network) so they lie evenly-spaced on a circle which approximates as closely as possible their unadjusted positions in the original sphere. Step 4 Add guys. Step 5 Using the coordinate values from the sphere as initial values, solve the tensegrity programming problem for the dome. Step 6 Make necessary adjustments to fix member force and interference problems.

To illustrate this method for truncating double-layer spheres, the tensegrity based on the 6ν octahedron is useful. It has a low-enough frequency to be pedagogically tractable and a high-enough frequency that the appearance of higher-frequency structures can be anticipated in studying it.

6.2.1 Dome Step 1: Compute the sphere

Figures 6.3 and 6.4 diagram the basic triangle network for the 6ν double-layer tensegrity octahedron sphere and a coordinate system for its analysis in the same manner as Figures 5.3 and 5.5 did for the 4ν version of the sphere in Section 5.3. The main difference is that, with the higher frequency, there is more of everything. For example, now the struts in Figure 6.4 are clustered about three basic t-tripods instead of two as in Figure 5.5.

Table 6.1 enumerates the members of this 6ν version of the double-layer sphere. The anomalous value of 1.5 for the length of Member #33 in Table 6.1 is chosen in light of experience with the 4ν structure.1

The weights for the inner and outer binding tendons in the objective function are derived using the formula $k ⁢ b1 +b2 2⁢ b1⁢ b2 2$ where the values used for $k$ are 1.2 and 0.5 respectively for the inner and outer binding tendons. $b1$ and $b2$ represent the spherical excess corresponding to the initial values of the two end points of the tendon.2 The spherical excess is the amount the sphere radius exceeds the distance of the unprojected point from the center of the octahedron. This number is calculated as a ratio and is always greater than or equal to 1.0. It is equal to 1.0 at the vertexes of the octahedron. Giving a smaller weight to the tendons distant from the vertexes of the basis octahedron allows them to be longer than they would otherwise be. This allows the octahedral faces to bulge out more than they would otherwise and gives the structure a more spherical, less faceted, look. The objective-function weights for the primary and secondary interlayer tendons are 2.0 and 1.4 respectively independent of any spherical excess values. Figure 6.3: 6ν T-Octahedron Sphere: Symmetry Regions Figure 6.4: 6ν T-Octahedron Sphere: Truss Members

Member
#
End Points   Weight   Constrained
Length
Comments
 1 2 3 4 5 6 7 8 9
 $P1'$ $P11$ $P2'$ $P7$ $P3'$ $P10$ $P4'$ $P12$ $P9'$ $P15$ $P8'$ $P2$ $P7'$ $P8$ $P5'$ $P3$ $P14'$ $P6$
 N/A N/A N/A N/A N/A N/A N/A N/A N/A
 3 3 3 3 3 3 3 3 3
Struts
 10 11 12 13 14 15 16 17 18
 $P2'$ $P11$ $P3'$ $P7$ $P1'$ $P10$ $P9'$ $P12$ $P8'$ $P15$ $P4'$ $P2$ $P13'$ $P8$ $P6'$ $P3$ $P7'$ $P6$
 2 2 2 2 2 2 2 2 2
 N/A N/A N/A N/A N/A N/A N/A N/A N/A
Primary Interlayer Tendons
 19 20 21 22 23 24 25 26 27
 $P1'$ $P2$ $P2'$ $P3$ $P3'$ $P1$ $P4'$ $P9$ $P9'$ $P8$ $P8'$ $P4$ $P7'$ $P13$ $P5'$ $P6$ $P14'$ $P7$
 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4
 N/A N/A N/A N/A N/A N/A N/A N/A N/A
Secondary Interlayer Tendons
 28 29 30 31 32 33 34 35 36
 $P2$ $P11$ $P3$ $P7$ $P1$ $P10$ $P9$ $P12$ $P8$ $P15$ $P4$ $P2$ $P13$ $P8$ $P6$ $P3$ $P7$ $P6$
 N/A N/A N/A N/A N/A N/A N/A N/A N/A
 1 1 1 1 1 1.5 1 1 1
Inner Convergence Tendons
 37 38 39 40 41 42 43 44 45
 $P1'$ $P2'$ $P2'$ $P3'$ $P3'$ $P1'$ $P4'$ $P9'$ $P9'$ $P8'$ $P8'$ $P4'$ $P7'$ $P13'$ $P5'$ $P6'$ $P14'$ $P7'$
 N/A N/A N/A N/A N/A N/A N/A N/A N/A
 1 1 1 1 1 1 1 1 1
Outer Convergence Tendons
 46 47 48 49 50 51 52 53 54
 $P2'$ $P11'$ $P3'$ $P7'$ $P1'$ $P10'$ $P9'$ $P12'$ $P8'$ $P15'$ $P4'$ $P2'$ $P13'$ $P8'$ $P6'$ $P3'$ $P7'$ $P6'$
 0.5 0.3065 0.4196 0.3065 0.2692 0.4196 0.3065 0.3065 0.2692
 N/A N/A N/A N/A N/A N/A N/A N/A N/A
Outer Binding Tendons
 55 56 57 58 59 60 61 62 63
 $P1$ $P2$ $P2$ $P3$ $P3$ $P1$ $P4$ $P9$ $P9$ $P8$ $P8$ $P4$ $P7$ $P13$ $P5$ $P6$ $P14$ $P7$
 1.2 1.0069 1.0069 0.7356 0.6462 0.7356 0.7356 0.7356 0.6462
 N/A N/A N/A N/A N/A N/A N/A N/A N/A
Inner Binding Tendons

Table 6.1: 6ν T-Octahedron Sphere: Truss Members

As with the 4ν version of this sphere, the derivation of the initial point values is facilitated by the use of the geodesic breakdown. Kenner's tables3 are used to generate initial point coordinates. Again, Kenner's table has to be expanded by rotating all the points about the $z$ axis by 90°. Table 6.2 outlines the correspondence between the basic points and his coordinate system. (Rotated points are indicated with an asterisk.)

 Point
 Kenner's Label
Coordinates
 $θ$ $φ$
 1,0 1,1 2,1 2,1* 3,0 3,1 3,2 3,1* 3,2*
 0 11.3099 90 11.3099 45 19.4712 135 19.4712 0 45 26.5651 36.6992 63.4349 36.6992 116.565 36.6992 153.435 36.6992

Table 6.2: 6ν T-Octahedron: Angular Point Coordinates

The initial coordinate values for inner and outer realizations of these points are summarized in Table 6.3. These are derived from the angular values in Table 6.2 with inner and outer radiuses applied. The inner radius (3.15) is chosen so the triangle tendon lengths average approximately 1 (0.995729). The outer radius (5.15) is chosen so strut lengths in the double-layer versions of the structure would initially average approximately 3 (2.94314). The implied initial lengths are summarized in Table 6.4.

 Point
Coordinates
 $x$ $y$ $z$
 $P1$ $P2$ $P3$ $P4$ $P5$ $P6$ $P7$ $P8$ $P9$
 0.6178 0 3.0888 0 0.6178 3.0888 0.7425 0.7425 2.9698 -0.7425 0.7425 2.9698 2.2274 0 2.2274 1.6837 0.8419 2.5256 0.8419 1.6837 2.5256 -0.8419 1.6837 2.5256 -1.6837 0.8419 2.5256
 $P1'$ $P2'$ $P3'$ $P4'$ $P5'$ $P6'$ $P7'$ $P8'$ $P9'$
 1.01 0 5.05 0 1.01 5.05 1.2139 1.2139 4.8555 -1.2139 1.2139 4.8555 3.6416 0 3.6416 2.7528 1.3764 4.1292 1.3764 2.7528 4.1292 -1.3764 2.7528 4.1292 -2.7528 1.3764 4.1292

Table 6.3: 6ν T-Octahedron Sphere: Initial Basic Point Coordinates

 Member # Length Member # Length Member # Length 1 2.5487 2 2.7450 3 2.7577 4 3.0672 5 3.3095 6 2.7450 7 2.9385 8 3.0672 9 3.3095 10 2.2908 11 2.4057 12 2.2248 13 2.4057 14 2.5135 15 2.2248 16 2.4057 17 2.4057 18 2.5135 19 2.2908 20 2.2248 21 2.2248 22 2.4057 23 2.5135 24 2.4057 25 2.4057 26 2.4057 27 2.5135 28 0.8737 29 1.0456 30 0.7622 31 1.0456 32 1.1906 33 0.7622 34 1.0456 35 1.0456 36 1.1906 37 1.4284 38 1.2461 39 1.2461 40 1.7094 41 1.9465 42 1.7094 43 1.7094 44 1.7094 45 1.9465 46 1.4284 47 1.7094 48 1.2461 49 1.7094 50 1.9465 51 1.2461 52 1.7094 53 1.7094 54 1.9465 55 0.8737 56 0.7622 57 0.7622 58 1.0456 59 1.1906 60 1.0456 61 1.0456 62 1.0456 63 1.1906

Table 6.4: 6ν T-Octahedron Sphere: Initial Member Lengths

The derivation of the symmetry points from the basic points is shown in Table 6.5. The symmetry transforms on which this table is based are enumerated in Table 5.5. Outer points follow the same symmetries as inner points.

 Point
Coordinates
 $x$ $y$ $z$
 Basic Point Transform Number
 $P10$ $P11$ $P12$ $P13$ $P14$ $P15$
 $-x4$ $-y4$ $z4$ $-x1$ $-y1$ $z1$ $-x5$ $-y5$ $z5$ $y5$ $z5$ $x5$ $y6$ $z6$ $x6$ $-y9$ $z9$ $-x9$
 $P4$ 4 $P1$ 4 $P5$ 4 $P5$ 2 $P6$ 2 $P9$ 5

Table 6.5: 6ν T-Octahedron Sphere: Symmetry Point Correspondences

The structure is computed by minimizing a weighted combination of the interlayer and binding tendons subject to constraints on the struts and convergence tendons. Two initial iterations are done using the penalty formulation $μ‾= 105$ in conjunction with Fletcher-Reeves to bring the initial points into approximate conformance with the constraints. After this five 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 $10 -3$.

Tables 6.6 and 6.7 show the values for the final lengths and relative forces.4 Table 6.8 shows the final values for the coordinates of the basic points. Figure 6.5 shows how the final version of the spherical structure appears as viewed from outside one of the octahedral vertices. For clarity, interlayer tendons have been excluded and members in the background have been eliminated by truncation. For reference, selected points in Figure 6.5 are labeled.

 Member # Length Member # Length Member # Length 1 3.0000 2 3.0000 3 3.0000 4 3.0000 5 3.0000 6 3.0000 7 3.0000 8 3.0000 9 3.0000 10 2.3545 11 2.3871 12 2.4881 13 2.2793 14 2.2883 15 2.3153 16 2.2212 17 2.2209 18 2.2354 19 2.1286 20 2.0833 21 2.1669 22 2.0342 23 2.0334 24 1.6827 25 2.0342 26 2.0454 27 2.0516 28 1.0000 29 1.0000 30 1.0000 31 1.0000 32 1.0000 33 1.5000 34 1.0000 35 1.0000 36 1.0000 37 1.0000 38 1.0000 39 1.0000 40 1.0000 41 1.0000 42 1.0000 43 1.0000 44 1.0000 45 1.0000 46 1.8502 47 2.4709 48 2.1613 49 2.6524 50 2.7414 51 2.4735 52 2.7549 53 2.6798 54 2.6482 55 1.2081 56 1.2653 57 1.2626 58 1.3406 59 1.6730 60 1.2480 61 1.9008 62 1.8434 63 1.9693

Table 6.6: 6ν T-Octahedron Sphere: Final Member Lengths

 Member # Relative  Force Member # Relative  Force Member # Relative  Force 1 -11.294 2 -9.788 3 -10.052 4 -10.125 5 -10.019 6 -9.925 7 -10.052 8 -10.064 9 -9.870 10 4.709 11 4.774 12 4.976 13 4.559 14 4.577 15 4.631 16 4.442 17 4.442 18 4.471 19 2.980 20 2.917 21 3.033 22 2.848 23 2.847 24 2.356 25 2.848 26 2.863 27 2.872 28 4.945 29 4.580 30 3.811 31 5.009 32 5.092 33 4.947 34 4.958 35 5.258 36 5.163 37 4.144 38 4.887 39 4.040 40 4.865 41 4.867 42 5.214 43 4.815 44 5.083 45 5.547 46 0.925 47 0.757 48 0.907 49 0.813 50 0.738 51 1.038 52 0.844 53 0.821 54 0.713 55 1.450 56 1.274 57 1.271 58 0.986 59 1.081 60 0.918 61 1.398 62 1.356 63 1.273

Table 6.7: 6ν T-Octahedron Sphere: Final Member Forces

 Point
Coordinates
 $x$ $y$ $z$
 $P1$ $P2$ $P3$ $P4$ $P5$ $P6$ $P7$ $P8$ $P9$
 1.0378 -0.236 3.5592 -0.064 0.2053 3.7844 0.8711 1.0149 3.5173 -1.0998 1.2224 3.4065 2.94 -0.4191 2.34 1.7538 0.7511 3.1285 1.3434 1.6303 2.8864 -1.4134 2.2919 2.8451 -2.3233 0.8934 2.9682
 $P1'$ $P2'$ $P3'$ $P4'$ $P5'$ $P6'$ $P7'$ $P8'$ $P9'$
 1.3525 0.2829 5.3714 0.3628 0.4068 5.444 0.9467 1.1801 5.1968 -1.561 1.7442 4.6513 3.7764 0.4745 3.0005 3.0842 0.8558 3.6132 1.439 2.929 3.5221 -2.1735 2.3144 4.1038 -2.4411 1.3815 4.345

Table 6.8: 6ν T-Octahedron: Final Basic Point Coordinates Figure 6.5: 6ν T-Octahedron Sphere: Vertex View

1 See Section 8.2.3 for details on this exception as it is introduced to the 4ν structure.

2 $b$ stands for bulge.

3 Kenner76, "Octahedron Class I Coordinates: Frequencies 12, 6, 3", column 6ν, p. 126.

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