Point x y z
P1 0.887555 -0.438450 0.455646 P2 0.677306 -0.505030 0.989215 P3 0.614181 -0.076748 0.705421 P4 0.710900 -0.048791 0.590190
Table 4.6: 4ν Diamond T-Tetrahedron: Preliminary Coordinate Values Member
ID Length
tab1 0.940409 tab2 0.448489 tbb1 0.455651 tbb2 0.601166
Table 4.7: 4ν Diamond T-Tetrahedron: Preliminary Objective Member Lengths lengths of the members in the objective function thus obtained.
This would be the end of the calculations, except that when the endogenous member forces are calculated, they indicate that “tendon” t12 is marginally in compression (see Table 7.1). This problem stems from the substitution of equalities for inequalities in the constraints. If inequalities had been used, this particular constraint would be found to be not effective. At this point the problem is dealt with by eliminating the member from the constraints which means the tendon doesn’t appear in the final structure.4 Eliminating this constraint also means a new selection of independent variables needs to be made since seven are now needed. Repartitioning results in z1 being added to the independent variables. Using the Parallel Tangents technique on this problem resulted in a final
objective-function value of 1.65174. Table 4.8 summarizes the corresponding point values;
Table 4.9 summarizes the objective function member lengths, and Figure 4.8 shows the final design where the location of the omitted tendon is indicated by a dashed line.
4.3 Zig-Zag Structures
4.3.1 Zig-Zag Structures: Descriptive Geometry
A zig-zag structure retains the struts and tendon triangles of the corresponding diamond structure; however, now adjacent tendon triangles are interconnected with only one tendon instead of two. This single tendon connects the “noses” of the two tendon triangles.
4Alternatively, its length could be shortened until it is effective.
Coordinates
Point x y z
P1 0.874928 -0.442843 0.484207 P2 0.675644 -0.506061 0.981906 P3 0.602311 -0.068420 0.715369 P4 0.699892 -0.049794 0.605188
Table 4.8: 4ν Diamond T-Tetrahedron: Final Coordinate Values Member
ID Length
tab1 0.937671 tab2 0.446946 tbb1 0.473042 tbb2 0.590748
Table 4.9: 4ν Diamond T-Tetrahedron: Final Objective Member Lengths
Figure 4.8: 4ν Diamond T-Tetrahedron: Final Design
4.3. ZIG-ZAG STRUCTURES 87
(
b
a b
P1
P2 P3
P4 P5
P6
P7
Figure 4.9: 4ν Zig-Zag T-Tetrahedron: Representative Struts
Examination of the structure from the struts’ point of view shows each strut is traversed by a “zig-zag” of three tendons. The simplest zig-zag tensegrity is the t-tetrahedron
examined in Section 2.4 (Figure 2.8). Again, since more complex zig-zag structures are not amenable to the treatment used in that simple structure, the general procedure is
illustrated using the zig-zag version of the 4ν t-tetrahedron examined in Section 4.2.
Figures 4.9 and 4.10 respectively show representative examples of the interconnecting struts and tendons. In these figures, the model has been expanded so that the struts are longer than in the initial geodesic calculation, while the tendon triangles remain the same size. This is done since, in the initial configuration, the noses of the tendon triangles touch each other and so the interconnecting zig-zag tendons have zero length. Expanding the structure without increasing the sizes of the tendon triangles gives the interconnecting tendons a non-zero length. The lengths of these tendons can be minimized to get a valid tensegrity. In the initial configuration, these tendons are certainly of minimum length, and the structure is theoretically a tensegrity in that configuration, but practically it isn’t an interesting solution since the sbb strut and its transformations intersect each other.
4.3.2 Zig-Zag Structures: Mathematical Model
The list of points is the same as that in Section 4.2.2, as is the list of constrained members. To avoid the problem of ending up with a solution in which the minimum of the
)
b
a b
P1
P2 P3 P4 P5
P6
P7
Figure 4.10: 4ν Zig-Zag T-Tetrahedron: Representative Tendons
Member
# ID End Points Comments 7 tab P3 P4 To be minimized 8 tbb P1 P5 To be minimized
Table 4.10: 4ν Zig-Zag T-Tetrahedron: Zig-Zag Tendon End Points
objective is zero, the struts sab and sbb are lengthened from √
2 and 0.919401 to 2 and √ 3 respectively. In the objective function the diamond tendons of Section 4.2.2, tab1, tab2, tbb1
and tbb2, are replaced by the zig-zag tendons tab and tbb. As mentioned, their initial lengths are zero. Table 4.10 enumerates the end points of these additional members.
The relevant mathematical programming problem becomes:
4.3. ZIG-ZAG STRUCTURES 89 minimize o ≡ |P3− P4|2+ |P1− P5|2
P1, P2, P3, P4
subject to Tendon constraints:
1
3 ≥ |P1− P2|2
tan (12π) ≥ |P1− P3|2 tan (12π) ≥ |P2− P3|2
1 ≥ |P4− P7|2
Strut constraints:
−4 ≥ −|P1− P7|2
−3 ≥ −|P2− P6|2
As before, only the coordinates of P1, P2, P3 and P4 are variables in the minimization process since the coordinates of P5, P6 and P7 are specified to be symmetry transforms of the coordinates of these points. Also, all inequality constraints are assumed to be met with equality.
4.3.3 Zig-Zag Structures: Solution
With the increased lengths of the struts, the initial values used for the problem no longer satisfy the constraints. With the best partitioning of the system (that used in Section 4.2.3), Newton’s method diverges when it is applied to the system to solve the constraint equations. So, in this case, the penalty formulation is used with a penalty value of µ = 105. The problem thus becomes:
minimize |P3− P4|2+ |P1− P5|2+ µ[13 − |P1− P2|2]2+ P1, P2, P3, P4 µ[tan (12π) − |P1− P3|2]2+ µ[tan (12π) − |P2− P3|2]2+
µ[1 − |P4− P7|2]2+ µ[4 − |P1− P7|2]2+ µ[3 − |P2 − P6|2]2 Ten iterations of the method of Fletcher-Reeves are applied to this reformulated objective function. These iterations bring the constraints close enough to a solution that the penalty formulation can be discarded for the exact formulation. Another ten iterations of Fletcher-Reeves bring the system to a solution.
The final values for the lengths of members in the objective function are summarized in Table 4.11. The corresponding point values are summarized in Table 4.12.
The value of the objective function is 1.03848. In this structure, there is no problem with non-effective constraints as there is in the previous structure. Figure 4.11 shows the final design.
Member
ID Length
tab 0.579238 tbb 0.838431
Table 4.11: 4ν Zig-Zag T-Tetrahedron: Final Objective Member Lengths Coordinates
Point x y z
P1 1.374465 -0.537613 1.081334 P2 1.008191 -0.399971 1.505871 P3 1.314861 -0.058122 1.267036 P4 1.067078 0.464915 1.243542
Table 4.12: 4ν Zig-Zag T-Tetrahedron: Final Coordinate Values
Figure 4.11: 4ν Zig-Zag T-Tetrahedron: Final Design
Chapter 5
Double-Layer Tensegrities
5.1 Double-Layer Tensegrities: Introduction
For most of the tensegrities discussed so far, the tensile members compose a single continuous spherical layer.1 Such structures are resilient, but are not very rigid and tend to vibrate too much for many practical applications. Also, it seems likely that large-frequency realizations of these structures, as can happen with geodesic domes, have little resistance to concentrated loads, so that it would be difficult to suspend substructures from the their roofs, and they might cave in excessively under an uneven load like snow.
These considerations are a strong motivation for the development of a space truss configuration for tensegrity structures. Such a configuration would be analogous to the space truss arrangements developed for the geodesic dome, like the Kaiser domes of Don Richter,2 or Fuller and Sadao’s Expo ’67 Dome,3 and serve the same purpose. Tensegrity space trusses are characterized by an outer and inner shell of tendons interconnected by a collection of struts and tendons. The result is a more rigid structure which is more
resistant to concentrated loads.
Designs for tensegrity trusses have been developed in a planar context by several authors. The trusses described in this book, especially the geodesic one described in Section 5.3, are akin to those experimented with by Kenneth Snelson in the 1950’s.4 Appendix A compares the truss of Section 5.3 with an example from Snelson’s work and another similar approach from other authors.
In Section 5.2, a general approach to the design of tensegrity trusses is outlined. Then, in Sections 5.3 and 5.4, two examples are given of geometries which implement this
approach. The second example demonstrates incidentally how icosahedral symmetries can be handled within the Cartesian framework.