Spatial rings and domes 73
.
(4.25)
Therefore, from Eqs (4.24) and (4.25), c′ and c′′ can be expressed in terms
of h and l,
.
(4.26)
To achieve a height differentiation h at a particular expanded configuration
when θ = θ f , c′ and c′′ can be determined from Eq. (4.26). Again they can be
negative in which case the tie appears similar to that of Figure 4.10(b).
4.3.3 Construction of domes
The alternative intermediate tie gives greater design flexibility. One can either
select a layout among those in Figure 4.5 for a ring or combine them for a
dome. The primary design variables for rings are n a , a and b. The domes
require two additional design variables: h, the height differentiation between
any of two neighbouring loops, and θ f , corresponded to the expanded configuration at which h is achieved. All of the dimensions of the assembly can be
obtained accordingly once those design variables are known.
Figure 4.11 shows the expansion sequence of a mobile frame for a dome
obtained by combining a concept B ring with one based on concept C using a
layout identical to that shown in Figure 4.8. From a bundle it expands to a
dome shape. It can also be folded to flat, revealing clearly its layout.
(a)
(b)
Figure 4.10 (a) A intermediate tie made of a pair of conventional scissor-like elements with height shift and (b) its variation.
.
(4.25)
Therefore, from Eqs (4.24) and (4.25), c′ and c′′ can be expressed in terms
of h and l,
.
(4.26)
To achieve a height differentiation h at a particular expanded configuration
when θ = θ f , c′ and c′′ can be determined from Eq. (4.26). Again they can be
negative in which case the tie appears similar to that of Figure 4.10(b).
4.3.3 Construction of domes
The alternative intermediate tie gives greater design flexibility. One can either
select a layout among those in Figure 4.5 for a ring or combine them for a
dome. The primary design variables for rings are n a , a and b. The domes
require two additional design variables: h, the height differentiation between
any of two neighbouring loops, and θ f , corresponded to the expanded configuration at which h is achieved. All of the dimensions of the assembly can be
obtained accordingly once those design variables are known.
Figure 4.11 shows the expansion sequence of a mobile frame for a dome
obtained by combining a concept B ring with one based on concept C using a
layout identical to that shown in Figure 4.8. From a bundle it expands to a
dome shape. It can also be folded to flat, revealing clearly its layout.
(a)
(b)
Figure 4.10 (a) A intermediate tie made of a pair of conventional scissor-like elements with height shift and (b) its variation.