Spatial rings and domes 67
Substituting the above equations into Eq. (4.4) yields
,
from which
.
(4.6)
Ratio b/a is a constant for a given n a due to Eq. (4.1).
Eq. (4.6) indicates that in order for the inner loop and outer loop to be
connected with a single intermediate element shown in Figure 4.4(c), the
ratio between the lengths of the elements forming the inner and outer loops
must be kept constant once the number of elements and the layout of ring
are determined.
4.2.3 Concept B
The project of this ring consists of 2n a triangles. The plan view, Figure
4.5(b), is a chain of isosceles triangles with alternate long bases, e.g. CD,
corresponding to a pair of rods of length 2b, and short bases, e.g. AB, corresponding to a pair of rods of length 2a. All the intermediate scissor- like
elements connecting the inner and outer loops are of length a + b with the
semi- length a near the inner loop just as those for concept A. The heights
of the intermediate elements match those of the elements in the inner and
outer loop, hence all that remains to be done is to find the geometric condition to complete the assembly. Here
(4.7)
and
.
(4.8)
During motion,
,
and
.
(4.9a, b, c)