Layouts of spatial motion structures 149
7.4 Assemblies of threefold- symmetric Bricard linkages
7.4.1 Case I
The threefold- symmetric Bricard linkage has six links of equal length. It is
natural to use (6
3
) tiling for the formation of motion structure.
Consider a portion of the assembly shown in Figure 7.12(a). Loops 1, 2
and 3 are all made of identical threefold- symmetric Bricard linkages with
twists of α and 2π – α alternating from one link to the other. Let
.
(7.7)
Then for loop 2,
.
(7.8)
Considering loop 3 gives
.
(7.9)
Now a problem arises, as for loop 1,
,
(7.10)
meaning that loop 1 violates the condition of the threefold- symmetric
Bricard linkage given in Eq. (6.1). Thus, the assembly consisting of more
than one threefold- symmetric Bricard linkage will become immobile.
This layout can be revived by using a 6R loop which has the same twist
for all of the six links, either α or 2π – α for loop 1 in the centre while keep
the rest the same. The 6R loop at 1 is in fact a very special line and plane
symmetric Bricard linkage that has mobility two, resulting in a discontinuous
Figure 7.12 An assembly of threefold-symmetric Bricard linkage.