88 The Bennett linkage
Note that some of lengths a i , b i (i = 1, 2, . . ., 8) can be negative, which
also lead to mobile assemblies as long as Eqs (5.1) and (5.15) are satisfied.
Some of these assemblies exhibit rather interesting profiles. Readers may
refer to Chen and You (2008b) for details.
5.3 Multi- layer assemblies of Bennett linkages
The assembly of Bennett linkages presented so far does not have multiple
layers. A typical portion of single- layer assembly of Bennett linkages surrounding a large Bennett linkage is redrawn as Figure 5.9(a). Again the
twists are marked alongside the links. The guideline passes through A, E,
G and C. In the expanded spiral profile, the top right Bennett linkage
PVND and the bottom left Bennett linkage JBKS are roughly placed along
the tangent of the cylindrical surface, with V and S pointing towards each
other. The following proof shows that they can be connected by a bridging
4R loop RSTV to form the assembly shown in Figure 5.9(b) while retaining mobility.
(a)
(b)
(c)
(d)
Figure 5.9 Formation of additional loops in a single-layer assembly. (a) A portion
of a single-layer assembly; (b) formation of loop RSTV at a layer different from loop ABCD and (c) its corresponding twists; (d) formation of
loop EFGH and its corresponding twists.