142 Layouts of spatial motion structures
(7.3b)
From Eqs (7.3a) and (7.3b), we obtain that
.
(7.3)
The solutions to Eq. (7.3) are, for any 0 ≤ α ≤ 2π and 0 ≤ β ≤ 2π,
,
,
,
,
leading to
, or
, and
, or
.
(7.4)
It can be concluded from Eq. (7.4) that only when α and β are 0 or π can
the assembly based on the tiling shown in Figure 7.4(b) remain mobile.
The Bennett linkage now degenerates to a planar 4R linkage for Bennett
linkages with twists other than 0 or π the mobility vanishes.
7.2.3 Case III
Now modify the building block further to that shown in Figure 7.5(a), in
which all of the links are extended at both ends. Using the pattern of
Figure 7.2(g) produces the assembly shown in Figure 7.5(b), which is identical to the layout given in Section 5.2.1. Hence, the assembly is mobile if
the conditions given in Section 5.2.2 are met.
7.2.4 Case IV
This is an extension of Case III. If only two corners of the Bennett linkage
are extended, Figure 7.6(a), the assembly shown in Figure 7.6(b) can be
obtained. Let the Bennett linkage ABCD have lengths a, b and twists α, β