44 Planar double chain linkages
or
. (3.12)
To have a mobile linkage, the above equation must be satisfied whatever θ
is, which leads to
(3.13)
and
.
(3.14)
Eqs (3.13) and (3.14) are the additional mobility conditions.
A close inspection of Eq. (3.13) reveals that geometrically it is equivalent to vectors p 1 , p 2 , . . ., p 5 forming a closed loop. Similarly, Eq. (3.14)
suggests that vectors q 1 , q 2 , . . ., q 5 also form a closed loop. Both cases are
shown in Figure 3.9(b). Moreover, if vectors p and q form closed loops,
change of the pivoting angle, θ, does not alter their relative positions. Only
the vector loops as a whole will rotate by the same amount, see Figure
3.9(b).
Hence, we can conclude that the conditions that ensure the formation of
a mobile closed double chain consisting of five intersecting pairs are as
follows.
,
(3.15a)
and
.
(3.15b)
The motion sequence of a model shown in Figure 3.10 illustrates the above
proof.
The above derivation can be extended to double chain linkage consisting of n intersecting pairs by adding or reducing items in equations. The
mobility condition for double chains composed of intersecting pairs under
the loop parallelogram constraint is that two vector sums of edges of the
pieces, defined as p and q, must be zero.
Note that here the number of pairs, n, can be either odd or even. We
shall explain the significance of this in the next section.