Planar double chain linkages 47
In order for the equation to hold for any θ, both the first and second items
must be zero. which, together with Eq. (3.16), have been found to be
equivalent to vectors p 1 , p 2 , p 3 , p 4 and q 1 , q 2 , q 3 , q 4 forming closed loops,
respectively, i.e.
,
(3.20a)
and
,
(3.20b)
as shown in Figure 3.12. Alteration of rotation angle θ only causes the
rotation of the closed vector loops as a whole.
This proof can be easily extended to double chain with any even number
of non- intersecting pairs.
Although the appearance of Eqs (3.20a) and (3.20b) is similar to what
we obtained for the double chain with intersecting pairs, they are in fact
different. This is due to the way the pieces were arranged under the loop
parallelogram constraint, see Figure 3.12(a). Here p 1 , p 2 , p 3 and p 4 represent the lengths of opposite pieces. So do q 1 , q 2 , q 3 and q 4 .
Now consider a double chain consisting of five non- intersecting pairs, see
Figure 3.13(a). Note that, unlike the previous double chain with four nonintersecting elements, the element on the left consists of a piece with edge
lengths of p 5 and q 1 whereas the other piece has edge lengths of q 5 and p 1 in
order to preserve the loop parallelogram constraint. The angles sustained by
p 5 and q 1, and by q 5 and p 1 , are represented by μ 5 and ν 5 , respectively.
Plotting all of the vectors ps and then qs, under the parallelogram constraint, see Figure 3.13(b), we obtain
,
(3.21a)
(a)
(b)
Figure 3.12 (a) Four non-intersecting elements form a closed double chain and (b)
geometrical representation of its mobility conditions.
In order for the equation to hold for any θ, both the first and second items
must be zero. which, together with Eq. (3.16), have been found to be
equivalent to vectors p 1 , p 2 , p 3 , p 4 and q 1 , q 2 , q 3 , q 4 forming closed loops,
respectively, i.e.
,
(3.20a)
and
,
(3.20b)
as shown in Figure 3.12. Alteration of rotation angle θ only causes the
rotation of the closed vector loops as a whole.
This proof can be easily extended to double chain with any even number
of non- intersecting pairs.
Although the appearance of Eqs (3.20a) and (3.20b) is similar to what
we obtained for the double chain with intersecting pairs, they are in fact
different. This is due to the way the pieces were arranged under the loop
parallelogram constraint, see Figure 3.12(a). Here p 1 , p 2 , p 3 and p 4 represent the lengths of opposite pieces. So do q 1 , q 2 , q 3 and q 4 .
Now consider a double chain consisting of five non- intersecting pairs, see
Figure 3.13(a). Note that, unlike the previous double chain with four nonintersecting elements, the element on the left consists of a piece with edge
lengths of p 5 and q 1 whereas the other piece has edge lengths of q 5 and p 1 in
order to preserve the loop parallelogram constraint. The angles sustained by
p 5 and q 1, and by q 5 and p 1 , are represented by μ 5 and ν 5 , respectively.
Plotting all of the vectors ps and then qs, under the parallelogram constraint, see Figure 3.13(b), we obtain
,
(3.21a)
(a)
(b)
Figure 3.12 (a) Four non-intersecting elements form a closed double chain and (b)
geometrical representation of its mobility conditions.