42 Planar double chain linkages
3.2.2 Double chains with intersecting elements
A double chain consisting of five intersecting elements prior to forming a
closed loop is shown in Figure 3.8. Two beams in each element, Figure 3.8(a),
can be represented by vectors p i , p i+1 , q i and q i+1 , respectively (i = 1, 2, . . ., 5.
When the subscript becomes 6, it is replaced by 1). Denote by μ i the inclination angle from p i to p i+1 and by ν i the inclination angle from q i to q i+1 . Both
of the angles range from –π to π and they are positive clockwise.
2
The chain is constructed subjected to the following restrictions:
a within four neighbouring pieces, four hinges, two from each piece,
form a parallelogram; and
b at both ends, AB and EF are equal and parallel; so are BC and DE, see
Figure 3.8(b).
(a)
(b)
(c)
Figure 3.7 (a) Intersecting and non-intersecting elements; (b) and (c) closed double
chains formed by intersecting and non-intersecting elements, respectively.
(a)
(b)
(c)
Figure 3.8 (a) Angulated beams forming an intersecting element; (b) a double chain
prior to closure and (c) relations among inclination angles.