Planar double chain linkages 43
The above restrictions are referred to as the loop parallelogram constraint
hereafter.
Under the loop parallelogram constraint, the inclination angles
,
(3.8a)
and
,
(3.8b)
as shown in Figure 3.8(c). Moreover, vectors
.
(3.9)
because both AC and DF are equal to p 1 – q 1 .
The condition for forming a closed loop is that the end connectors A
and D should always meet when the mechanism is activated, Figure 3.9(a),
and therefore,
.
(3.10)
3
Now assume that the motion of the mechanism is determined by pivoting
angle θ (–π ≤ θ ≤ π). It is the angle between vector q 1 and p 1 and is positive
clockwise as shown in Figure 3.9(a).
4
The vector equation (3.10) can be
replaced by complex number notations with the real and imaginary axes
parallel with and perpendicular to vector q 1 , respectively. There is
,
(3.11)
(a)
(b)
Figure 3.9 (a) Five intersecting elements form a closed double chain and (b) geometrical representation of its mobility conditions.