132 The Bricard linkages
Comparing Eqs (6.10)–(6.12) with Eq. (2.30), it can be concluded that this
original linkage is a line- symmetric Bricard linkage with symmetric line
passing through the centre and perpendicular to the paper for the configuration in Figure 6.21.
Note that, if either x 3 or x 6 is reversed, which is possible because they correspond to zero length links, the twists α 23 or α 56 have to change accordingly.
For instance, if x 6 reverses its direction, Eq. (6.11b) has to be replaced by
,
which indicates that this original linkage is actually a plane- symmetric
Bricard linkage by comparison with Eq. (2.31).
It is therefore concluded that the corresponding original linkage of the
6R frame is a special Bricard linkage with two-fold symmetry. Next, we
try to find the relationship between the geometric parameters of the original linkage and those of the alternative form.
Because of symmetry, only links 12 and 23 are studied, as shown in
Figure 6.22 with both the original linkage and the alternative form. Let
ED = l 1 and EG = l 2 . We have
.
DM is the link 12 in original linkage, MA is the offset of joint 2 and
∠FNH is the twist of link 23. So we have
Figure 6.21 The original Bricard linkage. Note that axes x 3 and x 6 are not
drawn.
Comparing Eqs (6.10)–(6.12) with Eq. (2.30), it can be concluded that this
original linkage is a line- symmetric Bricard linkage with symmetric line
passing through the centre and perpendicular to the paper for the configuration in Figure 6.21.
Note that, if either x 3 or x 6 is reversed, which is possible because they correspond to zero length links, the twists α 23 or α 56 have to change accordingly.
For instance, if x 6 reverses its direction, Eq. (6.11b) has to be replaced by
,
which indicates that this original linkage is actually a plane- symmetric
Bricard linkage by comparison with Eq. (2.31).
It is therefore concluded that the corresponding original linkage of the
6R frame is a special Bricard linkage with two-fold symmetry. Next, we
try to find the relationship between the geometric parameters of the original linkage and those of the alternative form.
Because of symmetry, only links 12 and 23 are studied, as shown in
Figure 6.22 with both the original linkage and the alternative form. Let
ED = l 1 and EG = l 2 . We have
.
DM is the link 12 in original linkage, MA is the offset of joint 2 and
∠FNH is the twist of link 23. So we have
Figure 6.21 The original Bricard linkage. Note that axes x 3 and x 6 are not
drawn.