The Bricard linkages 129
A close examination of the input–output curve reveals that, in the card
model, the folding process corresponds to a movement from D to F′ of the
input–output curve, instead of F, due to the fact that curve is periodic and
θ 1F 9 = 2π + θ 1F, θ 2F 9 = 2π + θ 2F . The reason for this is that α = π – arctan2, i.e.
116.57°, is sufficient close to 2π/3, i.e. 120°. For the 6R linkage with twist
2π/3, the input–output curve crosses at the point θ 1 = θ 2 = π. Hence, when the
force is applied to the linkage with α = π – arctan2, an imperfection is introduced to the twist of the linkage which alters to 2π/3. The folding process
reaches bifurcation point E. When the force is released, the twist of linkage
changes back to α = π – arctan2. Accordingly, (θ 1 , θ 2 ) reaches point F′.
Linkage II behaves differently. The folding process takes route from D
to F via A and C. There is no blockage during deployment and the structure can be folded up completely, as shown in Figure 6.15.
6.3 Line and plane symmetric Bricard linkage and its
alternative forms
Another Bricard linkage found to be useful in creation of motion structures
is that with both line and plane symmetry.
(a)
(b)
(c)
(d)
Figure 6.17 Card model of Linkage I. (a) At D, (b) B, (c) E and (d) F′ of the compatibility path.
A close examination of the input–output curve reveals that, in the card
model, the folding process corresponds to a movement from D to F′ of the
input–output curve, instead of F, due to the fact that curve is periodic and
θ 1F 9 = 2π + θ 1F, θ 2F 9 = 2π + θ 2F . The reason for this is that α = π – arctan2, i.e.
116.57°, is sufficient close to 2π/3, i.e. 120°. For the 6R linkage with twist
2π/3, the input–output curve crosses at the point θ 1 = θ 2 = π. Hence, when the
force is applied to the linkage with α = π – arctan2, an imperfection is introduced to the twist of the linkage which alters to 2π/3. The folding process
reaches bifurcation point E. When the force is released, the twist of linkage
changes back to α = π – arctan2. Accordingly, (θ 1 , θ 2 ) reaches point F′.
Linkage II behaves differently. The folding process takes route from D
to F via A and C. There is no blockage during deployment and the structure can be folded up completely, as shown in Figure 6.15.
6.3 Line and plane symmetric Bricard linkage and its
alternative forms
Another Bricard linkage found to be useful in creation of motion structures
is that with both line and plane symmetry.
(a)
(b)
(c)
(d)
Figure 6.17 Card model of Linkage I. (a) At D, (b) B, (c) E and (d) F′ of the compatibility path.