The Bennett linkage 83
lute joints remain straight. They expand along the longitudinal direction of
the cylinder. In the other diagonal direction, i.e. the direction from top
right to bottom left, joints generally deploy spirally on the surface of the
cylinder.
A model is shown in Figure 5.4 in which
,
(i = 1, 2, ···, 8),
(5.16)
and
(5.17)
The radius of the model, which inscribes the expanded assembly, can be
obtained geometrically. To do so, let us consider an assembly consisting of
nine large Bennett linkages, three in a row with three rows in total, Figure
5.5(a), where smaller intermediate Bennett linkages are not accounted for.
The assembly satisfies Eqs (5.16) and (5.17).
Denote by l 0 the distance between two rows along the guideline, and by r 0
the radius of the circle that inscribes the assembly of Bennett linkages, both of
which are indicated in Figures 5.5(a) and (b). To obtain l 0 and r 0 we need to
consider the geometry of a typical Bennett linkage in three dimensional space
first. Figure 5.6(a) shows a Bennett linkage ABCD being placed on a cylindrical surface in such a way that A, B, C and D are on the surface and line BD,
colinear with a guideline, is along the longitudinal direction of the cylinder.
The geometric parameters of Bennett linkage ABCD are
,
,
and
.
(a)
(b)
Figure 5.5 (a) Key geometrical parameters of an assembly and (b) the cross-section
of the expanded profile.
lute joints remain straight. They expand along the longitudinal direction of
the cylinder. In the other diagonal direction, i.e. the direction from top
right to bottom left, joints generally deploy spirally on the surface of the
cylinder.
A model is shown in Figure 5.4 in which
,
(i = 1, 2, ···, 8),
(5.16)
and
(5.17)
The radius of the model, which inscribes the expanded assembly, can be
obtained geometrically. To do so, let us consider an assembly consisting of
nine large Bennett linkages, three in a row with three rows in total, Figure
5.5(a), where smaller intermediate Bennett linkages are not accounted for.
The assembly satisfies Eqs (5.16) and (5.17).
Denote by l 0 the distance between two rows along the guideline, and by r 0
the radius of the circle that inscribes the assembly of Bennett linkages, both of
which are indicated in Figures 5.5(a) and (b). To obtain l 0 and r 0 we need to
consider the geometry of a typical Bennett linkage in three dimensional space
first. Figure 5.6(a) shows a Bennett linkage ABCD being placed on a cylindrical surface in such a way that A, B, C and D are on the surface and line BD,
colinear with a guideline, is along the longitudinal direction of the cylinder.
The geometric parameters of Bennett linkage ABCD are
,
,
and
.
(a)
(b)
Figure 5.5 (a) Key geometrical parameters of an assembly and (b) the cross-section
of the expanded profile.