94 The Bennett linkage
It is interesting to note that, if the two smallest Bennett linkages at the
centre of the model were removed, the assembly would show the largest
and second largest Bennett linkages connected by four smaller Bennett
linkages on each side, Figure 5.12(b), forming a mobile assembly. This
effectively shows that two Bennett linkages can also be connected together
just like the connection of two planar four- bar linkages illustrated in
Section 3.2.1. The difference is that smaller Bennett linkages have to be
used to facilitate the connection instead of single pins in planar cases. The
question of connecting of two Bennett linkages while retaining mobility
was first raised by Baker and Hu (1986). A solution to this question can be
extracted from the multi- layer assemblies just being created. Detailed proof
can be found in Chen and You (2002) and Chen and Baker (2005).
5.4 Alternative form of Bennett linkage
5.4.1 The equilateral Bennett linkage and its alternative forms
The Bennett linkage discussed so far has its four links spanning the shortest
distance between the axes of the neighbouring revolute joints. Although this
enables us to uniquely describe the linkage mathematically, in a physical
model constructed with these types of links, it is found that the linkage
cannot be folded up completely in both directions linking two diagonal joints
simultaneously. It duly disappoints readers who intend to build motion structures that fold to a compact bundle. However, in this section we demonstrate
that modifications can be carried out to an equilateral Bennett linkage so that
compact folding becomes possible (Chen and You, 2006).
To design for compact packaging, an equilateral Bennett linkage is
drawn in three dimensions in Figure 5.13(a) with its joints marked with
letters A, B, C and D, which correspond to 1, 2, 3 and 4 in Figure 5.1(a).
So the lengths and twists of linkage satisfy
,
(5.38)
,
,
(5.39)
due to Eq. (2.23). This linkage is symmetric both about the plane through
AC and perpendicular to BD and about the plane through BD and perpendicular to AC even though lines AC and BD may not cross each other. The
axes of revolute joints are marked at A, B, C and D by lines with arrows
which give the positive directions of the axes.
Denote M and N as the respective middle points of BD and AC, see
Figure 5.13(b). Obviously, ΔABD and ΔCDB are isosceles and identical
triangles due to Eq. (5.38). So are ΔBCA and ΔDAC. These lead to the
It is interesting to note that, if the two smallest Bennett linkages at the
centre of the model were removed, the assembly would show the largest
and second largest Bennett linkages connected by four smaller Bennett
linkages on each side, Figure 5.12(b), forming a mobile assembly. This
effectively shows that two Bennett linkages can also be connected together
just like the connection of two planar four- bar linkages illustrated in
Section 3.2.1. The difference is that smaller Bennett linkages have to be
used to facilitate the connection instead of single pins in planar cases. The
question of connecting of two Bennett linkages while retaining mobility
was first raised by Baker and Hu (1986). A solution to this question can be
extracted from the multi- layer assemblies just being created. Detailed proof
can be found in Chen and You (2002) and Chen and Baker (2005).
5.4 Alternative form of Bennett linkage
5.4.1 The equilateral Bennett linkage and its alternative forms
The Bennett linkage discussed so far has its four links spanning the shortest
distance between the axes of the neighbouring revolute joints. Although this
enables us to uniquely describe the linkage mathematically, in a physical
model constructed with these types of links, it is found that the linkage
cannot be folded up completely in both directions linking two diagonal joints
simultaneously. It duly disappoints readers who intend to build motion structures that fold to a compact bundle. However, in this section we demonstrate
that modifications can be carried out to an equilateral Bennett linkage so that
compact folding becomes possible (Chen and You, 2006).
To design for compact packaging, an equilateral Bennett linkage is
drawn in three dimensions in Figure 5.13(a) with its joints marked with
letters A, B, C and D, which correspond to 1, 2, 3 and 4 in Figure 5.1(a).
So the lengths and twists of linkage satisfy
,
(5.38)
,
,
(5.39)
due to Eq. (2.23). This linkage is symmetric both about the plane through
AC and perpendicular to BD and about the plane through BD and perpendicular to AC even though lines AC and BD may not cross each other. The
axes of revolute joints are marked at A, B, C and D by lines with arrows
which give the positive directions of the axes.
Denote M and N as the respective middle points of BD and AC, see
Figure 5.13(b). Obviously, ΔABD and ΔCDB are isosceles and identical
triangles due to Eq. (5.38). So are ΔBCA and ΔDAC. These lead to the