Fundamental concepts 33
2.3.6 Summary
There are a total of sixteen types of spatial overconstrained linkages with
revolution joints. A summary is given in Table 2.3. Only two of these linkages, namely the Bennett linkage and the Bricard linkage, can be regarded
as completely independent, whereas the rest are combinations or derivatives of the two linkages.
2.4 Mechanisms and structures
When only a unique solution exists for the closure equations (2.17), the
closed chain has zero mobility and is in fact a conventional structure
because none of n kinematic variables can change freely. The tetrahedron
given in Figure 2.2 is an example.
Structural engineers are also interested in finding out whether an assembly
is a structure or a mechanism. A criterion known as the Maxwell’s rule is
used to determine whether an assembly is a structure or mechanism (Calladine, 1978; Pellegrino and Calladine, 1986; Calladine and Pellegrino, 1991).
It stipulates that a truss in space with j spherical joints requires in general n
bars and r supports to render it stiff (or rigid) where n + r = 3j; when n + r > 3j,
the structure would have redundant members and when n + r < 3j, the assembly could be a mechanism. For trusses without ground support, r is taken as
6 to remove the rigid body motions associated to the trusses as a whole.
Table 2.3 Spatial overconstrained linkages with only revolute joints and their
dependent linkages.
Number of links Linkages
Dependent linkages
4
Bennett linkage
—
5
Goldberg 5R linkage
Bennett linkage
5
Myard linkage
Bennett linkage
6
Altmann linkage
Bricard linkage
6
Bennett 6R hybrid linkage
Bennett linkage
6
Bennett-joint 6R linkage
Bennett linkage
6
Bricard linkages
—
6
Dietmaier 6R linkage
Bennett linkage
6
Double-Hooke’s-joint linkage Bennett linkage
6
Goldberg 6R linkage
Bennett linkage
6
Sarrus linkage
Bennett linkage
6
Schatz linkage
Bricard linkage
6
Waldron hybrid linkages
Four-bar linkage with lower
joints
6
Wohlhart 6R linkage
Bricard linkage
6
Wohlhart double-Goldberg
linkage
Bennett linkage
6
Chen & You doubleGoldberg linkage
Bennett linkage
2.3.6 Summary
There are a total of sixteen types of spatial overconstrained linkages with
revolution joints. A summary is given in Table 2.3. Only two of these linkages, namely the Bennett linkage and the Bricard linkage, can be regarded
as completely independent, whereas the rest are combinations or derivatives of the two linkages.
2.4 Mechanisms and structures
When only a unique solution exists for the closure equations (2.17), the
closed chain has zero mobility and is in fact a conventional structure
because none of n kinematic variables can change freely. The tetrahedron
given in Figure 2.2 is an example.
Structural engineers are also interested in finding out whether an assembly
is a structure or a mechanism. A criterion known as the Maxwell’s rule is
used to determine whether an assembly is a structure or mechanism (Calladine, 1978; Pellegrino and Calladine, 1986; Calladine and Pellegrino, 1991).
It stipulates that a truss in space with j spherical joints requires in general n
bars and r supports to render it stiff (or rigid) where n + r = 3j; when n + r > 3j,
the structure would have redundant members and when n + r < 3j, the assembly could be a mechanism. For trusses without ground support, r is taken as
6 to remove the rigid body motions associated to the trusses as a whole.
Table 2.3 Spatial overconstrained linkages with only revolute joints and their
dependent linkages.
Number of links Linkages
Dependent linkages
4
Bennett linkage
—
5
Goldberg 5R linkage
Bennett linkage
5
Myard linkage
Bennett linkage
6
Altmann linkage
Bricard linkage
6
Bennett 6R hybrid linkage
Bennett linkage
6
Bennett-joint 6R linkage
Bennett linkage
6
Bricard linkages
—
6
Dietmaier 6R linkage
Bennett linkage
6
Double-Hooke’s-joint linkage Bennett linkage
6
Goldberg 6R linkage
Bennett linkage
6
Sarrus linkage
Bennett linkage
6
Schatz linkage
Bricard linkage
6
Waldron hybrid linkages
Four-bar linkage with lower
joints
6
Wohlhart 6R linkage
Bricard linkage
6
Wohlhart double-Goldberg
linkage
Bennett linkage
6
Chen & You doubleGoldberg linkage
Bennett linkage