Fundamental concepts 21
Bennett (1914) also identified some special cases.
a An equilateral linkage is obtained if α + β = π and a = b. Eq. (2.27c) then
becomes
.
(2.28)
b If α = β and a = b, the four links are congruent. The motion is discontinuous: θ 1 = π allows any value for θ 2 and θ 2 = π allows any value for
θ 1 .
c If α = β = 0, the linkage is a planar four- bar linkage.
d If α = 0 and β = π, the linkage becomes a 2D parallelogram.
e The linkage becomes a spherical 4R linkage if a = b = 0 (Phillips, 1990).
2.3.4 Linkages derived from the Bennett linkage
Attempts have also been made to build 5R or 6R three dimensional linkages based on the Bennett linkage. Most of the work was concentrated on
building new mobile chains with fewer than seven links rather than exploring the possibility of constructing large motion assemblies with the only
exception of Baker and Hu’s (1986) unsuccessful attempt to connect two
Bennett linkages, which we shall discuss in Chapter 5.
Goldberg (1943) arrived at the Goldberg 5R linkage by combining a
pair of Bennett linkages in such a way that a link common to both was
removed and a pair of adjacent links were rigidly attached to each other.
The techniques he developed can be summarised as the summation of two
Bennett loops to produce a 5R linkage, Figure 2.14(a), or the subtraction
of a primary composite loop from another Bennett chain to form a syncopated linkage, Figure 2.14(b).
Prior to Goldberg, Myard (1931) produced an overconstrained 5R
linkage as shown in Figure 2.15. It is a plane- symmetric 5R and has later
been reclassified as a special case of the Goldberg 5R linkage, for which
the two ‘rectangular’ Bennett chains with one pair of twists being π/2, are
symmetrically disposed and subsequently combined. Two Bennett linkages
are mirror images of each other where the mirror is coincident with the
plane of symmetry of the resultant linkage (Baker, 1979). The conditions
on its geometric parameters are as follows.
,
,
,
,
(2.29)
.
Bennett (1914) also identified some special cases.
a An equilateral linkage is obtained if α + β = π and a = b. Eq. (2.27c) then
becomes
.
(2.28)
b If α = β and a = b, the four links are congruent. The motion is discontinuous: θ 1 = π allows any value for θ 2 and θ 2 = π allows any value for
θ 1 .
c If α = β = 0, the linkage is a planar four- bar linkage.
d If α = 0 and β = π, the linkage becomes a 2D parallelogram.
e The linkage becomes a spherical 4R linkage if a = b = 0 (Phillips, 1990).
2.3.4 Linkages derived from the Bennett linkage
Attempts have also been made to build 5R or 6R three dimensional linkages based on the Bennett linkage. Most of the work was concentrated on
building new mobile chains with fewer than seven links rather than exploring the possibility of constructing large motion assemblies with the only
exception of Baker and Hu’s (1986) unsuccessful attempt to connect two
Bennett linkages, which we shall discuss in Chapter 5.
Goldberg (1943) arrived at the Goldberg 5R linkage by combining a
pair of Bennett linkages in such a way that a link common to both was
removed and a pair of adjacent links were rigidly attached to each other.
The techniques he developed can be summarised as the summation of two
Bennett loops to produce a 5R linkage, Figure 2.14(a), or the subtraction
of a primary composite loop from another Bennett chain to form a syncopated linkage, Figure 2.14(b).
Prior to Goldberg, Myard (1931) produced an overconstrained 5R
linkage as shown in Figure 2.15. It is a plane- symmetric 5R and has later
been reclassified as a special case of the Goldberg 5R linkage, for which
the two ‘rectangular’ Bennett chains with one pair of twists being π/2, are
symmetrically disposed and subsequently combined. Two Bennett linkages
are mirror images of each other where the mirror is coincident with the
plane of symmetry of the resultant linkage (Baker, 1979). The conditions
on its geometric parameters are as follows.
,
,
,
,
(2.29)
.