Draft:Bennett Mechanism

Bennett four-bar linkage

The Bennett mechanism, consisting of four links and four revolute joints, is the only movable spatial four-bar linkage proposed by Geoffrey Thomas Bennett in a 1903 paper called A New Mechanism. Bennett later published another article The Skew Isogram Mechanism that extended the previous findings. Since its discovery it has been deeply studied , and its kinematics description and synthesis can be seen for example in .

The Bennett linkage has 1 degree-of-freedom, which means that only a single motor is needed to actuate the mechanism. Its planar alternative is the well-known parallelogram four-bar linkage with the Chebychev–Grübler–Kutzbach criterion yielding theoretical mobility $n=1$. Note that the Bennett linkage is overconstrained with the the theoretical number of degrees of freedom $n=-2$, i.e. an arbitrary spatial four-bar linkage cannot even be assembled.

Motion of planar parallelograms

Given the Denavit–Hartenberg parameters of the Bennett linkage, where $a_{i}$ and $\alpha _{i}$ denote the length and twist angle of the link between joints $i$ and $i+1$, and $d_{i}$ the offset along the joint axis, its mobility is geometrically enforced by the Bennett condition: $${\begin{array}{c}d_{0}=d_{1}=d_{2}=d_{3}=0\\a_{0}=a_{2},\quad a_{1}=a_{3},\quad \alpha _{0}=\alpha _{2},\quad \alpha _{1}=\alpha _{3}\\{\dfrac {\sin \alpha _{0}}{\sin \alpha _{1}}}={\dfrac {a_{0}}{a_{1}}}\end{array}}$$ and it gives rise to the skew isogram (as named by Bennett himself), meaning that the opposite links are equal in length and opposite twist angles are equal in size.