Bennett Mechanism

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.

File:Parallel 4-bar Linkage 1.gif

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: $$\begin{array}{c}{d}_0=d_1=d_2=d_3=0\\ a_0=a_2,a_1=a_3,{\alpha }_0={\alpha }_2,{\alpha }_1={\alpha }_3\\ \frac{\sin {\alpha }_0}{\sin {\alpha }_1}=\frac{a_0}{a_1}\end{array}$$ 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.

References

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