Cognate linkage

In kinematics, cognate linkages are linkages that ensure the same input-output relationship or coupler curve geometry, while being dimensionally dissimilar. In case of four-bar linkage coupler cognates, the Roberts–Chebyschev Theorem, after Samuel Roberts and Pafnuty Chebyshev,[1] states that each coupler curve can be generated by three different four-bar linkages. These four-bar linkages can be constructed using similar triangles and parallelograms, and the Cayley diagram (named after Arthur Cayley).

Coupler-curve cognates of a crank-rocker four-bar linkage. Simulation done with MeKin2D.

Coupler-curve cognates of a crank-slider linkage. Simulation done with MeKin2D.

Overconstrained mechanisms can be obtained by connecting two or more cognate linkages together.

Roberts–Chebyschev theorem

The theorem states for a given coupler-curve there exist three four-bar linkages, three geared five-bar linkages, and more six-bar linkages which will generate the same path. The method for generating the additional two four bar linkages from a single four-bar mechanism is described below, using the Cayley diagram.

How to construct path cognate linkages

Cayley diagram for generating 4-bar coupler cognates.

Cayley diagram

From original triangle, ΔA₁,D,B₁

Sketch Cayley diagram
Using parallelograms, find A₂ and B₃ //O_A,A₁,D,A₂ and //O_B,B₁,D,B₃
Using similar triangles, find C₂ and C₃ ΔA₂,C₂,D and ΔD,C₃,B₃
Using a parallelogram, find O_C //O_C,C₂,D,C₃
Check similar triangles ΔO_A,O_C,O_B
Separate left and right cognate
Put dimensions on Cayley diagram

Dimensional relationships

Linkage dimensions.

The lengths of the four members can be found by using the law of sines. Both K_L and K_R are found as follows.

K_{L}={\frac {\sin(\alpha )}{\sin(\beta )}}

K_{R}={\frac {\sin(\gamma )}{\sin(\beta )}}

Linkage	Ground	Crank 1	Crank 2	Coupler
Original	R₁	R₂	R₃	R₄
Left cognate	K_LR₁	K_LR₃	K_LR₄	K_LR₂
Right cognate	K_RR₁	K_RR₂	K_RR₃	K_RR₄

Function cognates

3R-R-3R Watt II function cognates.
3R-P-3R Watt II function cognates.

Conclusions

If and only if the original is a Class I chain $(\ell +s)<(p+q)$ Both 4-bar cognates will be class I chains.
If the original is a drag-link (double crank), both cognates will be drag links.
If the original is a crank-rocker, one cognate will be a crank-rocker, and the second will be a double-rocker.
If the original is a double-rocker, the cognates will be crank-rockers.

References

Roberts and Chebyshev (Springer) Retrieved 2012-10-12

Uicker, John J.; Pennock, Gordon R.; Shigley, Joseph E. (2003). Theory of Machines and Mechanisms. Oxford University Press. ISBN 0-19-515598-X.
Samuel Roberts (1875) "On Three-bar Motion in Plane Space", Proceedings of the London Mathematical Society, vol 7.
Hartenberg, R.S. & J. Denavit (1964) Kinematic synthesis of linkages, p 169, New York: McGraw-Hill, weblink from Cornell University.

External links

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[1] Roberts and Chebyshev (Springer) Retrieved 2012-10-12