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–Chebychev theorem 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 parallelgrams, and the Cayley diagram (named after Arthur Cayley).

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 the four bar linkages from a single four-bar mechanism is described below, using the Cayley diagram.

How to construct path cognate linkages

Cayley diagram

Example Cayley diagram for generating cognate linkages.

From original triangle, ΔA₁,D,B₁

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

Dimensional relationships

Example linkage for dimensional analysis.

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	KLR1	KLR3	KLR4	KLR2
Right cognate	KRR1	KRR3	KRR4	KRR2

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

• 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 (1876) "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.

See also

• Chebychev
• Linkage (mechanical)
• Four-bar linkage
• Kinematic pair

External links

• Four- and six-bar function cognates and overconstrained mechanisms

• Applications of Watt II function generator cognates

• Coupler cognate mechanisms of certain parallelogram forms of Watt's six-link mechanism