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).
![](../I/m/Four-bar-curve-cognates.gif)
![](../I/m/Slide-Crank_Cognates.gif)
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
![](../I/m/CaleyDiagram.gif)
Cayley diagram
From original triangle, ΔA1,D,B1
- Sketch Cayley diagram
- Using parallelograms, find A2 and B3 //OA,A1,D,A2 and //OB,B1,D,B3
- Using similar triangles, find C2 and C3 ΔA2,C2,D and ΔD,C3,B3
- Using a parallelogram, find OC //OC,C2,D,C3
- Check similar triangles ΔOA,OC,OB
- Separate left and right cognate
- Put dimensions on Cayley diagram
Dimensional relationships
![](../I/m/CognateLinkagesDR.svg.png)
The lengths of the four members can be found by using the law of sines. Both KL and KR are found as follows.
Linkage | Ground | Crank 1 | Crank 2 | Coupler |
---|---|---|---|---|
Original | R1 | R2 | R3 | R4 |
Left cognate | KLR1 | KLR3 | KLR4 | KLR2 |
Right cognate | KRR1 | KRR2 | KRR3 | KRR4 |
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 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.