Cognate linkages

Three unique four-bar linkages can define the same coupler-curve. The linkages which describe the same coupler curve as a certain linkage are called 'cognate linkages'. This theorem in kinematics is called the Roberts-Chebychev theorem. The linkages can be constructed by using similar triangles and parallelgrams, and the Cayley diagram.

Roberts-Chebyschev Theorem

The theory states for a given coupler-curve there exists 3 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 Cognate Linkages

Cayley Diagram

"From Original Triangle, ΔA₁,D,B₁"

*1. Sketch Cayley Diagram
*2. Using Parallelograms, Find A₂ & B₃
//O_A,A₁,D,A₂ & //O_B,B₁,D,B₃
*3. Using Similar Triangles, Find C₂ & C₃
ΔA₂,C₂,D & 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 & Right Cognate
*7. Put Dimensions on Cayley Diagram

Dimensional Relationships

The lengths of the four members can be found through dimensional analysis. Using the Law of Sines both K_L and K_R are found as follows.

$K_L=frac{sin(alpha)}{sin(eta)}$ $K_R=frac{sin(gamma)}{sin(eta)}$

Conclusions

*If and only if the original is a Class I chain $(l+s)<(P+q)$ 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

ee also

* Chebychev
* Mechanical Linkages
* Four bar linkages
* Kinematic pairs

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