Cognate linkages

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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.

Contents

[edit] Roberts-Chebychev 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.


[edit] How to Construct Cognate Linkages

[edit] Cayley Diagram

Example Cayley Diagram for generating Cognate Linkages.
Example Cayley Diagram for generating Cognate Linkages.

From Original Triangle, ΔA1,D,B1

  • 1. Sketch Cayley Diagram
  • 2. Using Parallelograms, Find A2 & B3

//OA,A1,D,A2 & //OB,B1,D,B3

  • 3. Using Similar Triangles, Find C2 & C3

ΔA2,C2,D & ΔD,C3,B3

  • 4. Using a Parallelogram, Find OC

//OC,C2,D,C3

  • 5. Check Similar Triangles ΔOA,OC,OB
  • 6. Separate Left & Right Cognate
  • 7. Put Dimensions on Cayley Diagram

[edit] Dimensional Relationships

Example linkage for dimensional analysis.
Example linkage for dimensional analysis.

The lengths of the four members can be found through dimensional analysis. Using the Law of Sines both KL and KR 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

[edit] 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.


[edit] References

  1. Uicker, John J.; Pennock, Gordon R.; Shigley, Joseph E. (2003). Theory of Machines and Mechanisms. Oxford Univerisity Press. ISBN 0-19-515598-x. 

[edit] See also