World Library  
Flag as Inappropriate
Email this Article

Roberts–Chebyshev theorem

Article Id: WHEBN0025679416
Reproduction Date:

Title: Roberts–Chebyshev theorem  
Author: World Heritage Encyclopedia
Language: English
Subject: Pafnuty Chebyshev
Publisher: World Heritage Encyclopedia

Roberts–Chebyshev theorem

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).

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

From original triangle, ΔA1,D,B1

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

Dimensional relationships

The lengths of the four members can be found by 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


  • 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.

See also


  • Samuel Roberts (1876) "On Three-bar Motion in Plane Space", Proceedings of the London Mathematical Society, vol 7.
  • Hartenberg, R.S. & J. Denavit (1964) Cornell University.

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
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.