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| In [[kinematics]], '''cognate linkages''' are [[Linkage (mechanical)|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 (mathematician)|Samuel Roberts]] and [[Pafnuty Chebyshev]],<ref>[http://rd.springer.com/chapter/10.1007/978-94-007-4132-4_35 Roberts and Chebyshev (Springer)] Retrieved 2012-10-12</ref> 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]]).
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| [[Overconstrained mechanism]]s can be obtained by connecting two or more cognate linkages together.
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| == Roberts–Chebyschev theorem ==
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| 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.
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| == How to construct path cognate linkages ==
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| === Cayley diagram ===
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| [[Image:CaleyDiagram.gif|thumb|right|200px|Example Cayley diagram for generating cognate linkages.]]
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| ''From original triangle, ΔA<sub>1</sub>,D,B<sub>1</sub>''<br />
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| # Sketch Cayley diagram
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| # Using parallelograms, find ''A''<sub>2</sub> and ''B''<sub>3</sub> //O<sub>''A''</sub>,''A''<sub>1</sub>,''D'',''A''<sub>2</sub> and //O<sub>''B''</sub>,''B''<sub>1</sub>,''D'',''B''<sub>3</sub>
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| # Using similar triangles, find ''C''<sub>2</sub> and ''C''<sub>3</sub> Δ''A''<sub>2</sub>,''C''<sub>2</sub>,''D'' and Δ''D'',''C''<sub>3</sub>,''B''<sub>3</sub>
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| # Using a parallelogram, find O<sub>''C''</sub> //O<sub>''C''</sub>,''C''<sub>2</sub>,''D'',''C''<sub>3</sub>
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| # Check similar triangles ΔO<sub>''A''</sub>,O<sub>''C''</sub>,O<sub>''B''</sub>
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| # Separate left and right cognate
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| # Put dimensions on Cayley diagram
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| === Dimensional relationships ===
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| [[Image:CognateLinkagesDR.gif|thumb|right|100px|Example linkage for dimensional analysis.]]
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| The lengths of the four members can be found by using the [[law of sines]]. Both ''K''<sub>''L''</sub> and ''K''<sub>''R''</sub> are found as follows.
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| : <math>K_L=\frac{\sin(\alpha)}{\sin(\beta)}</math> <math>K_R=\frac{\sin(\gamma)}{\sin(\beta)}</math>
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| ! Linkage !! Ground !! Crank 1 !! Crank 2 !! Coupler
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| | ''R''<sub>1</sub> || ''R''<sub>2</sub> || ''R''<sub>3</sub> || ''R''<sub>4</sub>
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| |-
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| ! Left cognate
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| | ''K''<sub>''L''</sub>''R''<sub>1</sub> || ''K''<sub>''L''</sub>R<sub>3</sub> || ''K''<sub>''L''</sub>''R''<sub>4</sub> || ''K''<sub>''L''</sub>R<sub>2</sub>
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| |-
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| ! Right cognate
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| | ''K''<sub>''R''</sub>R<sub>1</sub> || ''K''<sub>''R''</sub>''R''<sub>3</sub> || ''K''<sub>''R''</sub>''R''<sub>4</sub> || ''K''<sub>''R''</sub>''R''<sub>2</sub>
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| |}
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| == Conclusions ==
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| *If and only if the original is a '''Class I''' chain <math>(\ell+s)<(P+q)</math> Both 4-bar cognates will be class I chains.
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| *If the original is a drag-link ('''double crank'''), both cognates will be drag links.
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| *If the original is a '''crank-rocker''', one cognate will be a crank-rocker, and the second will be a double-rocker.
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| *If the original is a double-rocker, the cognates will be crank-rockers.
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| ==See also==
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| * [[Chebychev]]
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| * [[Linkage (mechanical)]]
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| * [[Four-bar linkage]]
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| * [[Kinematic pair]]
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| ==References==
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| {{reflist}}
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| *{{cite book | author=Uicker, John J.; Pennock, Gordon R.; Shigley, Joseph E.| title=Theory of Machines and Mechanisms | publisher=Oxford University Press | year=2003 | id=ISBN 0-19-515598-X}}
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| * Samuel Roberts (1876) "On Three-bar Motion in Plane Space", ''Proceedings of the London Mathematical Society'', vol 7.
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| * Hartenberg, R.S. & J. Denavit (1964) [http://kmoddl.library.cornell.edu/bib.php?m=23 Kinematic synthesis of linkages], p 169, New York: McGraw-Hill, weblink from [[Cornell University]].
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| ==External links== | |
| *[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V46-43T2F9D-2&_user=10&_coverDate=08%2F31%2F2001&_rdoc=1&_fmt=high&_orig=gateway&_origin=gateway&_sort=d&_docanchor=&view=c&_searchStrId=1728995655&_rerunOrigin=scholar.google&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=6320a67fbaf29d1fe092bb5cb0ba3c62&searchtype=a Four- and six-bar function cognates and overconstrained mechanisms]
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| *[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V46-40TY38K-3&_user=10&_coverDate=11%2F01%2F2000&_rdoc=1&_fmt=high&_orig=gateway&_origin=gateway&_sort=d&_docanchor=&view=c&_searchStrId=1728998848&_rerunOrigin=scholar.google&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=dc411009c060929b9c293fb3266e8b34&searchtype=a Applications of Watt II function generator cognates]
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| *[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B756S-482GW8J-F&_user=10&_coverDate=08%2F31%2F1970&_rdoc=1&_fmt=high&_orig=gateway&_origin=gateway&_sort=d&_docanchor=&view=c&_searchStrId=1728998053&_rerunOrigin=scholar.google&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=b454679f90deb1e4c3aa5c5c61b77f84&searchtype=a Coupler cognate mechanisms of certain parallelogram forms of Watt's six-link mechanism]
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| [[Category:Linkages]]
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