Chebyshev lambda linkage: Difference between revisions

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{{Short description|Four-bar straight-line mechanism}}

[[Image:Hoekens linkage animated.gif|right|Hoeckens linkage]]

[[File:TchebyshevsChebyshev plantigradeLambda machineLinkage.gif|thumb|Tchebyshevs'''Animation plantigradeof machine]]the Chebyshev Lambda Linkage'''<br>

Dimensions (unit length {{mvar|a}}):

[[File:Lambda mechanism.svg|upright=0.5|thumb|Lambda Mechanism]]

{{legend-line|solid cyan|Link 2: {{mvar|a}}}}

{{legend-line|solid green|Link 4: {{math|2.5''a''}}}}

{{legend-line|solid yellow|Link 3: {{math|2.5''a'' + 2.5''a''}}}}

Link 1 (horizontal distance between ground joints): {{math|2''a''}}]]

[[File:Tchebyshevs plantigrade machine.gif|thumb|Chebyshev's plantigrade machine.]]

[[File:Chebyshev Table Linkage.gif|thumb|A Chebyshev Translating Table Linkage, which combines together two [[cognate linkage]]s: the [[Chebyshev linkage|Chebyshev Linkage]] and Chebyshev Lambda Linkage.]]

In [[kinematics]], the '''Chebyshev Lambda Linkage'''<ref name="Wolfram">{{Cite web|url=http://demonstrations.wolfram.com/ChebyshevsLambdaMechanism/|title=Chebyshev's Lambda Mechanism – Wolfram Demonstrations Project|website=demonstrations.wolfram.com}}</ref> is a [[four-bar mechanismlinkage]] that converts [[rotational motion]] to approximate [[straight-line motion]] with approximate constant velocity.<ref name=DOM>{{citeCite book |url=http://www.designofmachinery.com/DOM/ |title=Design of Machinery |year=2011}} </ref> It is so-named afterbecause Karlit Hoeckenlooks like a lowercase Greek letter [[lambda]] (1874−1962λ).<ref name=etudes /> The precise design trades off straightness, lack of [[acceleration]], and whatthe proportion of the driving rotation that is spent in the linear portion of the full curve.<ref name=DOM134>{{citeCite book |url=http://www.designofmachinery.com/DOM/Chap_03_3ed_p134.pdf |title=DOM p134 Hoecken linkage }}</ref>

The '''Hoeckens linkage'''<ref>{{cite book

|title=Distinguished Figures in Mechanism and Machine Science: Their Contributions and Legacies

|author=Ceccarelli, M.

|isbn=9789401789479

|series=History of mechanism and machine science

|url=http://books.google.de/books?id=yavIAwAAQBAJ

|year=2014

|publisher=Springer London, Limited

}}</ref> is a [[four-bar mechanism]] that converts rotational motion to approximate [[straight-line motion]] with approximate constant velocity.<ref name=DOM>{{cite book |url=http://www.designofmachinery.com/DOM/ |title=Design of Machinery |year=2011}} </ref> It is named after Karl Hoecken (1874−1962). The precise design trades off straightness, lack of acceleration, and what proportion of the driving rotation is spent in the linear portion of the full curve.<ref name=DOM134>{{cite book |url=http://www.designofmachinery.com/DOM/Chap_03_3ed_p134.pdf |title=DOM p134 Hoecken linkage }}</ref>

The example to the right spends over half of the cycle in the near -straight portion. The coupler (link 3) point stays within 1% positional tolerance while intersecting the ideal straight line 6 times.

The linkage was first shown in Paris on the [[Exposition Universelle (1878)]] as "The Plantigrade Machine".<ref>{{Cite web|url=http://mech.spbstu.ru/Dzenushko_Dainis:_Walking_mechanisms_survey#The_Chebyshev_Walking_Mechanism|title = Dzenushko Dainis: Walking mechanisms survey – Department of Theoretical and Applied Mechanics}}</ref><ref name=etudes>{{Cite web |url=https://etudes.ru/en/etudes/chebyshev-plantigrade-machine/ |title=Tchebyshev's plantigrade machine — Mathematical Etudes |accessdate=2014-11-16 |url-status=dead |archive-url=https://web.archive.org/web/20170728024836/http://www.etudes.ru/en/etudes/chebyshev-plantigrade-machine |archivedate=2017-07-28 }}</ref> The Chebyshev Lambda Linkage is a [[cognate linkage]] of the [[Chebyshev linkage]].

The Hoeckens linkage is a [[cognate linkage]] of the [[Chebyshev linkage]].

The Chebyshev Lambda Linkage is used in vehicle suspension mechanisms, walking robots, and rover wheel mechanisms. In 2004, a study completed as a Master of Science Thesis at [[Izmir Institute of Technology]] introduced a new mechanism design by combining two symmetrical Lambda linkages to distribute the force evenly on to ground with providing the straight vertical wheel motion.<ref>{{Cite thesis |last1=Barlas |first1=Fırat |title=Design of a Mars Rover suspension mechanism |url=https://openaccess.iyte.edu.tr/xmlui/handle/11147/3449 |publisher=Izmir Institute of Technology |access-date=3 April 2021 |date=June 2004 |hdl=11147/3449|type=Master Thesis }}</ref> It was then designed, manufactured, and tested in the Earth Rover Project of Los Angeles City College Electronics Club.<ref>{{Citation|title=Moxie Video Intro It|url=https://vimeo.com/117774490|language=en|access-date=2022-11-17}}</ref>

The linkage was first shown in Paris on the [[Exposition Universelle (1878)]] as "The Plantigrade Machine".<ref>http://mech.spbstu.ru/Dzenushko_Dainis:_Walking_mechanisms_survey#The_Chebyshev_Walking_Mechanism</ref><ref>http://www.etudes.ru/en/etudes/stopohod/</ref>

== See also ==

The Hoeckens linkage looks like the greek letter lambda, therefore the linkage is also known as Lambda Mechanism.<ref>http://www.etudes.ru/en/etudes/stopohod/</ref>

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The Hoeckens*[[Chebyshev linkage]], is athe [[cognateCognate linkage|cognate]] of the [[Chebyshev lambda linkage]].

==See also==

*[[Hoeckens linkage]], a linkage that produces a very similar locus using one less link but an additional slider.

*[[Leg mechanism]]

*[[Straight line mechanism]]

*[[Peaucellier–Lipkin linkage]] (an 8-bar linkage)

*[[Chebyshev linkage]]

*[[Four-bar linkage]]

== References ==

*[http://www.designofmachinery.com/DOM/ Design of Machinery -– esp' p134]

{{reflistReflist}}

== External links ==

{{Commons category|Chebyshev's Lambda Mechanism}}

{{commonscat|Hoeckens linkage}}

*[http://www.designofmachinery.com/DOM/Chap_03_3ed_p134.pdf Hoeckens approximate straight-line mechanism] (diagram and table of lengths)

*[http://flash.lakeheadu.ca/~kliu/hoekens.jpg Hoeckens straight line linkage] {{Webarchive|url=https://web.archive.org/web/20110706184142/http://flash.lakeheadu.ca/~kliu/hoekens.jpg |date=2011-07-06 }} (Homemade example)

*[https://web.archive.org/web/20141129015429/http://www.etudes.ru/en/etudes/stopohod/ Video of computer simulation of Tchebychev walking machine] (Стопоход Чебышева).

*[http://www.alexdenouden.nl/artikelen2/vierstangen.htm alexdenouden.nl] -– Rectilinear motion after "Tchebychev"

*[http://mw.concord.org/modeler1.3/mirror/mechanics/peaucellier.html A simulation] {{Webarchive|url=https://web.archive.org/web/20110725190957/http://mw.concord.org/modeler1.3/mirror/mechanics/peaucellier.html |date=2011-07-25 }} using the Molecular Workbench software

*[http://www.mekanizmalar.com/hoekens_linkage.html How does a Hoecken's Linkage Work?] (Interactive Flash Animation)

[[Category:Straight line mechanisms]]

[[Category:Linkages (mechanical)]]

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