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[[Image:Chebyshev linkage.gif|right|180px|Chebyshev linkage]] | |||
The '''Chebyshev linkage''' is a [[linkage (mechanical)|mechanical linkage]] that converts rotational motion to approximate [[straight-line motion]]. | |||
It was invented by the 19th century mathematician [[Pafnuty Chebyshev]] who studied theoretical problems in kinematic mechanisms. One of the problems was the construction of a linkage that converts a rotary motion into an approximate straight line motion. This was also studied by [[James Watt]] in his improvements to the [[steam engine]].<ref>[http://kmoddl.library.cornell.edu/model.php?m=140 Cornell university] - Cross link straight-line mechanism </ref> | |||
The straight line linkage confines the point P — the midpoint on the link ''L''<sub>3</sub> — on a straight line at the two extremes and at the center of travel. (''L''<sub>1</sub>, ''L''<sub>2</sub>, ''L''<sub>3</sub>, and ''L''<sub>4</sub> are as shown in the illustration.) Between those points, point P deviates slightly from a perfect straight line. The proportions between the links are | |||
: <math>L_1 : L_2 : L_3 = 2 : 2.5 : 1 = 4 : 5 : 2. \, </math> | |||
Point P is in the middle of ''L''<sub>3</sub>. This relationship assures that the link ''L''<sub>3</sub> lies vertically when it is at the extremes of its travel.<ref>[http://www.brockeng.com/mechanism/Tchebicheff.htm Brock institute for advanced studies] - Tchebicheff's linkage</ref> | |||
The lengths are related mathematically as follows: | |||
: <math>L_4=L_3+\sqrt{L_2^2 - L_1^2}. \, </math> | |||
==See also== | |||
*[[Watt's linkage]], a similar straight-line mechanism with the direction of one of the arms reversed. | |||
*[[Straight line mechanism]] | |||
*[[Hoekens linkage]] (4-bar linkage that converts rotational motion to approximate straight-line motion) | |||
*[[Peaucellier–Lipkin linkage]] ( an 8-bar linkage that gives perfect linear motion) | |||
*[[Four-bar linkage]] | |||
==References== | |||
{{Reflist}} | |||
==External links== | |||
*[http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=Kemp009&view=50&frames=0&seq=21 Cornell university, ''"How to draw a straight line, by A.B. Kempe, B.A."''] | |||
*[http://mw.concord.org/modeler1.3/mirror/mechanics/peaucellier.html A simulation] using the Molecular Workbench software | |||
[[Category:Linkages]] | |||
[[Category:Linear motion]] | |||
{{technology-stub}} | |||
Revision as of 17:57, 14 November 2013
The Chebyshev linkage is a mechanical linkage that converts rotational motion to approximate straight-line motion.
It was invented by the 19th century mathematician Pafnuty Chebyshev who studied theoretical problems in kinematic mechanisms. One of the problems was the construction of a linkage that converts a rotary motion into an approximate straight line motion. This was also studied by James Watt in his improvements to the steam engine.[1]
The straight line linkage confines the point P — the midpoint on the link L3 — on a straight line at the two extremes and at the center of travel. (L1, L2, L3, and L4 are as shown in the illustration.) Between those points, point P deviates slightly from a perfect straight line. The proportions between the links are
Point P is in the middle of L3. This relationship assures that the link L3 lies vertically when it is at the extremes of its travel.[2]
The lengths are related mathematically as follows:
See also
- Watt's linkage, a similar straight-line mechanism with the direction of one of the arms reversed.
- Straight line mechanism
- Hoekens linkage (4-bar linkage that converts rotational motion to approximate straight-line motion)
- Peaucellier–Lipkin linkage ( an 8-bar linkage that gives perfect linear motion)
- Four-bar linkage
References
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External links
- Cornell university, "How to draw a straight line, by A.B. Kempe, B.A."
- A simulation using the Molecular Workbench software
- ↑ Cornell university - Cross link straight-line mechanism
- ↑ Brock institute for advanced studies - Tchebicheff's linkage