Catenary
A catenary is a type of curve. An ideal chain hanging between two supports and acted on by a uniform gravitational force makes the shape of a catenary.[1] (An ideal chain is one that can bend perfectly, cannot be stretched and has the same density throughout.[2]) The supports can be at different heights and the shape will still be a catenary.[3] A catenary looks a bit like a parabola, but they are different.[4]
The equation for a catenary in Cartesian coordinates is[2][5]
- [math]\displaystyle{ y = a \cosh \left(\frac{x}{a}\right) }[/math]
where [math]\displaystyle{ a }[/math] is a parameter that determines the shape of the catenary[5] and [math]\displaystyle{ \cosh }[/math] is the hyperbolic cosine function, which is defined as[6]
- [math]\displaystyle{ \cosh x = \frac {e^x + e^{-x}} {2} }[/math].
Hence, we can also write the catenary equation as
- [math]\displaystyle{ y = \frac{a\left(e^\frac{x}{a} + e^{-\frac{x}{a}}\right)}{2} }[/math].
The word "catenary" comes from the Latin word catena, which means "chain".[6] A catenary is also called called an alysoid and a chainette.[1]
Catenary Media
A chain hanging from points forms a catenary.
Freely-hanging overhead power lines also form a catenary (most prominently visible with high-voltage lines, and with some imperfection near to the insulators).
The silk on a spider's web forming multiple elastic catenaries.
Antoni Gaudí's catenary model at Casa Milà
Analogy between an arch and a hanging chain and comparison to the dome of Saint Peter's Basilica in Rome (Giovanni Poleni, 1748)
Simple suspension bridges are essentially thickened cables, and follow a catenary curve.
Stressed ribbon bridges, like the Leonel Viera Bridge in Maldonado, Uruguay, also follow a catenary curve, with cables embedded in a rigid deck.
A heavy anchor chain forms a catenary, with a low angle of pull on the anchor.
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References
- ↑ 1.0 1.1 "Catenary". Wolfram Research. Retrieved 2016-10-30.
- ↑ 2.0 2.1 "The Catenary - The "Chain" Curve". California State University. Archived from the original on 2017-06-30. Retrieved 2019-01-01.
{{cite web}}: CS1 maint: bot: original URL status unknown (link) - ↑ Rosbjerg, Bo. "Catenary" (PDF). Aalborg University. Retrieved 2016-10-30.
- ↑ "Catenary and Parabola Comparison". Drexel University. Retrieved 2016-11-05.
- ↑ 5.0 5.1 "Equation of Catenary". Math24.net. Archived from the original on 2016-10-21. Retrieved 2016-10-30.
- ↑ 6.0 6.1 Stroud, K. A.; Booth, Dexter J. (2013). Engineering Mathematics (7th ed.). Palgrave Macmillan. p. 438. ISBN 978-1-137-03120-4.