Catenary

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Catenary
Field: Geometry
Image Created By: Mtpaley
Website: Wikipedia

Catenary

A catenary is the curve created by a theoretical representation of a hanging chain or cable held at both ends.


Contents

Basic Description

Notice the shape of the individual segments of the spider web. Held by vertical segments, the curves are formed due to their own weight. These curves are catenaries. Below is another example of a catenary.



The hanging chain is only holding its own weight. The curve that it creates is a catenary.

Generally, a catenary is the shape of a string hanging from two points. It approximates the shape of most string-like objects, such as ropes, chains, necklaces, and even spider webs.

The Catenary and the Parabola Conceptually

The shape of a catenary resembles greatly the shape of a parabola. In fact, in general the visual difference between these two curves is almost imperceptible. However, there is a clear conceptual as well as mathematical difference between these two curves.

A great way to explain the conceptual difference between a parabola and a catenary is by comparing a simple suspension bridge and a suspended deck bridge.

Bridge 1: Simple Suspension Bridge
Bridge 1: Simple Suspension Bridge
Bridge 2: Suspended Deck Bridge
Bridge 2: Suspended Deck Bridge


The bridge labeled as Bridge 1 is the simple suspension bridge; these bridges are rarely seen anymore. The one labeled as Bridge 2 is the suspended deck bridge; most suspended bridges are designed using this model nowadays. Both bridges show a similar u-shaped curve. However, bridge 1 is holding its own weight, whereas bridge 2 is holding its weight as well as that of the horizontal deck attached. Which one is the catenary and which one is the parabola?

A catenary is limited to those curves that are free hanging; there are no other forces than gravity and its own weight acting on the curve.


If you can see this message, you do not have the Java software required to view the applet.

This link has a great applet that allows the user to overlay three different curves (a catenary, a parabola, and a chain) over the picture of a real chain and adjust these curves to see which one matches with the chain: java applet

A More Mathematical Explanation

Note: understanding of this explanation requires: *Calculus, Algebra, Basic Dynamics

A Catenary Mathematically

This curve is a theoretical representation of a palpable object. Thus, [...]

A Catenary Mathematically

This curve is a theoretical representation of a palpable object. Thus, there exists a formula for it. Catenaries are the graph of the equation below:

  • y=a\cosh(\frac{x}{a})

where

  • cosh(x)=\frac{e^x+e^{-x}}{2}


Cosh stands for ''Hyperbolic Cosine''and it is the function that represents the catenary.The differences between catenaries arises from the scaling factor a in the first equation above, which determines the width and steepness of the catenary.

The proof of these equations is complicated; as it relates to the tension and external forces present on the hanging string or chain, it requires an understanding of Physics. To see a derivation of the catenary equation click here.

Other Properties

  • A series of catenaries attached at their endpoints form a roulette. It is created by using a straight line as the fixed line and and a parabola as the rolling curve. The focus of the parabola acts as the fixed point on the rolling curve. This creates a catenary.
  • A series of inverted catenaries creates a surface that allows any polygon-shaped wheels to roll perfectly smoothly, as long as the proportions of the catenary are appropriate for the measurements of the wheels.
Image:Roll4gon.gif




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