Catenary

In mathematics, the catenary is the shape of a hanging flexible chain or cable when supported at its ends and acted upon by a uniform gravitational force (its own weight). The slope of the chain is largest near the points of suspension because this part of the chain has the most weight pulling down on it. Toward the bottom, the slope of the chain decreases because the chain is supporting less weight.

The word catenary is derived from the Latin word catena, which means "chain." The curve is also called the alysoid, funicular, and chainette. Galileo claimed that the curve of a chain hanging under gravity would be a parabola, but this was disproved by Jungius in a work published in 1669. In 1691, Leibniz, Christiaan Huygens, and Johann Bernoulli derived the equation in response to a challenge by Jakob Bernoulli. Huygens first used the term 'catenaria' in a letter to Leibniz in 1690, and David Gregory wrote a treatise on the catenary in 1690. However Thomas Jefferson is usually credited with the English word 'catenary' ^*.

If you roll a parabola along a straight line, its focus traces out a catenary (see roulette). As proved by Euler in 1744, the catenary is also the curve which, when rotated about the x axis, gives the surface of minimum surface area (the catenoid) for the given bounding circle.

Square wheels can roll perfectly smoothly if the road has evenly spaced bumps in the shape of a series of inverted catenary curves. The wheels can be any regular polygon, but one must use the correct catenary, corresponding correctly to the shape and dimensions of the wheels.

The intrinsic equation of the shape of the catenary is given by the hyperbolic function and exponential equivalent

y = a \cdot \cosh \left ({x \over a} \right ) = {a \over 2} \cdot \left (e^{x/a} + e^{-x/a} \right ).

Suspension bridges

While free-hanging chains follow the curve of the hyperbolic function above, suspension bridge chains or cables, which are tied to the bridge deck at uniform intervals, follow a parabolic curve, much as Galileo originally claimed (derivation).

It is interesting to note that when suspension bridges are constructed, the suspension cables initially sag hyperbolically, before being tied to the deck below, and then gradually assume a parabolic curve as additional connecting cables are tied to connect the main suspension cables with the bridge deck below.

The inverted catenary arch

The catenary is the ideal form for an arch which supports only itself. If made of individual elements whose contacting surfaces are perpendicular to the curve of the arch, no significant shear forces will be present at the joints, and the thrust into the ground will be directly along the line of the arch.

The Gateway Arch in Saint Louis, Missouri, United States follows the form of an inverted catenary. It is 630 feet wide at the base and 630 feet tall. The exact formula

y = -127.7 \cdot \cosh({x / 127.7}) + 757.7

is displayed inside the arch.

In structural engineering a catenary shell is a structural form, usually made of concrete, that follows a catenary curve. The profile for the shell is obtained by using flexible material subjected to gravity, converting it into a rigid formwork for pouring the concrete and then using it as required, usually in an inverted manner.

A kiln, a kind of oven for firing pottery, may be made from firebricks with a body in the shape of a catenary arch, usually nearly as wide as it is high, with the ends closed off with a permanent wall in the back and a temporary wall in the front. The bricks (mortared with fireclay) are stacked upon a temporary form in the shape of an inverted catenary, which is removed upon completion. The form is designed with a simple length of light chain, whose shape is traced onto an end panel of the form, which is inverted for assembly. A particular advantage of this shape is that it does not tend to dismantle itself over repeated heating and cooling cycles — most other forms such as the vertical cylinder must be held together with steel bands.

The Catalan architect Antoni Gaudí made extensive use of catenary shapes in his cathedral Sagrada Familia. In order to solve for the ideal vault lines he built inverted scale models of the numerous domes by using threads under tension to represent stones under compression.

Towed Cables

Drag forces affect the catenary of a cable towed in steady state, leading to more general shapes. A cable having radius $a$ and specific gravity $\sigma$ , and towed at speed $v$ in a medium (e.g., air or water) with density $\rho
_{0}$ , will have an $(x,y)$ position described by the following equations (Dowling 1988):

\frac=-\rho _{0}\left( {\sigma -1}\right) \pi a^{2}g\sin \phi -\rho _{0}v^{2}\pi aC_{T}\cos \phi ;

T\frac=-\rho _{0}\left( {\sigma -1}\right) \pi a^{2}g\cos

\phi +\rho _{0}av^{2}\left{C_{D}\sin \phi +\pi C_{N}}\right \sin \phi ;

\frac=\cos \phi ;

\frac=-\sin \phi .

Here $T$ is the tension, $\phi$ is the incident angle, $g=9.81 {m}/
{s}^{2}$ , and $s$ is the cable scope. There are three drag coefficients: the normal drag coefficient $C_{D}$ ( $\approx 1.5$ for a smooth cylindrical cable); the tangential drag coefficient $C_{T}$ ( $\approx
0.0025$ ), and $C_{N}$ ( $=0.75C_{T}$ ).

The system of equations has four equations and four unknowns: $T$ , $\phi$ , $x$ and $y$ , and is typically solved numerically.

Critical Angle Tow

Critical angle tow occurs when the incident angle doesn't change. In practice, critical angle tow is common, and occurs far from significant point forces.

Setting $\frac=0$ leads to an equation for the critical angle:

\rho _{0}\left( {\sigma -1}\right) \pi a^{2}g\cos \phi =\rho _{0}av^{2}\left[ {C_{D}\sin \phi +\pi C_{N}}\right] \sin \phi .

If $\pi C_{N}<, the formula for the critical angle becomes$

\rho _{0}\left( {\sigma -1}\right) \pi a^{2}g\cos \phi =\rho _{0}av^{2}{ C_{D}\sin }^{2}{\phi ;}

\left( {\sigma -1}\right) \pi ag\cos \phi =v^{2}{C_{D}\sin }^{2}{\phi =}v^{2} {C_{D}}\left( 1-\cos ^{2}\phi \right) ;

\cos ^{2}\phi +\frac{\left( {\sigma -1}\right) \pi ag}{v^{2}{C_{D}}}\cos \phi -1=0;

leading to the rule-of-thumb formula

\cos \phi =-\frac{\left( {\sigma -1}\right) \pi ag}{2v^{2}{C_{D}}}+\sqrt{1+ \frac{\left( {\sigma -1}\right) ^{2}\pi ^{2}a^{2}g^{2}}{4v^{4}{C_{D}^{2}}}}. The drag coefficients of a faired cable are more complicated, involving loading functions that account for drag variation as a function of incidence angle.

Other uses of the term

In railway engineering, a catenary structure consists of overhead lines used to deliver electricity to a railway locomotive, multiple unit, railcar, tram or trolleybus through a pantograph or a trolleypole. These structures consist of an upper structural wire in the form of a shallow catenary, short suspender wires, which may or may not contain insulators, and a lower conductive contact wire. By adjusting the tension in various elements the conductive wire is kept parallel to the centerline of the track, reducing the tendency of the pantograph or trolley to bounce or sway, which could cause a disengagement at high speed.
In semi-rigid airships, a catenary curtain is a fabric and cable internal structure used to distribute the weight of the gondola across a large area of the ship's envelope.
In conveyor systems, the catenary is the portion of the belt underneath the conveyor that is traveling back to the top. It is the weight of the catenary that keeps tension in the belt.

References

A.P. Dowling, The dynamics of towed flexible cylinders. Part 2. Negatively buoyant elements (1988). Journal of Fluid Mechanics, 187, 533-571.

External links

Hanging With Galileo - mathematical derivation of formula for suspended and free-hanging chains; interactive graphical demo of parabolic vs. hyperbolic suspensions.
Catenary Demonstration Experiment - An easy way to demonstrate the Mathematical properties of a cosh using the hanging cable effect. Devised by Jonathan Lansey

Curves | Exponentials