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Encyclopedia > Catenary
Catenaries for different values of the parameter 'a'

## Contents

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.[1] 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' [1]. A broad metal chain made of torus-shaped links. ... Galileo redirects here. ... A parabola A graph showing the reflective property, the directrix (light blue), and the lines connecting the focus and directrix to the parabola (blue) In mathematics, the parabola (from the Greek: Ï€Î±ÏÎ±Î²Î¿Î»Î®) (IPA pronunciation: ) is a conic section generated by the intersection of a right circular conical surface and a plane... Events March 5 - French troops under Marshal Louis-Francois de Boufflers besiege the Spanish-held town of Mons March 20 - Leislers Rebellion - New governor arrives in New York - Jacob Leisler surrenders after standoff of several hours March 29 - Siege of Mons ends to the cityâ€™s surrender May 6... Gottfried Leibniz Gottfried Wilhelm von Leibniz (July 1, 1646 in Leipzig - November 14, 1716 in Hannover) was a German philosopher, scientist, mathematician, diplomat, librarian, and lawyer of Sorb descent. ... Christiaan Huygens (pronounced in English (IPA): ; in Dutch: ) (April 14, 1629 â€“ July 8, 1698), was a Dutch mathematician, astronomer and physicist; born in The Hague as the son of Constantijn Huygens. ... Johann Bernoulli (Basel, July 27, 1667 - January 1, 1748) was a Swiss mathematician. ... An equation is a mathematical statement, in symbols, that two things are the same (or equivalent). ... To meet Wikipedias quality standards, this article or section may require cleanup. ... David Gregory (June 3, 1659â€”October 10, 1708) was a Savilian Professor of astronomy at Oxford and a commentator on Isaac Newtons Principia. ... Thomas Jefferson (13 April 1743 N.S.â€“4 July 1826) was the third President of the United States (1801â€“09), the principal author of the Declaration of Independence (1776), and one of the most influential Founding Fathers for his promotion of the ideals of Republicanism in the United States. ...

The intrinsic equation of the shape of the catenary is given by the hyperbolic cosine function or its exponential equivalent A ray through the origin intercepts the hyperbola in the point , where is the area between the ray, its mirror image with respect to the -axis, and the hyperbola (see animated version with comparison with the trigonometric (circular) functions). ...

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

in which

where To is the horizontal component of the tension (a constant) and P is the weight per length unit.

If you roll a parabola along a straight line, its focus traces out a catenary (see roulette). (The curve traced by one point of a wheel (circle) as it makes one rotation rolling along a horizontal line is not an inverted catenary but a cycloid.) Finally, 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. In geometry, the focus (pl. ... In the differential geometry of curves, a roulette is the general concept behind cycloids, epicycloids, hypocycloids, and involutes. ... Leonhard Euler aged 49 (oil painting by Emanuel Handmann, 1756) Leonhard Euler (April 15, 1707 - September 18, 1783) (pronounced oiler) was a Swiss mathematician and physicist. ... Area is the measure of how much exposed area any two dimensional object has. ... A catenoid A catenoid is a three-dimensional shape made by rotating a catenary curve around the x axis. ... Circle illustration This article is about the shape and mathematical concept of 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 save for a triangle, but one must use the correct catenary, corresponding correctly to the shape and dimensions of the wheels [2]. A literal square wheel is a wheel that, instead of being circular, has the shape of a square. ...

A charge in a uniform electric field moves along a catenary (which tends to a parabola if the charge velocity is much less than the speed of light c). Look up charge in Wiktionary, the free dictionary. ... In physics, the space surrounding an electric charge or in the presence of a time-varying magnetic field has a property called an electric field. ... A parabola A graph showing the reflective property, the directrix (light blue), and the lines connecting the focus and directrix to the parabola (blue) In mathematics, the parabola (from the Greek: Ï€Î±ÏÎ±Î²Î¿Î»Î®) (IPA pronunciation: ) is a conic section generated by the intersection of a right circular conical surface and a plane... The speed of light in a vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness.[1] It is the speed of all electromagnetic radiation, including visible light, in a vacuum. ...

## Suspension bridges

Ponte Hercilio Luz, Florianópolis, Brazil. Suspension bridges follow a parabolic, not catenary, curve.

When suspension bridges are constructed, the suspension cables initially sag as the catenaric function, 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 curve for an arch which supports only its own weight. When the centerline of an arch is made to follow the curve of an up-side-down (ie. inverted) catenary, the arch endures almost pure compression, in which no significant bending moment occurs inside the material. If the arch is made of individual elements (eg., stones) whose contacting surfaces are perpendicular to the curve of the arch, no significant shear forces are be present at these contacting surfaces. (Shear stress is still present inside each stone, as it resists the compressive force along the shear sliding plane.) The thrust (including the weight) of the arch at its two ends is tangent to its centerline. For other uses, see Arch (disambiguation). ... Physical compression is the result of the subjection of a material to compressive stress, resulting in reduction of volume. ... Figure 1. ... Fig. ... Shear stress is a stress state where the stress is parallel or tangential to a face of the material, as opposed to normal stress when the stress is perpendicular to the face. ... For other uses, see tangent (disambiguation). ...

The throne room of the Taq-i Kisra in 1824.

In Antiquity, the curvature of the inverted catenary was intuitively discovered and found to lead to stable arches and vaults. A spectacular example remains in the Taq-i Kisra in Ctesiphon, which was once a great city of Mesopotamia. In ancient Greek and Roman cultures, the less efficient curvature of the circle was more commonly used in arches and vaults. The efficient curvature of inverted catenary was perhaps forgotten in Europe from the fall of Rome to the Middle-Ages and the Renaissance, where it was almost never used, although the pointed arch was perhaps a fortuitous approximation of it. Palace in Ctesiphon File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Palace in Ctesiphon File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Antiquity means different things: Generally it means ancient history, and may be used of any period before the Middle Ages. ... Picture from 1921 The Taq-i Kisra (Persian Ø·Ø§Ù‚ ÙƒØ³Ø±Ù‰ , meaning Iwan of Khusraw) is a monument in Al-Madain, associated to the city of Ctesiphon. ... Ctesiphon, 1932 Ctesiphon (Parthian and Pahlavi: Tyspwn as well as Tisfun, Persian: â€Ž, also known as in Arabic Madain, Maden or Al-Madain: Ø§Ù„Ù…Ø¯Ø§Ø¦Ù†) is one of the great cities of ancient Mesopotamia and the capital of the Parthian Empire and its successor, the Sassanid Empire, for more than 800 years... Mesopotamia was a cradle of civilization geographically located between the Tigris and Euphrates rivers, largely corresponding to modern-day Iraq. ... For other uses, see Arch (disambiguation). ...

Catenary arches under the roof of Gaudí's Casa Milá, Barcelona, Spain

The Gateway Arch (looking East).

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 Gateway Arch, 2001, by Rick Dikeman File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Gateway Arch, 2001, by Rick Dikeman File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The Jefferson National Expansion Memorial (also known as the Gateway Arch or simply The Arch) is located in St. ... Nickname: Location in the state of Missouri Coordinates: , Country State County Independent City Government  - Mayor Francis G. Slay (D) Area  - City  66. ... Official language(s) English Capital Jefferson City Largest city Kansas City Largest metro area St Louis[1] Area  Ranked 21st  - Total 69,709 sq mi (180,693 kmÂ²)  - Width 240 miles (385 km)  - Length 300 miles (480 km)  - % water 1. ...

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. Taipei 101, the worlds tallest building as of 2004. ... This article is about the construction material. ... Gravity is a force of attraction that acts between bodies that have mass. ... Panelized ceiling slab forming system with temporary support structures on a university dorm project. ...

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. Charcoal Kilns, California Gold Kiln, Victoria, Australia Hop kiln. ... Unfired green ware pottery on a traditional drying rack at Conner Prairie living history museum. ... A Fire brick or refractory brick is a block of ceramic material used in lining furnaces and kilns. ... Fire clay is a specific kind of clay used in the manufacture of ceramics. ... A right circular cylinder An elliptic cylinder In mathematics, a cylinder is a quadric surface, with the following equation in Cartesian coordinates: This equation is for an elliptic cylinder, a generalization of the ordinary, circular cylinder (a = b). ...

## Towed cables

A truss arch bridge designed by Gustav Eiffel employing an inverted catenary arch

$frac{{dT}}{{ds}}=-rho _{0}left( {sigma -1}right) pi a^{2}gsin phi -rho _{0}v^{2}pi aC_{T}cos phi ;$
$Tfrac{{dphi }}{{ds}}=-rho _{0}left( {sigma -1}right) pi a^{2}gcos phi +rho _{0}av^{2}left[ {C_{D}sin phi +pi C_{N}}right] sin phi ;$
$frac{{dx}}{{ds}}=cos phi ;$
$frac{{dy}}{{ds}}=-sin phi .$

Here T is the tension, φ is the incident angle, g = 9.81m / s2, and s is the cable scope. There are three drag coefficients: the normal drag coefficient CD ($approx 1.5$ for a smooth cylindrical cable); the tangential drag coefficient CT ($approx 0.0025$), and CN ( = 0.75CT).

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

### Critical angle tow

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

Setting $frac{{dphi }}{{ds}}=0$ leads to an equation for the critical angle:

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

If πCN < < CDsinφ, the formula for the critical angle becomes

or

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

or

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

$cos phi =-frac{left( {sigma -1}right) pi ag}{2v^{2}{C_{D}}}+sqrt{1+ frac{left( {sigma -1}right) ^{2}pi ^{2}a^{2}g^{2}}{4v^{4}{C_{D}^{2}}}}.$

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 chain or belt underneath the conveyor that is traveling back to the start. It is the weight of the catenary that keeps tension in the chain or belt.

## References

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

1. ^ Swetz, Faauvel, Bekken, "Learn from the Masters", 1997, MAA ISBN 0883857030, pp.128-9

Results from FactBites:

 Catenary Summary (1581 words) In 1691, Leibniz, Christiaan Huygens, and Johann Bernoulli derived the equation in response to a challenge by Jakob Bernoulli. 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. 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.
 Catenary - LoveToKnow 1911 (326 words) The simple catenary is shown in the figure. the catenary solves the problem: to find a curve joining two given points, which when revolved about a line co-planar with the points traces a surface of minimum area (see Variations, Calculus Of). x The involute of the catenary is called the tractory, tractrix or antifriction curve; it has a cusp at the vertex of the catenary, and is asymptotic to the directrix.
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