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

In physics, 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 chain is steepest 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 overhead lines of a Swiss Federal Railways track. ... In mathematics, a commutative ring R is catenary if for any pair of prime ideals p, q, any two strictly increasing chains p=p0 ⊂p1 . ... Image File history File links Catenary for different parameters of a. ... Image File history File links Catenary for different parameters of a. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... Look up shape in Wiktionary, the free dictionary. ... A broad metal chain made of torus-shaped links. ... 6 or 15cm outside diameter, oil-cooled cables, traversing the Grand Coulee Dam throughout. ... Gravity is a force of attraction that acts between bodies that have mass. ... This article is about the mathematical term. ...

Contents

Overview

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: παραβολή) (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 city’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 – 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, 1659—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.–4 July 1826) was the third President of the United States (1801–09), 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: παραβολή) (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.
Ponte Hercilio Luz, Florianópolis, Brazil. Suspension bridges follow a parabolic, not catenary, curve.

Free-hanging chains follow the curve of the hyperbolic function above, but 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). Image File history File links Download high resolution version (947x760, 98 KB) City of Florianópolis, state of Santa Catarina (SC), Brazil Hercilio Luz Bridge - December 1996 picture taken by: Sérgio Schmiegelow Cesarious 01:35, 11 May 2005 (UTC) (Cesario Simões Junior) File links The following pages link... Image File history File links Download high resolution version (947x760, 98 KB) City of Florianópolis, state of Santa Catarina (SC), Brazil Hercilio Luz Bridge - December 1996 picture taken by: Sérgio Schmiegelow Cesarious 01:35, 11 May 2005 (UTC) (Cesario Simões Junior) File links The following pages link... Nickname: Location in Brazil Coordinates: , Country Region State Santa Catarina Founded March 23, 1726 Government  - Mayor Dario Elias Berger (PMDB) Area  - City 436. ... A suspension bridge is a type of bridge that has been created since ancient times as early as 100 AD. Simple suspension bridges, for use by pedestrians and livestock, are still constructed, based upon the ancient Inca rope bridge. ...


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 طاق كسرى , 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: ‎, also known as in Arabic Madain, Maden or Al-Madain: المدائن) 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
Catenary arches under the roof of Gaudí's Casa Milá, Barcelona, Spain

The Catalan architect Antoni Gaudí made extensive use of catenary shapes in most of his work. In order to find the best curvature for the arches and ribs that he desired to use in the crypt of the Church of Colònia Güell, Gaudí constructed inverted scale models made of numerous threads under tension to represent stones under compression. This technique worked well to solve angled columns, arches, and single-curvature vaults, but could not be used to solve the more complex, double-curvature vaults that he intended to use in the nave of the church of the Sagrada Familia. The idea that Gaudi used thread models to solve the nave of the Sagrada Familia is a common misconception, although it could have been used in the solution of the bell towers. Image File history File links Download high resolution version (1564x2346, 414 KB) Summary Parabolic or catenary arches under the terrace of Casa Milá (a. ... Image File history File links Download high resolution version (1564x2346, 414 KB) Summary Parabolic or catenary arches under the terrace of Casa Milá (a. ... Casa Milà, Barcelona. ... Capital Barcelona Official languages Spanish and Catalan In Val dAran, also Aranese. ... For other uses, see Architect (disambiguation). ... Antoni Gaudí i Cornet (Riudoms or Reus, 25 June 1852 – Barcelona, 10 June 1926) – sometimes referred to by the Spanish translation of his name, Antonio Gaudí – was a Spanish architect from Catalonia, who belonged to the Modernisme (Art Nouveau) movement and was famous for his unique style and highly individualistic... Antoni Gaudis great unfinished work. ... The Sagrada Familia by night in March 2006 La Sagrada Familia (The Holy Family) is a large Roman Catholic basilica under construction in Barcelona, Catalonia, Spain. ...

The Gateway Arch (looking East).
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 km²)  - 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
A truss arch bridge designed by Gustav Eiffel employing an inverted catenary arch

When a cable is subject to wind or water flows, the drag forces lead to more general shapes, since the forces are not distributed in the same way as the weight. A cable having radius a and specific gravity σ, and towed at speed v in a medium (e.g., air or water) with density ρ0, will have an (x,y) position described by the following equations (Dowling 1988): Image File history File links Download high resolution version (1984x1488, 833 KB) Summary Viaduc de Garabit (juillet 2005) from fr:wiki <2005> (C) <J. THURION> Licensing File links The following pages link to this file: Garabit viaduct ... Image File history File links Download high resolution version (1984x1488, 833 KB) Summary Viaduc de Garabit (juillet 2005) from fr:wiki <2005> (C) <J. THURION> Licensing File links The following pages link to this file: Garabit viaduct ... A truss arch bridge combines elements of a truss and an arch. ... Alexandre Gustave Eiffel (December 15, 1832 – December 27, 1923; French pronunciation in IPA, in English usually pronounced in the German manner ) was a French structural engineer and architect and a specialist of metallic structures. ...

 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;

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 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.

Engineering is the discipline of acquiring and applying knowledge of design, analysis, and/or construction of works for practical purposes. ... The overhead lines of a Swiss Federal Railways track. ... Great Western Railway No. ... This article is about Multiple Units vehicles. ... A railcar (not to be confused with a railway car) is a self-propelled railway vehicle designed to transport passengers. ... This article refers to public transport vehicles running on rails. ... Further information: electric bus A trolleybus (also known as trolley bus, trolley coach, trackless trolley, trackless tram or simply trolley) is an electric bus powered by two overhead wires, from which it draws electricity using two trolley poles. ... A pantograph is a device that collects electric current from overhead lines for electric trains or trams. ... Trolley poles are usually tapered cylindrical poles of wood or metal, used to transfer electricity from a live overhead wire to the control and propulsion equipment of a trolley car, tram or trolley bus. ... USS Akron (ZRS-4) in flight, November 2, 1931 An airship or dirigible is a buoyant lighter-than-air aircraft that can be steered and propelled through the air. ... A Venetian gondola A gòndola is a traditional Venetian sculling boat. ... Front of an envelope mailed in the U.S. in 1906 contains postage stamp and address. ... Point of contact between a power transmission belt and its pulley A conveyor belt or belt conveyor consists of two end pulleys, with a continuous loop of material that rotates about them. ...

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

External links

Wikimedia Commons has media related to:
Catenary
  • 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
  • Horizontal Conveyor Arrangement - Diagrams of different horizontal conveyor layouts showing options for the catenary section both supported and unsupported
  • Catenary curve derived - The shape of a catenary is derived, plus examples of a chain hanging between 2 points of unequal height, including C program to calculate the curve.

  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.
  More results at FactBites »

 
 

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