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Catenary
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{{short description|Plane curve formed by a hanging cable}}{{About|the mathematical curve}}{{Redirect|Chainette|the wine grape also known as Chainette|Cinsaut}}(File:Kette Kettenkurve Catenary 2008 PD.JPG|thumb|180px|right|A chain hanging from points forms a catenary.)File:PylonsSunset-5982.jpg|thumb|180px|right|Freely-hanging electric power cablepower cableFile:SpiderCatenary.jpg|thumb|180px|right|The silk on a spider's web forming multiple elastic catenaries.]]In physics and geometry, a catenary ({{IPAc-en|US|Ëˆ|k|Ã¦|t|É™n|É›r|i}}, {{IPAc-en|UK|k|É™|Ëˆ|t|iË|n|É™r|i}}) is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends.The catenary curve has a U-like shape, superficially similar in appearance to a parabolic arch, but it is not a parabola.The curve appears in the design of certain types of arches and as a cross section of the catenoidâ€”the shape assumed by a soap film bounded by two parallel circular rings.The catenary is also called the alysoid, chainette,MathWorld or, particularly in the materials sciences, funicular.e.g.: BOOK, Shodek, Daniel L., Structures, 5th, 2004, Prentice Hall, 978-0-13-048879-4, 148137330, 22, Mathematically, the catenary curve is the graph of the hyperbolic cosine function. The surface of revolution of the catenary curve, the catenoid, is a minimal surface, specifically a minimal surface of revolution. A hanging chain will assume a shape of least potential energy which is a catenary.WEB,weblink "The Calculus of Variations", 2015, 2019-05-03, The mathematical properties of the catenary curve were first studied by Robert Hooke in the 1670s, and its equation was derived by Leibniz, Huygens and Johann Bernoulli in 1691.Catenaries and related curves are used in architecture and engineering, in the design of bridges and arches, so that forces do not result in bending moments. In the offshore oil and gas industry, "catenary" refers to a steel catenary riser, a pipeline suspended between a production platform and the seabed that adopts an approximate catenary shape. In the rail industry it refers to the overhead wiring which transfers power to trains due to the curve of the supporting cable. (Since this supports a lighter contact wire it does not follow a true catenary curve.)In optics and electromagnetics, the hyperbolic cosine and sine functions are basic solutions to Maxwell's equations.BOOK, Luo, Xiangang, Catenary optics, 2019, Springer, Singapore, 978-981-13-4818-1, 10.1007/978-981-13-4818-1, The symmetric modes consisting of two evanescent waves would form a catenary shape.JOURNAL, Bourke, Levi, Blaikie, Richard J., 2017-12-01, Herpin effective media resonant underlayers and resonant overlayer designs for ultra-high NA interference lithography,weblink JOSA A, EN, 34, 12, 2243â€“2249, 10.1364/JOSAA.34.002243, 29240100, 1520-8532, JOURNAL, Pu, Mingbo, Guo, Yinghui, Li, Xiong, Ma, Xiaoliang, Luo, Xiangang, 2018-07-05, Revisitation of Extraordinary Young's Interference: from Catenary Optical Fields to Spinâ€“Orbit Interaction in Metasurfaces, ACS Photonics, 5, 8, 3198â€“3204, en, 10.1021/acsphotonics.8b00437, 2330-4022, JOURNAL, Pu, Mingbo, Ma, XiaoLiang, Guo, Yinghui, Li, Xiong, Luo, Xiangang, 2018-07-23, Theory of microscopic meta-surface waves based on catenary optical fields and dispersion,weblink Optics Express, EN, 26, 15, 19555â€“19562, 10.1364/OE.26.019555, 30114126, 1094-4087, - the content below is remote from Wikipedia
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History
File:GaudiCatenaryModel.jpg|thumb|250px|Antoni GaudÃ's catenary model at Casa MilÃ Casa MilÃ The word "catenary" is derived from the Latin word catÄ“na, which means "chain". The English word "catenary" is usually attributed to Thomas Jefferson,WEB,weblink "Catenary" at Math Words, Pballew.net, 1995-11-21, 2010-11-17, BOOK, Barrow, John D., 100 Essential Things You Didn't Know You Didn't Know: Math Explains Your World, 2010, W. W. Norton & Company, 978-0-393-33867-6, 27, who wrote in a letter to Thomas Paine on the construction of an arch for a bridge:}}It is often said that Galileo thought the curve of a hanging chain was parabolic. In his Two New Sciences (1638), Galileo says that a hanging cord is an approximate parabola, and he correctly observes that this approximation improves as the curvature gets smaller and is almost exact when the elevation is less than 45Â°.BOOK, Fahie, John Joseph, Galileo, His Life and Work,weblink 1903, J. Murray, 359â€“360, That the curve followed by a chain is not a parabola was proven by Joachim Jungius (1587â€“1657); this result was published posthumously in 1669.Lockwood p. 124The application of the catenary to the construction of arches is attributed to Robert Hooke, whose "true mathematical and mechanical form" in the context of the rebuilding of St Paul's Cathedral alluded to a catenary.JOURNAL, 532102, Monuments and Microscopes: Scientific Thinking on a Grand Scale in the Early Royal Society, Notes and Records of the Royal Society of London, 55, 2, 289â€“308, Lisa, Jardine, 2001, 10.1098/rsnr.2001.0145, Some much older arches approximate catenaries, an example of which is the Arch of Taq-i Kisra in Ctesiphon.BOOK, Denny, Mark, Super Structures: The Science of Bridges, Buildings, Dams, and Other Feats of Engineering, 2010, JHU Press, 978-0-8018-9437-4, 112â€“113, In 1671, Hooke announced to the Royal Society that he had solved the problem of the optimal shape of an arch, and in 1675 published an encrypted solution as a Latin anagramcf. the anagram for Hooke's law, which appeared in the next paragraph. in an appendix to his Description of Helioscopes,WEB,weblink Arch Design, Lindahall.org, 2002-10-28, 2010-11-17, yes,weblink" title="web.archive.org/web/20101113210736weblink">weblink 2010-11-13, where he wrote that he had found "a true mathematical and mechanical form of all manner of Arches for Building." He did not publish the solution to this anagramThe original anagram was abcccddeeeeefggiiiiiiiillmmmmnnnnnooprrsssttttttuuuuuuuux: the letters of the Latin phrase, alphabetized. in his lifetime, but in 1705 his executor provided it as ut pendet continuum flexile, sic stabit contiguum rigidum inversum, meaning "As hangs a flexible cable so, inverted, stand the touching pieces of an arch."In 1691, Gottfried Leibniz, Christiaan Huygens, and Johann Bernoulli derived the equation in response to a challenge by Jakob Bernoulli; their solutions were published in the Acta Eruditorum for June 1691.{{citation|first=C.|last=Truesdell|title=The Rotational Mechanics of Flexible Or Elastic Bodies 1638â€“1788: Introduction to Leonhardi Euleri Opera Omnia Vol. X et XI Seriei Secundae|location=ZÃ¼rich|url=https://books.google.co.uk/books?id=gxrzm6y10EwC&pg=PA66#v=onepage&q&f=false|page=66|publisher=Orell FÃ¼ssli|date=1960|isbn=9783764314415}}{{citation|first=C. R.|last=Calladine|title=An amateur's contribution to the design of Telford's Menai Suspension Bridge: a commentary on Gilbert (1826) 'On the mathematical theory of suspension bridges'|journal=Philosophical Transactions of the Royal Society A|date=2015-04-13|volume=373|issue=2039|page=20140346|doi=10.1098/rsta.2014.0346|pmid=25750153|pmc=4360092}} David Gregory wrote a treatise on the catenary in 1697{{citation|first=Davidis|last=Gregorii|title=Catenaria|journal=Philosophical Transactions|volume=19|issue=231|date=August 1697|pages=637â€“652|doi=10.1098/rstl.1695.0114}} in which he provided an incorrect derivation of the correct differential equation.Euler proved in 1744 that the catenary is the curve which, when rotated about the {{mvar|x}}-axis, gives the surface of minimum surface area (the catenoid) for the given bounding circles. Nicolas Fuss gave equations describing the equilibrium of a chain under any force in 1796.Routh Art. 455, footnoteInverted catenary arch
Catenary arches are often used in the construction of kilns. To create the desired curve, the shape of a hanging chain of the desired dimensions is transferred to a form which is then used as a guide for the placement of bricks or other building material.BOOK, Minogue, Coll, Sanderson, Robert, Wood-fired Ceramics: Contemporary Practices, 2000, University of Pennsylvania, 978-0-8122-3514-2, 42, BOOK, Peterson, Susan, Peterson, Jan, The Craft and Art of Clay: A Complete Potter's Handbook,weblink 2003, Laurence King, 978-1-85669-354-7, 224, The Gateway Arch in St. Louis, Missouri, United States is sometimes said to be an (inverted) catenary, but this is incorrect.{{Citation | last1=Osserman | first1=Robert | title=Mathematics of the Gateway Arch | url=http://www.ams.org/notices/201002/index.html | year=2010 | journal=Notices of the American Mathematical Society | issn=0002-9920 | volume=57 | issue=2 | pages=220â€“229}} It is close to a more general curve called a flattened catenary, with equation {{math|1=y = Aâ€‰cosh(Bx)}}, which is a catenary if {{math|1=AB = 1}}. While a catenary is the ideal shape for a freestanding arch of constant thickness, the Gateway Arch is narrower near the top. According to the U.S. National Historic Landmark nomination for the arch, it is a "weighted catenary" instead. Its shape corresponds to the shape that a weighted chain, having lighter links in the middle, would form.JOURNAL, Hicks, Clifford B., The Incredible Gateway Arch: America's Mightiest National Monument,weblink 120, December 1963, 89, 0032-4558, 6, Popular Mechanics, {{citation|url={{NHLS url|id=87001423}}|title=National Register of Historic Places Inventory-Nomination: Jefferson National Expansion Memorial Gateway Arch / Gateway Arch; or "The Arch"|year=1985 |first=Laura SoulliÃ¨re |last=Harrison |publisher=National Park Service }} and {{NHLS url|id=87001423|title=Accompanying one photo, aerial, from 1975|photos=y}} {{small|(578 KB)}}Catenary bridges
File:Soderskar-bridge.jpg|thumb|right|250px|Simple suspension bridgeSimple suspension bridgeFile:Puentedelabarra(below).jpg|thumb|right|250px|Stressed ribbon bridges, like this one in Maldonado, UruguayMaldonado, UruguayIn free-hanging chains, the force exerted is uniform with respect to length of the chain, and so the chain follows the catenary curve.BOOK, Owen, Byer, Felix, Lazebnik, Deirdre L., Smeltzer, Methods for Euclidean Geometry,weblink 2010-09-02, MAA, 978-0-88385-763-2, 210, The same is true of a simple suspension bridge or "catenary bridge," where the roadway follows the cable.BOOK, Leonardo, FernÃ¡ndez Troyano, Bridge Engineering: A Global Perspective,weblink 2003, Thomas Telford, 978-0-7277-3215-6, 514, BOOK, W., Trinks, M. H., Mawhinney, R. A., Shannon, R. J., Reed, J. R., Garvey, Industrial Furnaces,weblink 2003-12-05, Wiley, 978-0-471-38706-0, 132, A stressed ribbon bridge is a more sophisticated structure with the same catenary shape.BOOK, John S., Scott, Dictionary Of Civil Engineering, 1992-10-31, Springer, 978-0-412-98421-1, 433, JOURNAL, The Architects' Journal, 207, 1998, 51, However, in a suspension bridge with a suspended roadway, the chains or cables support the weight of the bridge, and so do not hang freely. In most cases the roadway is flat, so when the weight of the cable is negligible compared with the weight being supported, the force exerted is uniform with respect to horizontal distance, and the result is a parabola, as discussed below (although the term "catenary" is often still used, in an informal sense). If the cable is heavy then the resulting curve is between a catenary and a parabola.Lockwood p. 122WEB, Hanging With Galileo, June 30, 2006, Paul, Kunkel, Whistler Alley Mathematics,weblink March 27, 2009, File:Comparison catenary parabola.svg|thumb|none|400px|Comparison of a catenary arch (black dotted curve) and a 1=y = cosh(x)}} and {{math|1=y =x2}}Anchoring of marine objects
File:Catenary.PNG|thumb|right|250px|A heavy anchoranchorThe catenary produced by gravity provides an advantage to heavy (wikt:rode#Noun|anchor rodes). An anchor rode (or anchor line) usually consists of chain or cable or both. Anchor rodes are used by ships, oil rigs, docks, floating wind turbines, and other marine equipment which must be anchored to the seabed.When the rode is slack, the catenary curve presents a lower angle of pull on the weblink Chain, Rope, and Catenary â€“ Anchor Systems For Small Boats, Petersmith.net.nz, 2010-11-17,Mathematical description
Equation
missing image!
- catenary-pm.svg">thumb|350px|right|Catenaries for different values of {{mvar|a}}Catenary-tension.png -
Î»}} mass per unit length.The equation of a catenary in Cartesian coordinates has the form
- catenary-pm.svg">thumb|350px|right|Catenaries for different values of {{mvar|a}}Catenary-tension.png -
y = a cosh left(frac{x}{a}right) = frac{a}{2}left(e^frac{x}{a} + e^{-frac{x}{a}}right)
where {{math|cosh}} is the hyperbolic cosine function. All catenary curves are similar to each other; changing the parameter {{mvar|a}} is equivalent to a uniform scaling of the curve.WEB,weblink Catenary, Xahlee.org, 2003-05-28, 2010-11-17, The Whewell equation for the catenary is
tan varphi = frac{s}{a},.
Differentiating gives
frac{dvarphi}{ds} = frac{cos^2varphi}{a}
and eliminating {{mvar|Ï†}} gives the CesÃ ro equationMathWorld, eq. 7
kappa=frac{a}{s^2+a^2},.
The radius of curvature is then
rho = a sec^2 varphi
which is the length of the line normal to the curve between it and the {{mvar|x}}-axis.Routh Art. 444Relation to other curves
When a parabola is rolled along a straight line, the roulette curve traced by its focus is a catenary. The envelope of the directrix of the parabola is also a catenary.Yates p. 80 The involute from the vertex, that is the roulette formed traced by a point starting at the vertex when a line is rolled on a catenary, is the tractrix.Another roulette, formed by rolling a line on a catenary, is another line. This implies that square wheels can roll perfectly smoothly on a road made of a series of bumps in the shape of an inverted catenary curve. The wheels can be any regular polygon except a triangle, but the catenary must have parameters corresponding to the shape and dimensions of the wheels."MEMBERWIDE">FIRST1=LEONFIRST2=STANYEAR=1992, Roads and Wheelsvolume=65pages=283â€“301 doi=10.2307/2691240,Geometrical properties
Over any horizontal interval, the ratio of the area under the catenary to its length equals {{mvar|a}}, independent of the interval selected. The catenary is the only plane curve other than a horizontal line with this property. Also, the geometric centroid of the area under a stretch of catenary is the midpoint of the perpendicular segment connecting the centroid of the curve itself and the {{mvar|x}}-axis.JOURNAL, Parker, Edward, 2010, A Property Characterizing the Catenary, Mathematics Magazine, 83, 63â€“64, 10.4169/002557010X485120,Science
A moving charge in a uniform electric field travels along a catenary (which tends to a parabola if the charge velocity is much less than the speed of light {{mvar|c}}).BOOK, Landau, Lev Davidovich,weblink The Classical Theory of Fields, 1975, Butterworth-Heinemann, 978-0-7506-2768-9, 56, The surface of revolution with fixed radii at either end that has minimum surface area is a catenary revolved about the {{mvar|x}}-axis.BOOK, Curves and their Propertieslast=Yatesyear=1952, 13,Analysis
Model of chains and arches
In the mathematical model the chain (or cord, cable, rope, string, etc.) is idealized by assuming that it is so thin that it can be regarded as a curve and that it is so flexible any force of tension exerted by the chain is parallel to the chain.Routh Art. 442, p. 316 The analysis of the curve for an optimal arch is similar except that the forces of tension become forces of compression and everything is inverted.BOOK, Church, Irving Porter, Mechanics of Engineering,weblink 1890, Wiley, 387, An underlying principle is that the chain may be considered a rigid body once it has attained equilibrium.Whewell p. 65 Equations which define the shape of the curve and the tension of the chain at each point may be derived by a careful inspection of the various forces acting on a segment using the fact that these forces must be in balance if the chain is in static equilibrium.Let the path followed by the chain be given parametrically by {{math|1=r = (x, y) = (x(s), y(s))}} where {{mvar|s}} represents arc length and {{math|r}} is the position vector. This is the natural parameterization and has the property that
frac{dmathbf{r}}{ds}=mathbf{u}
where {{math|u}} is a unit tangent vector.(File:CatenaryForceDiagram.svg|thumb|Diagram of forces acting on a segment of a catenary from {{math|c}} to {{math|r}}. The forces are the tension {{math|T0}} at {{math|c}}, the tension {{math|T}} at {{math|r}}, and the weight of the chain {{math|(0, âˆ’Î»gs)}}. Since the chain is at rest the sum of these forces must be zero.)A differential equation for the curve may be derived as follows.Following Routh Art. 443 p. 316 Let {{math|c}} be the lowest point on the chain, called the vertex of the catenary.Routh Art. 443 p. 317 The slope {{math|1=dy/dx}} of the curve is zero at C since it is a minimum point. Assume {{math|r}} is to the right of {{math|c}} since the other case is implied by symmetry. The forces acting on the section of the chain from c to r are the tension of the chain at {{math|c}}, the tension of the chain at r, and the weight of the chain. The tension at {{math|c}} is tangent to the curve at {{math|c}} and is therefore horizontal without any vertical component and it pulls the section to the left so it may be written {{math|(âˆ’T0, 0)}} where {{math|T0}} is the magnitude of the force. The tension at {{math|r}} is parallel to the curve at {{math|r}} and pulls the section to the right. The tension at {{math|r}} can be split into two components so it may be written {{math|1=Tu = (T cos Ï†, T sin Ï†)}}, where {{mvar|T}} is the magnitude of the force and {{mvar|Ï†}} is the angle between the curve at {{math|r}} and the {{mvar|x}}-axis (see tangential angle). Finally, the weight of the chain is represented by {{math|(0, âˆ’Î»gs)}} where {{mvar|Î»}} is the mass per unit length, {{mvar|g}} is the acceleration of gravity and {{mvar|s}} is the length of the segment of chain between {{math|c}} and {{math|r}}.The chain is in equilibrium so the sum of three forces is {{math|0}}, therefore
T cos varphi = T_0
and
T sin varphi = lambda gs,,
and dividing these gives
frac{dy}{dx}=tan varphi = frac{lambda gs}{T_0},.
It is convenient to write
a = frac{T_0}{lambda g}
which is the length of chain whose weight on Earth is equal in magnitude to the tension at {{math|c}}.Whewell p. 67 Then
frac{dy}{dx}=frac{s}{a}
is an equation defining the curve.The horizontal component of the tension, {{math|1=T cos Ï† = T0}} is constant and the vertical component of the tension, {{math|1=T sin Ï† = Î»gs}} is proportional to the length of chain between the {{math|r}} and the vertex.Derivation of equations for the curve
The differential equation given above can be solved to produce equations for the curve.Following Routh Art. 443 p/ 317From
frac{dy}{dx} = frac{s}{a},,
the formula for arc length gives
frac{ds}{dx} = sqrt{1+left(dfrac{dy}{dx}right)^2} = frac{sqrt{a^2+s^2}}{a},.
Then
frac{dx}{ds} = frac{1}{frac{ds}{dx}} = frac{a}{sqrt{a^2+s^2}}
and
frac{dy}{ds} = frac{frac{dy}{dx}}{frac{ds}{dx}} = frac{s}{sqrt{a^2+s^2}},.
The second of these equations can be integrated to give
y = sqrt{a^2+s^2} + beta
and by shifting the position of the {{mvar|x}}-axis, {{mvar|Î²}} can be taken to be 0. Then
y = sqrt{a^2+s^2},,quad y^2=a^2+s^2,.
The {{mvar|x}}-axis thus chosen is called the directrix of the catenary.It follows that the magnitude of the tension at a point {{math|(x, y)}} is {{math|1=T = Î»gy}}, which is proportional to the distance between the point and the directrix.Routh Art. 443 p. 318The integral of the expression for {{mvar|{{sfrac|dx|ds}}}} can be found using standard techniques, givingUse of hyperbolic functions follows Maurer p. 107
x = aoperatorname{arsinh}left(frac{s}{a}right) + alpha,.
and, again, by shifting the position of the {{mvar|y}}-axis, {{mvar|Î±}} can be taken to be 0. Then
x = aoperatorname{arsinh}left(frac{s}{a}right),,quad s=a sinhleft(frac{x}{a}right),.
The {{mvar|y}}-axis thus chosen passes through the vertex and is called the axis of the catenary.These results can be used to eliminate {{mvar|s}} giving
y = a coshleft(frac{x}{a}right),.
Alternative derivation
The differential equation can be solved using a different approach.Following Lamb p. 342 From
s = a tan varphi
it follows that
frac{dx}{dvarphi} = frac{dx}{ds}frac{ds}{dvarphi}=cos varphi cdot a sec^2 varphi= a sec varphi
and
frac{dy}{dvarphi} = frac{dy}{ds}frac{ds}{dvarphi}=sin varphi cdot a sec^2 varphi= a tan varphi sec varphi,.
Integrating gives,
x = a ln(sec varphi + tan varphi) + alpha
and
y = a sec varphi + beta,.
As before, the {{mvar|x}} and {{mvar|y}}-axes can be shifted so {{mvar|Î±}} and {{mvar|Î²}} can be taken to be 0. Then
sec varphi + tan varphi = e^frac{x}{a},,
and taking the reciprocal of both sides
sec varphi - tan varphi = e^{-frac{x}{a}},.
Adding and subtracting the last two equations then gives the solution
y = a sec varphi = a coshleft(frac{x}{a}right),,
and
s = a tan varphi = a sinhleft(frac{x}{a}right),.
Determining parameters
In general the parameter {{mvar|a}} is the position of the axis. The equation can be determined in this case as follows:Following Todhunter Art. 186Relabel if necessary so that {{math|P1}} is to the left of {{math|P2}} and let {{mvar|h}} be the horizontal and {{mvar|v}} be the vertical distance from {{math|P1}} to {{math|P2}}. Translate the axes so that the vertex of the catenary lies on the {{mvar|y}}-axis and its height {{mvar|a}} is adjusted so the catenary satisfies the standard equation of the curve
y = a coshleft(frac{x}{a}right)
and let the coordinates of {{math|P1}} and {{math|P2}} be {{math|(x1, y1)}} and {{math|(x2, y2)}} respectively. The curve passes through these points, so the difference of height is
v = a coshleft(frac{x_2}{a}right) - a coshleft(frac{x_1}{a}right),.
and the length of the curve from {{math|P1}} to {{math|P2}} is
s = a sinhleft(frac{x_2}{a}right) - a sinhleft(frac{x_1}{a}right),.
When {{math|s2 âˆ’ v2}} is expanded using these expressions the result is
s^2-v^2=2a^2left(coshleft(frac{x_2-x_1}{a}right)-1right)=4a^2sinh^2left(frac{h}{2a}right),,
so
sqrt{s^2-v^2}=2asinhleft(frac{h}{2a}right),.
This is a transcendental equation in {{mvar|a}} and must be solved numerically. It can be shown with the methods of calculusSee Routh art. 447 that there is at most one solution with {{math|a > 0}} and so there is at most one position of equilibrium.Generalizations with vertical force
Nonuniform chains
If the density of the chain is variable then the analysis above can be adapted to produce equations for the curve given the density, or given the curve to find the density.Following Routh Art. 450Let {{mvar|w}} denote the weight per unit length of the chain, then the weight of the chain has magnitude
int_mathbf{c}^mathbf{r} w, ds,,
where the limits of integration are {{math|c}} and {{math|r}}. Balancing forces as in the uniform chain produces
T cos varphi = T_0
and
T sin varphi = int_mathbf{c}^mathbf{r} w, ds,,
and therefore
frac{dy}{dx}=tan varphi = frac{1}{T_0} int_mathbf{c}^mathbf{r} w, ds,.
Differentiation then gives
w=T_0 frac{d}{ds}frac{dy}{dx} = frac{T_0 dfrac{d^2y}{dx^2}}{sqrt{1+left(dfrac{dy}{dx}right)^2}},.
In terms of {{mvar|Ï†}} and the radius of curvature {{mvar|Ï}} this becomes
w= frac{T_0}{rho cos^2 varphi},.
Suspension bridge curve
File:Golden Gate Bridge, SF.jpg|thumb|right|480px|Golden Gate Bridge. Most suspension bridgesuspension bridgeA similar analysis can be done to find the curve followed by the cable supporting a suspension bridge with a horizontal roadway.Following Routh Art. 452 If the weight of the roadway per unit length is {{mvar|w}} and the weight of the cable and the wire supporting the bridge is negligible in comparison, then the weight on the cable from {{math|c}} to {{math|r}} is {{mvar|wx}} where {{mvar|x}} is the horizontal distance between {{math|c}} and {{math|r}}. Proceeding as before gives the differential equation
frac{dy}{dx}=tan varphi = frac{w}{T_0}x,.
This is solved by simple integration to get
y=frac{w}{2T_0}x^2 + beta
and so the cable follows a parabola. If the weight of the cable and supporting wires is not negligible then the analysis is more complex.Ira Freeman investigated the case where only the cable and roadway are significant, see the External links section. Routh gives the case where only the supporting wires have significant weight as an exercise.Catenary of equal strength
In a catenary of equal strength, the cable is strengthened according to the magnitude of the tension at each point, so its resistance to breaking is constant along its length. Assuming that the strength of the cable is proportional to its density per unit length, the weight, {{mvar|w}}, per unit length of the chain can be written {{mvar|{{sfrac|T|c}}}}, where {{mvar|c}} is constant, and the analysis for nonuniform chains can be applied.Following Routh Art. 453In this case the equations for tension are
T cos varphi = T_0,,
T sin varphi = frac{1}{c}int T, ds,.
Combining gives
c tan varphi = int sec varphi, ds
and by differentiation
c = rho cos varphi
where {{mvar|Ï}} is the radius of curvature.The solution to this is
y = c lnleft(secleft(frac{x}{c}right)right),.
In this case, the curve has vertical asymptotes and this limits the span to {{math|Ï€c}}. Other relations are
x = cvarphi,,quad s = lnleft(tanleft(frac{pi+2varphi}{4}right)right),.
The curve was studied 1826 by Davies Gilbert and, apparently independently, by Gaspard-Gustave Coriolis in 1836.Recently, it was shown that this type of catenary could act as a building block of electromagnetic metasurface and was known as "catenary of equal phase gradient".PU>FIRST1=MINGBOFIRST2=XIONGFIRST3=XIAOLIANGFIRST4=XIANGANGTITLE=CATENARY OPTICS FOR ACHROMATIC GENERATION OF PERFECT OPTICAL ANGULAR MOMENTUMVOLUME=1PAGES=E1500396 DOI=10.1126/SCIADV.1500396PMC=4646797, Elastic catenary
In an elastic catenary, the chain is replaced by a spring which can stretch in response to tension. The spring is assumed to stretch in accordance with Hooke's Law. Specifically, if {{math|p}} is the natural length of a section of spring, then the length of the spring with tension {{math|T}} applied has length
s=left(1+frac{T}{E}right)p,,
where {{math|E}} is a constant equal to {{math|kp}}, where {{math|k}} is the stiffness of the spring.Routh Art. 489 In the catenary the value of {{math|T}} is variable, but ratio remains valid at a local level, soRouth Art. 494
frac{ds}{dp}=1+frac{T}{E},.
The curve followed by an elastic spring can now be derived following a similar method as for the inelastic spring.Following Routh Art. 500The equations for tension of the spring are
T cos varphi = T_0,,
and
T sin varphi = lambda_0 gp,,
from which
frac{dy}{dx}=tan varphi = frac{lambda_0 gp}{T_0},,quad T=sqrt{T_0^2+lambda_0^2 g^2p^2},,
where {{mvar|p}} is the natural length of the segment from {{math|c}} to {{math|r}} and {{math|Î»0}} is the mass per unit length of the spring with no tension and {{mvar|g}} is the acceleration of gravity. Write
a = frac{T_0}{lambda_0 g}
so
frac{dy}{dx}=tan varphi = frac{p}{a},,quad T=frac{T_0}{a}sqrt{a^2+p^2},.
Then
frac{dx}{ds} = cos varphi = frac{T_0}{T}
and
frac{dy}{ds} = sin varphi = frac{lambda_0 gp}{T},,
from which
frac{dx}{dp} = frac{T_0}{T}frac{ds}{dp} = T_0left(frac{1}{T}+frac{1}{E}right)=frac{a}{sqrt{a^2+p^2}}+frac{T_0}{E}
and
frac{dy}{dp} = frac{lambda_0 gp}{T}frac{ds}{dp} = frac{T_0p}{a}left(frac{1}{T}+frac{1}{E}right)=frac{p}{sqrt{a^2+p^2}}+frac{T_0p}{Ea},.
Integrating gives the parametric equations
x=aoperatorname{arcsinh}left(frac{p}{a}right)+frac{T_0}{E}p + alpha,,
y=sqrt{a^2+p^2}+frac{T_0}{2Ea}p^2+beta,.
Again, the {{mvar|x}} and {{mvar|y}}-axes can be shifted so {{mvar|Î±}} and {{mvar|Î²}} can be taken to be 0. So
x=aoperatorname{arcsinh}left(frac{p}{a}right)+frac{T_0}{E}p,,
y=sqrt{a^2+p^2}+frac{T_0}{2Ea}p^2
are parametric equations for the curve. At the rigid limit where {{mvar|E}} is large, the shape of the curve reduces to that of a non-elastic chain.Other generalizations
Chain under a general force
With no assumptions have been made regarding the force {{math|G}} acting on the chain, the following analysis can be made.Follows Routh Art. 455First, let {{math|1=T = T(s)}} be the force of tension as a function of {{mvar|s}}. The chain is flexible so it can only exert a force parallel to itself. Since tension is defined as the force that the chain exerts on itself, {{math|T}} must be parallel to the chain. In other words,
mathbf{T} = T mathbf{u},,
where {{mvar|T}} is the magnitude of {{math|T}} and {{math|u}} is the unit tangent vector.Second, let {{math|1=G = G(s)}} be the external force per unit length acting on a small segment of a chain as a function of {{mvar|s}}. The forces acting on the segment of the chain between {{mvar|s}} and {{math|s + Î”s}} are the force of tension {{math|T(s + Î”s)}} at one end of the segment, the nearly opposite force {{math|âˆ’T(s)}} at the other end, and the external force acting on the segment which is approximately {{math|GÎ”s}}. These forces must balance so
mathbf{T}(s+Delta s)-mathbf{T}(s)+mathbf{G}Delta s approx mathbf{0},.
Divide by {{math|Î”s}} and take the limit as {{math|Î”s â†’ 0}} to obtain
frac{dmathbf{T}}{ds} + mathbf{G} = mathbf{0},.
These equations can be used as the starting point in the analysis of a flexible chain acting under any external force. In the case of the standard catenary, {{math|1=G = (0, âˆ’Î»g)}} where the chain has mass {{mvar|Î»}} per unit length and {{mvar|g}} is the acceleration of gravity.See also
- Catenary arch
- Chain fountain or self-siphoning beads
- Overhead line â€“ power lines suspended over rail or tram vehicles
- Roulette (curve) â€“ an elliptic/hyperbolic catenary
- Troposkein â€“ the shape of a spun rope
- Weighted catenary
Notes
{{Reflist}}Bibliography
- {{anchor|Lockwood}}BOOK, A Book of Curves, E.H., Lockwood, Cambridge, 1961
- BOOK, Salmon, George, Higher Plane Curves
- {{anchor|Routh}}BOOK, Routh, Edward John, Edward Routh, A Treatise on Analytical Statics,weblink 1891, University Press, Chapter X: On Strings,
- BOOK, Maurer, Edward Rose, Technical Mechanics,weblink 1914, J. Wiley & Sons, Art. 26 Catenary Cable,
- BOOK, Lamb, Sir Horace, An Elementary Course of Infinitesimal Calculus,weblink 1897, University Press, Art. 134 Transcendental Curves; Catenary, Tractrix,
- BOOK, Todhunter, Isaac, Isaac Todhunter, A Treatise on Analytical Statics,weblink 1858, Macmillan, XI Flexible Strings. Inextensible, XII Flexible Strings. Extensible,
- {{anchor|Whewell}}BOOK, Whewell, William, William Whewell, Analytical Statics,weblink 1833, J. & J.J. Deighton, 65, Chapter V: The Equilibrium of a Flexible Body,
- {{anchor|MathWorld}}{{mathworld|Catenary|Catenary}}
Further reading
- BOOK, Swetz, Frank, Learn from the Masters,weblink 1995, MAA, 978-0-88385-703-8, 128â€“9,
- BOOK, Venturoli, Giuseppe, Trans. Daniel Cresswell, Elements of the Theory of Mechanics,weblink 1822, J. Nicholson & Son, Chapter XXIII: On the Catenary,
External links
{{Commons category}}{{Wikisource1911Enc|Catenary}}- {{MacTutor|class=Curves|id=Catenary|title=Catenary}}
- {{PlanetMath|urlname=Catenary|title=Catenary}}
- Catenary curve calculator
- Catenary at The Geometry Center
- "Catenary" at Visual Dictionary of Special Plane Curves
- The Catenary - Chains, Arches, and Soap Films.
- Cable Sag Error Calculator â€“ Calculates the deviation from a straight line of a catenary curve and provides derivation of the calculator and references.
- Dynamic as well as static cetenary curve equations derived â€“ The equations governing the shape (static case) as well as dynamics (dynamic case) of a centenary is derived. Solution to the equations discussed.
- The straight line, the catenary, the brachistochrone, the circle, and Fermat Unified approach to some geodesics.
- Ira Freeman "A General Form of the Suspension Bridge Catenary" Bulletin of the AMS
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