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In
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
and
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a catenary (, ) is the curve that an idealized hanging
chain A chain is a serial assembly of connected pieces, called links, typically made of metal, with an overall character similar to that of a rope in that it is flexible and curved in compression but linear, rigid, and load-bearing in tension. ...
or cable assumes under its own
weight In science and engineering, the weight of an object is the force acting on the object due to gravity. Some standard textbooks define weight as a vector quantity, the gravitational force acting on the object. Others define weight as a scalar qua ...
when supported only at its ends in a uniform gravitational field. The catenary curve has a U-like shape, superficially similar in appearance to a
parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descri ...
, which it is not. The curve appears in the design of certain types of
arch An arch is a vertical curved structure that spans an elevated space and may or may not support the weight above it, or in case of a horizontal arch like an arch dam, the hydrostatic pressure against it. Arches may be synonymous with vau ...
es and as a cross section of the catenoid—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. Rope statics describes catenaries in a classic statics problem involving a hanging rope. Mathematically, the catenary curve is the graph of the
hyperbolic cosine In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the ...
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.
Galileo Galilei Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642) was an Italian astronomer, physicist and engineer, sometimes described as a polymath. Commonly referred to as Galileo, his name was pronounced (, ). He ...
in 1638 discussed the catenary in the book '' Two New Sciences'' recognizing that it was different from a
parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descri ...
. The mathematical properties of the catenary curve were studied by
Robert Hooke Robert Hooke FRS (; 18 July 16353 March 1703) was an English polymath active as a scientist, natural philosopher and architect, who is credited to be one of two scientists to discover microorganisms in 1665 using a compound microscope that ...
in the 1670s, and its equation was derived by
Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ma ...
, Huygens and Johann Bernoulli in 1691. Catenaries and related curves are used in architecture and engineering (e.g., 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 that transfers power to trains. (This often supports a lighter contact wire, in which case 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. The symmetric modes consisting of two evanescent waves would form a catenary shape.


History

The word "catenary" is derived from the Latin word ''catēna'', which means "
chain A chain is a serial assembly of connected pieces, called links, typically made of metal, with an overall character similar to that of a rope in that it is flexible and curved in compression but linear, rigid, and load-bearing in tension. ...
". The English word "catenary" is usually attributed to
Thomas Jefferson Thomas Jefferson (April 13, 1743 – July 4, 1826) was an American statesman, diplomat, lawyer, architect, philosopher, and Founding Fathers of the United States, Founding Father who served as the third president of the United States from 18 ...
, who wrote in a letter to
Thomas Paine Thomas Paine (born Thomas Pain; – In the contemporary record as noted by Conway, Paine's birth date is given as January 29, 1736–37. Common practice was to use a dash or a slash to separate the old-style year from the new-style year. In th ...
on the construction of an arch for a bridge: It is often said that
Galileo Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642) was an Italian astronomer, physicist and engineer, sometimes described as a polymath. Commonly referred to as Galileo, his name was pronounced (, ). He was ...
thought the curve of a hanging chain was parabolic. However, in his '' Two New Sciences'' (1638), Galileo wrote that a hanging cord is only an approximate parabola, correctly observing that this approximation improves in accuracy as the curvature gets smaller and is almost exact when the elevation is less than 45°. The fact that the curve followed by a chain is not a parabola was proven by
Joachim Jungius Joachim Jungius (born Joachim Junge; 22 October 1587 – 23 September 1657) was a German mathematician, logician and philosopher of science. Life Jungius was a native of Lübeck. He studied metaphysics at the Universities of Rostock and Giess ...
(1587–1657); this result was published posthumously in 1669. Lockwood p. 124 The application of the catenary to the construction of arches is attributed to
Robert Hooke Robert Hooke FRS (; 18 July 16353 March 1703) was an English polymath active as a scientist, natural philosopher and architect, who is credited to be one of two scientists to discover microorganisms in 1665 using a compound microscope that ...
, whose "true mathematical and mechanical form" in the context of the rebuilding of
St Paul's Cathedral St Paul's Cathedral is an Anglicanism, Anglican cathedral in London and is the seat of the Bishop of London. The cathedral serves as the mother church of the Diocese of London. It is on Ludgate Hill at the highest point of the City of London ...
alluded to a catenary. Some much older arches approximate catenaries, an example of which is the Arch of Taq-i Kisra in
Ctesiphon Ctesiphon ( ; Middle Persian: 𐭲𐭩𐭮𐭯𐭥𐭭 ''tyspwn'' or ''tysfwn''; fa, تیسفون; grc-gre, Κτησιφῶν, ; syr, ܩܛܝܣܦܘܢThomas A. Carlson et al., “Ctesiphon — ܩܛܝܣܦܘܢ ” in The Syriac Gazetteer last modi ...
. In 1671, Hooke announced to the
Royal Society The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, re ...
that he had solved the problem of the optimal shape of an arch, and in 1675 published an encrypted solution as a Latin
anagram An anagram is a word or phrase formed by rearranging the letters of a different word or phrase, typically using all the original letters exactly once. For example, the word ''anagram'' itself can be rearranged into ''nag a ram'', also the word ...
in an appendix to his ''Description of Helioscopes,'' 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 anagram 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 Christiaan Huygens, Lord of Zeelhem, ( , , ; also spelled Huyghens; la, Hugenius; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, astronomer, and inventor, who is regarded as one of the greatest scientists o ...
, and Johann Bernoulli derived the
equation In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in F ...
in response to a challenge by Jakob Bernoulli; their solutions were published in the '' Acta Eruditorum'' for June 1691. David Gregory wrote a treatise on the catenary in 1697 in which he provided an incorrect derivation of the correct differential equation.
Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ...
proved in 1744 that the catenary is the curve which, when rotated about the -axis, gives the surface of minimum
surface area The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of ...
(the catenoid) for the given bounding circles.
Nicolas Fuss Nicolas Fuss (29 January 1755 – 4 January 1826), also known as Nikolai Fuss, was a Swiss mathematician, living most of his life in Imperial Russia. Biography Fuss was born in Basel, Switzerland. He moved to Saint Petersburg to serve as a mat ...
gave equations describing the equilibrium of a chain under any
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a ...
in 1796.


Inverted catenary arch

Catenary arches A catenary arch is a type of architectural arch that follows an inverted catenary curve. The catenary curve has been employed in buildings since ancient times. It forms an underlying principle to the overall system of vaults and buttresses in ...
are often used in the construction of
kiln A kiln is a thermally insulated chamber, a type of oven, that produces temperatures sufficient to complete some process, such as hardening, drying, or chemical changes. Kilns have been used for millennia to turn objects made from clay int ...
s. 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. The Gateway Arch in St. Louis, Missouri, United States is sometimes said to be an (inverted) catenary, but this is incorrect. It is close to a more general curve called a flattened catenary, with equation , which is a catenary if . 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 A National Historic Landmark (NHL) is a building, district, object, site, or structure that is officially recognized by the United States government for its outstanding historical significance. Only some 2,500 (~3%) of over 90,000 places liste ...
nomination for the arch, it is a "
weighted catenary A weighted catenary is a catenary curve, but of a special form. A "regular" catenary has the equation :y = a \, \cosh \left(\frac\right) = \frac for a given value of ''a''. A ''weighted catenary'' has the equation :y = b \, \cosh \left(\frac\ ...
" instead. Its shape corresponds to the shape that a weighted chain, having lighter links in the middle, would form. and   The logo for
McDonald's McDonald's Corporation is an American multinational fast food chain, founded in 1940 as a restaurant operated by Richard and Maurice McDonald, in San Bernardino, California, United States. They rechristened their business as a hambur ...
, the Golden Arches, while intended to be two joined parabolas, is also based on the catenary. File:LaPedreraParabola.jpg, Catenary arches under the roof of Gaudí's '' Casa Milà'',
Barcelona Barcelona ( , , ) is a city on the coast of northeastern Spain. It is the capital and largest city of the autonomous community of Catalonia, as well as the second most populous municipality of Spain. With a population of 1.6 million within c ...
, Spain. File:Sheffield Winter Garden.jpg, The Sheffield Winter Garden is enclosed by a series of
catenary arches A catenary arch is a type of architectural arch that follows an inverted catenary curve. The catenary curve has been employed in buildings since ancient times. It forms an underlying principle to the overall system of vaults and buttresses in ...
. File:Gateway Arch.jpg, The Gateway Arch ( St. Louis, Missouri) is a flattened catenary. File:CatenaryKilnConstruction06025.JPG, Catenary arch kiln under construction over temporary form


Catenary bridges

In free-hanging chains, the force exerted is uniform with respect to length of the chain, and so the chain follows the catenary curve. The same is true of a simple suspension bridge or "catenary bridge," where the roadway follows the cable. A stressed ribbon bridge is a more sophisticated structure with the same catenary shape. However, in a
suspension bridge A suspension bridge is a type of bridge in which the deck is hung below suspension cables on vertical suspenders. The first modern examples of this type of bridge were built in the early 1800s. Simple suspension bridges, which lack vertical ...
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 In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descri ...
, 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. 122


Anchoring of marine objects

The catenary produced by gravity provides an advantage to heavy 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 rope is slack, the catenary curve presents a lower angle of pull on the
anchor An anchor is a device, normally made of metal , used to secure a vessel to the bed of a body of water to prevent the craft from drifting due to wind or current. The word derives from Latin ''ancora'', which itself comes from the Greek � ...
or mooring device than would be the case if it were nearly straight. This enhances the performance of the anchor and raises the level of force it will resist before dragging. To maintain the catenary shape in the presence of wind, a heavy chain is needed, so that only larger ships in deeper water can rely on this effect. Smaller boats also rely on catenary to maintain maximum holding power.


Mathematical description


Equation

The equation of a catenary in
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
has the form y = a \cosh \left(\frac\right) = \frac\left(e^\frac + e^\right), where is the hyperbolic cosine function, and where is measured from the lowest point. All catenary curves are similar to each other, since changing the parameter is equivalent to a
uniform scaling In affine geometry, uniform scaling (or isotropic scaling) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a ''scale factor'' that is the same in all directions. The result of uniform scaling is similar ...
of the curve. The
Whewell equation The Whewell equation of a plane curve is an equation that relates the tangential angle () with arclength (), where the tangential angle is the angle between the tangent to the curve and the -axis, and the arc length is the distance along the cu ...
for the catenary is \tan \varphi = \frac, where \varphi is the
tangential angle In geometry, the tangential angle of a curve in the Cartesian plane, at a specific point, is the angle between the tangent line to the curve at the given point and the -axis. (Some authors define the angle as the deviation from the direction of ...
and the
arc length ARC may refer to: Business * Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s * Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services * ...
. Differentiating gives \frac = \frac, and eliminating \varphi gives the Cesàro equation \kappa=\frac, where \kappa is the
curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the can ...
. The radius of curvature is then \rho = a \sec^2 \varphi, which is the length of the normal between the curve and the -axis.


Relation to other curves

When a
parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descri ...
is rolled along a straight line, the
roulette Roulette is a casino game named after the French word meaning ''little wheel'' which was likely developed from the Italian game Biribi''.'' In the game, a player may choose to place a bet on a single number, various groupings of numbers, the ...
curve traced by its focus is a catenary. The
envelope An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter or card. Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a ...
of the directrix of the parabola is also a catenary. The
involute In mathematics, an involute (also known as an evolvent) is a particular type of curve that is dependent on another shape or curve. An involute of a curve is the locus of a point on a piece of taut string as the string is either unwrapped from o ...
from the vertex, that is the roulette traced by a point starting at the vertex when a line is rolled on a catenary, is the
tractrix In geometry, a tractrix (; plural: tractrices) is the curve along which an object moves, under the influence of friction, when pulled on a horizontal plane by a line segment attached to a pulling point (the ''tractor'') that moves at a right ...
. Another roulette, formed by rolling a line on a catenary, is another line. This implies that
square wheel A square wheel is a wheel that, instead of being circular, has the shape of a square. While literal square wheels exist, a more common use is as an idiom meaning feeling bad and naive (see reinventing the wheel). A square wheel can roll smoot ...
s 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.


Geometrical properties

Over any horizontal interval, the ratio of the area under the catenary to its length equals , 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 -axis.


Science

A moving charge in a uniform
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field ...
travels along a catenary (which tends to a
parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descri ...
if the charge velocity is much less than the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit fo ...
). The surface of revolution with fixed radii at either end that has minimum surface area is a catenary revolved about the -axis.


Analysis


Model of chains and arches

In the
mathematical model A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, ...
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. 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. An underlying principle is that the chain may be considered a rigid body once it has attained equilibrium. 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 In classical mechanics, a particle is in mechanical equilibrium if the net force on that particle is zero. By extension, a physical system made up of many parts is in mechanical equilibrium if the net force on each of its individual parts is z ...
. Let the path followed by the chain be given parametrically by where represents
arc length ARC may refer to: Business * Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s * Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services * ...
and is the
position vector In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a point ''P'' in space in relation to an arbitrary reference origin ''O''. Usually denoted x, r, or ...
. This is the natural parameterization and has the property that \frac=\mathbf where is a
unit tangent vector Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (al ...
. A
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
for the curve may be derived as follows. Let be the lowest point on the chain, called the vertex of the catenary. The slope of the curve is zero at since it is a minimum point. Assume is to the right of since the other case is implied by symmetry. The forces acting on the section of the chain from to are the tension of the chain at , the tension of the chain at , and the weight of the chain. The tension at is tangent to the curve at and is therefore horizontal without any vertical component and it pulls the section to the left so it may be written where is the magnitude of the force. The tension at is parallel to the curve at and pulls the section to the right. The tension at can be split into two components so it may be written , where is the magnitude of the force and is the angle between the curve at and the -axis (see
tangential angle In geometry, the tangential angle of a curve in the Cartesian plane, at a specific point, is the angle between the tangent line to the curve at the given point and the -axis. (Some authors define the angle as the deviation from the direction of ...
). Finally, the weight of the chain is represented by where is the mass per unit length, is the gravitational field strength and is the length of the segment of chain between and . The chain is in equilibrium so the sum of three forces is , therefore T \cos \varphi = T_0 and T \sin \varphi = \lambda gs\,, and dividing these gives \frac=\tan \varphi = \frac\,. It is convenient to write a = \frac which is the length of chain whose weight is equal in magnitude to the tension at . Then \frac=\frac is an equation defining the curve. The horizontal component of the tension, is constant and the vertical component of the tension, is proportional to the length of chain between and the vertex.


Derivation of equations for the curve

The differential equation given above can be solved to produce equations for the curve. From \frac = \frac\,, the formula for
arc length ARC may refer to: Business * Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s * Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services * ...
gives \frac = \sqrt = \frac\,. Then \frac = \frac = \frac and \frac = \frac = \frac\,. The second of these equations can be integrated to give y = \sqrt + \beta and by shifting the position of the -axis, can be taken to be 0. Then y = \sqrt\,,\quad y^2=a^2+s^2\,. The -axis thus chosen is called the ''directrix'' of the catenary. It follows that the magnitude of the tension at a point is , which is proportional to the distance between the point and the directrix. Routh Art. 443 p. 318 This tension may also be expressed as . The integral of the expression for can be found using standard techniques, giving x = a\operatorname\left(\frac\right) + \alpha\,. and, again, by shifting the position of the -axis, can be taken to be 0. Then x = a\operatorname\left(\frac\right)\,,\quad s=a \sinh\left(\frac\right)\,. The -axis thus chosen passes through the vertex and is called the axis of the catenary. These results can be used to eliminate giving y = a \cosh\left(\frac\right)\,.


Alternative derivation

The differential equation can be solved using a different approach. From s = a \tan \varphi it follows that \frac = \frac\frac=\cos \varphi \cdot a \sec^2 \varphi= a \sec \varphi and \frac = \frac\frac=\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 and -axes can be shifted so and can be taken to be 0. Then \sec \varphi + \tan \varphi = e^\frac\,, and taking the reciprocal of both sides \sec \varphi - \tan \varphi = e^\,. Adding and subtracting the last two equations then gives the solution y = a \sec \varphi = a \cosh\left(\frac\right)\,, and s = a \tan \varphi = a \sinh\left(\frac\right)\,.


Determining parameters

In general the parameter is the position of the axis. The equation can be determined in this case as follows: Relabel if necessary so that is to the left of and let be the horizontal and be the vertical distance from to .
Translate Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transl ...
the axes so that the vertex of the catenary lies on the -axis and its height is adjusted so the catenary satisfies the standard equation of the curve y = a \cosh\left(\frac\right) and let the coordinates of and be and respectively. The curve passes through these points, so the difference of height is v = a \cosh\left(\frac\right) - a \cosh\left(\frac\right)\,. and the length of the curve from to is s = a \sinh\left(\frac\right) - a \sinh\left(\frac\right)\,. When is expanded using these expressions the result is s^2-v^2=2a^2\left(\cosh\left(\frac\right)-1\right)=4a^2\sinh^2\left(\frac\right)\,, so \sqrt=2a\sinh\left(\frac\right)\,. This is a transcendental equation in and must be solved
numerically Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods t ...
. It can be shown with the methods of calculus that there is at most one solution with and so there is at most one position of equilibrium. However, if both ends of the curve ( and ) are at the same level (), it can be shown that a = \frac \, where L is the total length of the curve between and and is the sag (vertical distance between , and the vertex of the curve). It can also be shown that L = 2a \sinh \frac \, and H = 2a \operatorname \frac \, where H is the horizontal distance between and which are located at the same level (). The horizontal traction force at and is , where is the mass per unit length of the chain or cable.


Variational formulation

Consider a chain of length L suspended from two points of equal height and at distance D. The curve has to minimize its potential energy U = \int_0^D g\rho y\sqrt dx and is subject to the constraint \int_0^D \sqrt dx = L\,. The modified Lagrangian is therefore \mathcal = (g\rho y - \lambda )\sqrt where \lambda is the Lagrange multiplier to be determined. As the independent variable x does not appear in the Lagrangian, we can use the Beltrami identity \mathcal-y' \frac = C where C is an integration constant, in order to obtain a first integral \frac = -C This is an ordinary first order differential equation that can be solved by the method of
separation of variables In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs ...
. Its solution is the usual hyperbolic cosine where the parameters are obtained from the constraints.


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. Let denote the weight per unit length of the chain, then the weight of the chain has magnitude \int_\mathbf^\mathbf w\, ds\,, where the limits of integration are and . Balancing forces as in the uniform chain produces T \cos \varphi = T_0 and T \sin \varphi = \int_\mathbf^\mathbf w\, ds\,, and therefore \frac=\tan \varphi = \frac \int_\mathbf^\mathbf w\, ds\,. Differentiation then gives w=T_0 \frac\frac = \frac\,. In terms of and the radius of curvature this becomes w= \frac\,.


Suspension bridge curve

A similar analysis can be done to find the curve followed by the cable supporting a
suspension bridge A suspension bridge is a type of bridge in which the deck is hung below suspension cables on vertical suspenders. The first modern examples of this type of bridge were built in the early 1800s. Simple suspension bridges, which lack vertical ...
with a horizontal roadway. If the weight of the roadway per unit length is and the weight of the cable and the wire supporting the bridge is negligible in comparison, then the weight on the cable (see the figure in Catenary#Model of chains and arches) from to is where is the horizontal distance between and . Proceeding as before gives the differential equation \frac=\tan \varphi = \fracx\,. This is solved by simple integration to get y=\fracx^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.


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, , per unit length of the chain can be written , where is constant, and the analysis for nonuniform chains can be applied. In this case the equations for tension are \begin T \cos \varphi &= T_0\,,\\ T \sin \varphi &= \frac\int T\, ds\,. \end Combining gives c \tan \varphi = \int \sec \varphi\, ds and by differentiation c = \rho \cos \varphi where is the radius of curvature. The solution to this is y = c \ln\left(\sec\left(\frac\right)\right)\,. In this case, the curve has vertical asymptotes and this limits the span to . Other relations are x = c\varphi\,,\quad s = \ln\left(\tan\left(\frac\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".


Elastic catenary

In an
elastic Elastic is a word often used to describe or identify certain types of elastomer, elastic used in garments or stretchable fabrics. Elastic may also refer to: Alternative name * Rubber band, ring-shaped band of rubber used to hold objects togethe ...
catenary, the chain is replaced by a
spring Spring(s) may refer to: Common uses * Spring (season), a season of the year * Spring (device), a mechanical device that stores energy * Spring (hydrology), a natural source of water * Spring (mathematics), a geometric surface in the shape of a h ...
which can stretch in response to tension. The spring is assumed to stretch in accordance with
Hooke's Law In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of t ...
. Specifically, if is the natural length of a section of spring, then the length of the spring with tension applied has length s=\left(1+\frac\right)p\,, where is a constant equal to , where is the
stiffness Stiffness is the extent to which an object resists deformation in response to an applied force. The complementary concept is flexibility or pliability: the more flexible an object is, the less stiff it is. Calculations The stiffness, k, of a ...
of the spring. In the catenary the value of is variable, but ratio remains valid at a local level, so \frac=1+\frac\,. The curve followed by an elastic spring can now be derived following a similar method as for the inelastic spring. The equations for tension of the spring are T \cos \varphi = T_0\,, and T \sin \varphi = \lambda_0 gp\,, from which \frac=\tan \varphi = \frac\,,\quad T=\sqrt\,, where is the natural length of the segment from to and is the mass per unit length of the spring with no tension and is the gravitational field strength. Write a = \frac so \frac=\tan \varphi = \frac \quad\text\quad T=\frac\sqrt\,. Then \begin \frac &= \cos \varphi = \frac \\ pt\frac &= \sin \varphi = \frac\,, \end from which \begin \frac &= \frac\frac &&= T_0\left(\frac+\frac\right) &&= \frac+\frac \\ pt\frac &= \frac\frac &&= \frac\left(\frac+\frac\right) &&= \frac+\frac\,. \end Integrating gives the parametric equations \begin x&=a\operatorname\left(\frac\right)+\fracp + \alpha\,, \\ pty&=\sqrt+\fracp^2+\beta\,. \end Again, the and -axes can be shifted so and can be taken to be 0. So \begin x&=a\operatorname\left(\frac\right)+\fracp\,, \\ pty&=\sqrt+\fracp^2 \end are parametric equations for the curve. At the rigid
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
where 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 being made regarding the force acting on the chain, the following analysis can be made.Follows Routh Art. 455 First, let be the force of tension as a function of . 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, must be parallel to the chain. In other words, \mathbf = T \mathbf\,, where is the magnitude of and is the unit tangent vector. Second, let be the external force per unit length acting on a small segment of a chain as a function of . The forces acting on the segment of the chain between and are the force of tension at one end of the segment, the nearly opposite force at the other end, and the external force acting on the segment which is approximately . These forces must balance so \mathbf(s+\Delta s)-\mathbf(s)+\mathbf\Delta s \approx \mathbf\,. Divide by and take the limit as to obtain \frac + \mathbf = \mathbf\,. 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, where the chain has mass per unit length and is the gravitational field strength.


See also

* Catenary arch *
Chain fountain The chain fountain phenomenon, also known as the self-siphoning beads, or Newton's beads is a physical phenomenon observed with a chain placed inside a jar, when one end of the chain is pulled from the jar and is allowed to fall under the influen ...
or self-siphoning beads *
Overhead catenary An overhead line or overhead wire is an electrical cable that is used to transmit electrical energy to electric locomotives, trolleybuses or trams. It is known variously as: * Overhead catenary * Overhead contact system (OCS) * Overhead equipm ...
– power lines suspended over rail or tram vehicles * Roulette (curve) – an elliptic/hyperbolic catenary * Troposkein – the shape of a spun rope *
Weighted catenary A weighted catenary is a catenary curve, but of a special form. A "regular" catenary has the equation :y = a \, \cosh \left(\frac\right) = \frac for a given value of ''a''. A ''weighted catenary'' has the equation :y = b \, \cosh \left(\frac\ ...


Notes


Bibliography

* * * * * * * *


Further reading

* *


External links

* *
Catenary curve calculator


at The Geometry Center
"Catenary" at Visual Dictionary of Special Plane Curves

The Catenary - Chains, Arches, and Soap Films.


– 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
– 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''
{{Mathematics and art Roulettes (curve) Differential equations Exponentials Analytic geometry