Inphysics andgeometry, acatenary (US:/ˈkætənɛri/KAT-ən-err-ee,UK:/kəˈtiːnəri/kə-TEE-nər-ee) is thecurve that an idealized hangingchain orcable assumes under its ownweight when supported only at its ends in a uniformgravitational field.
The catenary curve has a U-like shape, superficially similar in appearance to aparabola.
The curve appears in the design of certain types ofarches and as a cross section of thecatenoid—the shape assumed by a soap film bounded by two parallel circular rings.
The catenary is also called thealysoid,chainette,[1] or, particularly in the materials sciences, an example of afunicular.[2]Rope statics describes catenaries in a classic statics problem involving a hanging rope.[3]
Mathematically, the catenary curve is thegraph of thehyperbolic cosine function. Thesurface of revolution of the catenary curve, thecatenoid, is aminimal surface, specifically aminimal surface of revolution. A hanging chain will assume a shape of least potential energy which is a catenary.[4]Galileo Galilei in 1638 discussed the catenary in the bookTwo New Sciences recognizing that it was different from aparabola. The mathematical properties of the catenary curve were studied byRobert Hooke in the 1670s, and its equation was derived byLeibniz,Huygens andJohann Bernoulli in 1691.
Catenaries and related curves are used in architecture and engineering (e.g., in the design of bridges andarches so that forces do not result in bending moments). In the offshore oil and gas industry, "catenary" refers to asteel 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 theoverhead wiring that transfers power to trains. (This often supports a 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.[5] The symmetric modes consisting of twoevanescent waves would form a catenary shape.[6][7][8]
The word "catenary" is derived from the Latin wordcatēna, which means "chain". The English word "catenary" is usually attributed toThomas Jefferson,[9][10]who wrote in a letter toThomas Paine on the construction of an arch for a bridge:
I have lately received from Italy a treatise on theequilibrium of arches, by the Abbé Mascheroni. It appears to be a very scientific work. I have not yet had time to engage in it; but I find that the conclusions of his demonstrations are, that every part of the catenary is in perfect equilibrium.[11]
It is often said[12] thatGalileo thought the curve of a hanging chain was parabolic. However, in hisTwo 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°.[13] The fact that the curve followed by a chain is not a parabola was proven byJoachim Jungius (1587–1657); this result was published posthumously in 1669.[12]
The application of the catenary to the construction of arches is attributed toRobert Hooke, whose "true mathematical and mechanical form" in the context of the rebuilding ofSt Paul's Cathedral alluded to a catenary.[14] Some much older arches approximate catenaries, an example of which is the Arch ofTaq-i Kisra inCtesiphon.[15]
In 1671, Hooke announced to theRoyal Society that he had solved the problem of the optimal shape of an arch, and in 1675 published an encrypted solution as a Latinanagram[16] in an appendix to hisDescription of Helioscopes,[17] 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[18] in his lifetime, but in 1705 his executor provided it asut 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, andJohann Bernoulli derived theequation in response to a challenge byJakob Bernoulli;[12] their solutions were published in theActa Eruditorum for June 1691.[19][20]David Gregory wrote a treatise on the catenary in 1697[12][21] in which he provided an incorrect derivation of the correct differential equation.[20]
Leonhard Euler proved in 1744 that the catenary is the curve which, when rotated about thex-axis, gives the surface of minimumsurface area (thecatenoid) for the given bounding circles.[1]Nicolas Fuss gave equations describing the equilibrium of a chain under anyforce in 1796.[22]
Catenary arches are often used in the construction ofkilns. 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.[23][24]
TheGateway Arch inSt. Louis, Missouri, United States, is sometimes said to be an (inverted) catenary, but this is incorrect.[25] It is close to a more general curve called a flattened catenary, with equationy =A cosh(Bx), which is a catenary ifAB = 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.[26][27]
![Catenary[28] arches under the roof of Gaudí's Casa Milà, Barcelona, Spain.](/mt/?noimg=&dark=on&url=http%3a%2f%2fen.wikipedia.org%2fwiki%2fFile%3aLaPedreraParabola.jpg)
![The Sheffield Winter Garden is enclosed by a series of catenary arches.[29]](/mt/?noimg=&dark=on&url=http%3a%2f%2fen.wikipedia.org%2fwiki%2fFile%3aSheffield_Winter_Garden.jpg)
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In free-hanging chains, the force exerted is uniform with respect to length of the chain, and so the chain follows the catenary curve.[30] The same is true of asimple suspension bridge or "catenary bridge," where the roadway follows the cable.[31][32]
Astressed ribbon bridge is a more sophisticated structure with the same catenary shape.[33][34]
However, in asuspension 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 aparabola, 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.[35][36]
The catenary produced by gravity provides an advantage to heavyanchor 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 theanchor 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.[37]
Cable ferries andchain boats present a special case of marine vehicles moving although moored by the two catenaries each of one or more cables (wire ropes or chains) passing through the vehicle and moved along by motorized sheaves. The catenaries can be evaluated graphically.[38]
The equation of a catenary inCartesian coordinates has the form[35]
wherecosh is thehyperbolic cosine function, and wherea is the distance of the lowest point above the x axis.[39] All catenary curves aresimilar to each other, since changing the parametera is equivalent to auniform scaling of the curve.
TheWhewell equation for the catenary is[35]where is thetangential angle ands thearc length.
Differentiating givesand eliminating gives theCesàro equation[40]where is thecurvature.
Theradius of curvature is thenwhich is the length of thenormal between the curve and thex-axis.[41]
When aparabola is rolled along a straight line, theroulette curve traced by itsfocus is a catenary.[42] Theenvelope of thedirectrix of the parabola is also a catenary.[43] Theinvolute from the vertex, that is the roulette traced by a point starting at the vertex when a line is rolled on a catenary, is thetractrix.[42]
Another roulette, formed by rolling a line on a catenary, is another line. This implies thatsquare 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 anyregular polygon except a triangle, but the catenary must have parameters corresponding to the shape and dimensions of the wheels.[44]
Over any horizontal interval, the ratio of the area under the catenary to its length equalsa, 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 thex-axis.[45]
A movingcharge in a uniformelectric field travels along a catenary (which tends to aparabola if the charge velocity is much less than thespeed of lightc).[46]
Thesurface of revolution with fixed radii at either end that has minimum surface area is a catenary
revolved about the-axis.[42]
In themathematical model the chain (or cord, cable, rope, string, etc.) is idealized by assuming that it is so thin that it can be regarded as acurve and that it is so flexible any force oftension exerted by the chain is parallel to the chain.[47] The analysis of the curve for an optimal arch is similar except that the forces of tension become forces ofcompression and everything is inverted.[48]An underlying principle is that the chain may be considered a rigid body once it has attained equilibrium.[49] 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 instatic equilibrium.
Let the path followed by the chain be givenparametrically byr = (x,y) = (x(s),y(s)) wheres representsarc length andr is theposition vector. This is thenatural parameterization and has the property that
whereu is aunit tangent vector.
Adifferential equation for the curve may be derived as follows.[50] Letc be the lowest point on the chain, called the vertex of the catenary.[51] The slopedy/dx of the curve is zero atc since it is a minimum point. Assumer is to the right ofc since the other case is implied by symmetry. The forces acting on the section of the chain fromc tor are the tension of the chain atc, the tension of the chain atr, and the weight of the chain. The tension atc is tangent to the curve atc and is therefore horizontal without any vertical component and it pulls the section to the left so it may be written(−T0, 0) whereT0 is the magnitude of the force. The tension atr is parallel to the curve atr and pulls the section to the right. The tension atr can be split into two components so it may be writtenTu = (T cosφ,T sinφ), whereT is the magnitude of the force andφ is the angle between the curve atr and thex-axis (seetangential angle). Finally, the weight of the chain is represented by(0, −ws) wherew is the weight per unit length ands is the length of the segment of chain betweenc andr.
The chain is in equilibrium so the sum of three forces is0, therefore
and
and dividing these gives
It is convenient to write
which is the length of chain whose weight is equal in magnitude to the tension atc.[52] Then
is an equation defining the curve.
The horizontal component of the tension,T cosφ =T0 is constant and the vertical component of the tension,T sinφ =ws is proportional to the length of chain betweenr and the vertex.[53]
The differential equation, given above, can be solvedto produce equations for the curve.[54]We will solve the equation using the boundary condition thatthe vertex is positioned at and.
First, invoke the formula forarc lengthto getthenseparate variablesto obtain
A reasonably straightforward approach to integrate this is to usehyperbolic substitution,which gives(where is aconstant of integration),and hence
But, sowhichintegrates as(with being the constant of integration satisfying the boundary condition).
Since the primary interest here is simply the shape of the curve,the placement of the coordinate axes are arbitrary;so make the convenient choice ofto simplify the result to
For completeness, the relation can be derived bysolving each of the and relations for, giving:sowhichcan be rewritten as
The differential equation can be solved using a different approach.[55] From
it follows that
and
Integrating gives,
and
As before, thex andy-axes can be shifted soα andβ can be taken to be 0. Then
and taking the reciprocal of both sides
Adding and subtracting the last two equations then gives the solutionand
The parametera is the minimaly coordinate of the points of the catenary. In practical applications, neither this parameter, nor the axes where the catenary has its standard form are knowna priori.
These data can be determined from the position of two given points and and the lengthL of the catenary between these points as follows:[56]
Relabel if necessary so thatP1 is to the left ofP2 and letH be the horizontal andv be the vertical distance fromP1 toP2.Translate the axes so that the vertex of the catenary lies on they-axis and its heighta is adjusted so the catenary satisfies the standard equation of the curve
and let the coordinates ofP1 andP2 be(x1,y1) and(x2,y2) respectively. The curve passes through these points, so the difference of height is
and the length of the curve fromP1 toP2 is
WhenL2 −v2 is expanded using these expressions the result is
so
This is a transcendental equation ina and must be solvednumerically. Since is strictly monotonic on,[57] there is at most one solution witha > 0 and so there is at most one position of equilibrium.
However, if both ends of the curve (P1 andP2) are at the same level (y1 =y2), it can be shown that[58]where L is the total length of the curve betweenP1 andP2 andh is the sag (vertical distance betweenP1,P2 and the vertex of the curve).
It can also be shown thatandwhere H is the horizontal distance betweenP1 andP2 which are located at the same level (H =x2 −x1).
The horizontal traction force atP1 andP2 isT0 =wa, wherew is the weight per unit length of the chain or cable.
There is a simple relationship between the tension in the cable at a point and itsx- and/ory- coordinate. Begin by combining the squares of the vector components of the tension:which (recalling that) can be rewritten asBut,as shown above, (assuming that), so we get the simple relations[59]
Consider a chain of length suspended from two points of equal height and at distance. The curve has to minimize its potential energy(wherew is the weight per unit length) and is subject to the constraint
The modifiedLagrangian is thereforewhere is theLagrange multiplier to be determined. As the independent variable does not appear in the Lagrangian, we can use theBeltrami identitywhere is an integration constant, in order to obtain a first integral
This is an ordinary first order differential equation that can be solved by the method ofseparation of variables. Its solution is the usual hyperbolic cosine where the parameters are obtained from the constraints.
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.[60]
Letw denote the weight per unit length of the chain, then the weight of the chain has magnitude
where the limits of integration arec andr. Balancing forces as in the uniform chain produces
andand therefore
Differentiation then gives
In terms ofφ and the radius of curvatureρ this becomes
A similar analysis can be done to find the curve followed by the cable supporting asuspension bridge with a horizontal roadway.[61] If the weight of the roadway per unit length isw 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 inCatenary#Model of chains and arches) fromc tor iswx wherex is the horizontal distance betweenc andr. Proceeding as before gives the differential equation
This is solved by simple integration to get
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.[62]
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,w, per unit length of the chain can be writtenT/c, wherec is constant, and the analysis for nonuniform chains can be applied.[63]
In this case the equations for tension are
Combining gives
and by differentiation
whereρ is the radius of curvature.
The solution to this is
In this case, the curve has vertical asymptotes and this limits the span toπc. Other relations are
The curve was studied 1826 byDavies Gilbert and, apparently independently, byGaspard-Gustave Coriolis in 1836.
Recently, it was shown that this type of catenary could act as a building block ofelectromagnetic metasurface and was known as "catenary of equal phase gradient".[64]
In anelastic catenary, the chain is replaced by aspring which can stretch in response to tension. The spring is assumed to stretch in accordance withHooke's law. Specifically, ifp is the natural length of a section of spring, then the length of the spring with tensionT applied has length
whereE is a constant equal tokp, wherek is thestiffness of the spring.[65] In the catenary the value ofT is variable, but ratio remains valid at a local level, so[66]The curve followed by an elastic spring can now be derived following a similar method as for the inelastic spring.[67]
The equations for tension of the spring are
and
from which
wherep is the natural length of the segment fromc tor andw0 is the weight per unit length of the spring with no tension. Writeso
Thenfrom which
Integrating gives the parametric equations
Again, thex andy-axes can be shifted soα andβ can be taken to be 0. So
are parametric equations for the curve. At the rigidlimit whereE is large, the shape of the curve reduces to that of a non-elastic chain.
With no assumptions being made regarding the forceG acting on the chain, the following analysis can be made.[68]
First, letT =T(s) be the force of tension as a function ofs. 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,T must be parallel to the chain. In other words,
whereT is the magnitude ofT andu is the unit tangent vector.
Second, letG =G(s) be the external force per unit length acting on a small segment of a chain as a function ofs. The forces acting on the segment of the chain betweens ands + Δs are the force of tensionT(s + Δs) at one end of the segment, the nearly opposite force−T(s) at the other end, and the external force acting on the segment which is approximatelyGΔs. These forces must balance so
Divide byΔs and take the limit asΔs → 0 to obtain
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,G = (0, −w) where the chain has weightw per unit length.
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