Catenary Curve in Mathematics: Types, Properties & Uses

In Mathematics, a catenary is a curve that an idealized hanging chain or cable assumes when supported only at its ends under its own weight. The common catenary curve has a U-shaped form that resembles a parabolic arch on the surface, but it is not a parabola.

Catenary, in science, a bend that depicts the state of an adaptable hanging chain or link—the name gets from the Latin catenaria ("chain"). Any openly hanging link or string accepts this shape, likewise called a chainette, assuming the body is of uniform mass per unit of length and is followed up on exclusively by gravity.

From the get-go in the seventeenth century, the German space expert Johannes Kepler applied the oval to the portrayal of planetary circles, and the Italian researcher Galileo Galilei utilized the parabola to depict shot movement without a trace of air obstruction. Roused by the extraordinary achievement of conic areas in these settings, Galileo mistakenly accepted that a hanging chain would take the state of a parabola. It was later in the seventeenth century that the Dutch mathematician Christiaan Huygens showed that the chain bend can't be given by a logarithmic condition (one including just number juggling tasks along with powers and roots); he additionally begat the term catenary. Notwithstanding Huygens, the Swiss mathematician Jakob Bernoulli and the German mathematician Gottfried Leibniz added to the total depiction of the situation of the catenary.

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The catenary is also known as the alysoid, chainette, or funicular, particularly in the materials sciences. Catenaries in a classic statics problem involving a hanging rope are defined by rope statics.

The curve can be seen in the design of some forms of arches and as a cross-section of a catenoid, a soap film bounded by two parallel circular circles that take on the form of a catenoid.

The catenary curve is the graph of the hyperbolic cosine function. The catenoid, the catenary curve's surface of revolution, is a minimal surface of revolution. A hanging chain can take the form of a catenary, which has the lowest potential energy. 

History of Catenary

• The word "catenary" comes from the Latin word "catena," which simply translates "chain."

• Galileo assumed a hanging chain's curve was parabolic. Galileo states in his Two New Sciences (1638) that a hanging cord is an approximate parabola, and he correctly notes that as the curvature gets smaller, the approximation increases and becomes almost exactly when the elevation is less than 45°.

• Joachim Jungius (1587–1657) proved that the curve followed by a chain is not a parabola in a posthumous publication in 1669.

• Robert Hooke, whose true mathematical and mechanical type in the sense of the restoration of St Paul's Cathedral referred to as a catenary, is credited with the application of the catenary to the creation of arches.

• Hooke declared to the Royal Society in 1671 that he had solved the problem of the best arch design, and in 1675 he published an encrypted solution as a Latin anagram in an appendix to his Description of Helioscopes, where he wrote that he had discovered a true mathematical and mechanical type of all kinds of Arches for Building.

• In response to a challenge from Jakob Bernoulli, Gottfried Leibniz, Christiaan Huygens, and Johann Bernoulli derived the equation in 1691, and their solutions were published in the Acta Eruditorum for June 1691.

• In 1697, David Gregory published a treatise on the catenary in which he incorrectly deduced the correct differential equation.

• The catenary is the curve that, when rotated around the x-axis, gives the surface with the smallest surface area (the catenoid) for the given bounding circles, as Euler proved in 1744.

• In 1796, Nicolas Fuss published equations that described the equilibrium of a chain under any power.

Inverted Catenary

Under its own weight, the inverted catenary is the arch's approximate optimum form. The greater the height of inverted catenary arches of equal length, the lower the horizontal force in the built-in ends and in the center of the arch. The slenderest arches are those that are subjected to the least amount of lateral force.

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Weighted Catenary

A weighted catenary is a special type of catenary curve. A rectangular frame containing the selected fragment of the curve potentially continuing to infinity is defined by the aspect ratio of a weighted catenary.

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Mathematical Equations of a Catenary

The catenary equation in the cartesian coordinate system is 

y = acosh \[\frac{x}{a}\]

y = \[\frac{a}{2}\] \[\left (e^{\frac{x}{a}}+e^{\frac{-x}{a}}  \right )\]

Where a is the changing parameter.

Cosh is the hyperbolic cosine function

The value of x is measured from the lowest point.

In 1849, William Whewell derived the catenary equation which is named after him is known as the Whewell equation. The equation of Whewell in a plane curve is an equation relating the tangential angle (φ) to arclength (s), with the tangential angle being the angle between the corner of the tangent and the x-axis and the length of the arc being the distance from a fixed point along the curve. 

Whewell equation of catenary is given as follows:

Tanφ = \[\frac{s}{a}\]

Differentiating the equation we get, 

\[\frac{d\phi}{ds}\] = \[\frac{cos^{2}\phi}{a}\]

By differentiating the Whewell equation, we will get the Cesaro equation. A plane curve's Cesaro equation is an equation that connects the curvature (κ) at a point on the curve to the arc length (s) from the curve's start to the given point. It can also be expressed as an equation relating arc length (s) to the radius of curvature (R). The Cesaro equation would be the same for two congruent curves. Cesaro equations are named after Ernesto Cesaro.

k= \[\frac{a}{s^{2}+a^{2}}\]

Geometrical Properties of Catenary and Its Relation With Other Curves

• The ratio of the area under the catenary to its length equals an over any horizontal interval, regardless of the interval chosen. 

• The catenary is the only plane curve with this property that is not a horizontal line. 

• The geometric centroid of the area under a catenary stretch is also the midpoint of the perpendicular section linking the curve's centroid and the x-axis.

• The roulette curve traced by the attention of a parabola when rolled along a straight line is called a catenary. 

• The envelope of the parabola's directrix is also a catenary. 

• The tractrix is the involute from the vertex, which is the roulette created when a line is rolled on a catenary and a point starts at the vertex.

• Another roulette is a line created by rolling a line on a catenary. 

• This means that square wheels will roll over a series of bumps in the form of an inverted catenary curve perfectly smoothly. 

• The wheels can be any regular polygon other than a triangle, but the catenary must have parameters that conform to the wheels' shape and dimensions.

Applications of Catenary

• Catenaries and associated curves are used in architecture and engineering to prevent forces from causing bending moments in bridges and arches, for example.

• A steel catenary riser suspended between a production platform and the seabed has an approximate catenary shape in the offshore oil and gas industry.

• The common catenary is the overhead wiring that passes power to trains in the rail industry.

• The hyperbolic cosine and sine functions are simple solutions to Maxwell's equations in optics and electromagnetics. A catenary shape would be formed by the symmetric modes, which are made up of two evanescent waves.

• The catenary generated by gravity gives heavy anchor rodes an advantage. Chain, wire, or both are commonly used in anchor rodes and anchor lines. Ships, oil rigs, docks, floating wind turbines, and other marine equipment that must be anchored to the seabed use anchor rodes. When the rope is loose, the catenary curve pulls on the anchor or mooring device at a lower angle than it would if it were nearly straight. This improves the anchor's performance and increases the amount of force it can withstand before dragging. Only larger ships in deeper water may rely on this effect because it requires a heavy chain to retain the catenary shape in the presence of wind.

• A moving charge in a uniform electric field moves along a catenary which tends to a parabola if the charge velocity is much less than the speed of light c.

• A catenary revolving around the x-axis is a surface of revolution with fixed radii at either end and the smallest surface area.

Catenary Software

In this section, we will learn about different types of catenary software available in the market. This catenary software is used to draw different catenary shapes and to calculate the various properties of catenary structures.

• GM Catenary is a single-line 2D catenary analysis software that can be used to analyze mooring or towing lines. The programme was created for the marine marketplace, where mooring and towing services are being offered. GM Catenary is a useful toolkit for surveyors and those responsible for anchor running and towing operations. When addressing pay-out and clearance issues, GM Catenary is an invaluable tool on tugs, anchor handling, and all spread-moored vessels, where it supports mooring analysis and provides fast, reliable, and comprehensive results of catenary line problems.

• STEVCAT is a catenary calculator developed by Deep Sea Mooring for calculating and plotting related catenary properties for pre-lay operations, rig mooring, towing, and marine operations.

• The Catenary App of Mooring Lines is a tool that measures the catenary profile and tension in a multi-segmented line. It can be used to simulate a free-hanging catenary line or a grounded catenary line.

• Through integration with EIVA NaviPac and the 3D visualization option, NaviCat software can run catenary simulations related to anchor handling tug operations. NaviCat allows the design, alteration, and display of catenaries extremely simple and intuitive by using a predefined cable, MLB, and anchor libraries.

Conclusion

The catenary is a curve that defines the form of a flexible hanging chain or cable. It gets its name from the Latin word catenaria, which means "chain". If the body is of uniform mass per unit of length and is acted upon solely by gravity, any freely hanging cable or string assumes this shape, also known as a chainette. Despite the fact that the catenary curve is not a parabola, it is important to note that it is similar to one: a catenary is a curve traced in the plane by the focus of a parabola as it rolls along a straight line. A catenoid is the surface of revolution generated when an upward-opening catenary is rotated around the horizontal axis.

Definitively, the bend in the xy-plane of such a chain suspended from equivalent statues at its closures and dropping at x = 0 to its least tallness y = is given by the situation y = (a/2) (ex/a + e−x/a). It can likewise be communicated as far as the exaggerated cosine work as y = a cosh(x/a). See the figure.

Albeit the catenary bend neglects to be depicted by a parabola, it is important to take note of that it is connected with a parabola: the bend followed in the plane by the focal point of a parabola as it rolls along a straight line is a catenary. The outer layer of transformation produced when a vertical opening catenary is rotated around the level pivot is known as a catenoid. The catenoid was found in 1744 by the Swiss mathematician Leonhard Euler and it is the main insignificant surface, other than the plane, that can be acquired as a surface of unrest.

The catenary and the connected exaggerated capacities assume parts in different applications. A rearranged hanging link gives the shape to a steady self-standing curve, for example, the Gateway Arch situated in St. Louis, Missouri. The exaggerated capacities additionally emerge in the depiction of waveforms, temperature circulations, and the movement of falling bodies subject to air obstruction relative to the square of the speed of the body.