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For the railroad term see Overhead lines
For its use in ring theory, see Catenary ring. Overhead lines or overhead wires are used to transmit Electrical energy to Trams Trolleybuses or Trains at a distance from the In Mathematics, a Commutative ring R is catenary if for any pair of Prime ideals p, q, any two strictly
Catenaries for different values of the parameter 'a'
Catenaries for different values of the parameter 'a'

In physics, the catenary is the shape of a hanging flexible chain or cable when supported at its ends and acted upon by a uniform gravitational force (its own weight). Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. The shape ( OE sceap Eng created thing) of an object located in some space refers to the part of space occupied by the object as determined A chain is a series of connected links. This article is about the literal physical chain A cable is one or more Wires or Optical fibers bound together typically in a common protective jacket or sheath Gravitation is a natural Phenomenon by which objects with Mass attract one another The chain is steepest near the points of suspension because this part of the chain has the most weight pulling down on it. Toward the bottom, the slope of the chain decreases because the chain is supporting less weight. Slope is used to describe the steepness incline gradient or grade of a straight line.

Contents

History

The word catenary is derived from the Latin word catena, which means "chain". A chain is a series of connected links. This article is about the literal physical chain The curve is also called the "alysoid", "funicular", and "chainette". Galileo claimed that the curve of a chain hanging under gravity would be a parabola, but this was disproved by Jungius in a work published in 1669. Galileo Galilei (15 February 1564 &ndash 8 January 1642 was a Tuscan ( Italian) Physicist, Mathematician, Astronomer, and Philosopher In Mathematics, the parabola (pəˈræbələ from the Greek παραβολή) is a Conic section, the intersection of a right circular [1]

In 1691, Leibniz, Christiaan Huygens, and Johann Bernoulli derived the equation in response to a challenge by Jakob Bernoulli. Christiaan Huygens (ˈhaɪgənz in English ˈhœyɣəns in Dutch) ( April 14, 1629 &ndash July 8, 1695) was a Dutch Johann Bernoulli ( Basel, 27 July 1667 - 1 January 1748 was a Swiss Mathematician. An equation is a mathematical statement, in symbols, that two things are exactly the same (or equivalent For other family members named Jacob see Bernoulli family. Jacob Bernoulli (also known as James or Jacques) ( Basel Huygens first used the term 'catenaria' in a letter to Leibniz in 1690, and David Gregory wrote a treatise on the catenary in 1690. David Gregory may refer to David Gregory (mathematician, Scottish mathematician David Gregory (journalist, American journalist at However Thomas Jefferson is usually credited with the English word 'catenary'[2]. Thomas Jefferson (April 13 1743 – July 4 1826 was the third President of the United States (1801–1809 the principal author of the Declaration of Independence

The application of the catenary to the construction of arches is ancient, as described below; the modern rediscovery and statement is due to Robert Hooke, who discovered it in the context of the rebuilding of St Paul's Cathedral[3], possibly having seen Huygen's work on the catenary. Robert Hooke, FRS (18 July 1635 – 3 March 1703 was an English Natural philosopher and Polymath who played an important role in the St Paul's Cathedral, is the Anglican Cathedral on Ludgate Hill, in the City of London, and the seat of the Bishop of London. 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 anagram[4] in an appendix to his Description of Helioscopes,[5] 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 of this anagram[6] in his lifetime, but in 1705 his executor provided it as:

“Ut pendet continuum flexile, sic stabit contiguum rigidum inversum. The Royal Society of London for the Improvement of Natural Knowledge, known simply as The Royal Society, is a Learned society for science that was founded in 1660 Year 1705 ( MDCCV) was a Common year starting on Thursday (link will display the full calendar of the Gregorian calendar (or a ”

meaning

“As hangs a flexible cable, so inverted, stand the touching pieces of an arch. ”

Mathematical Description

Derivation

To derive the equation for the shape of a catenary in a uniform gravitational field \vec{g}, we begin with the condition of static equilibrium for a link at position s along the catenary; the sum of all forces must be 0. A gravitational field is a model used within Physics to explain how gravity exists in the universe

\vec{0}=F_0+\int_0^s ds'\lambda(s')\hat{g}+\vec{\tau}(s),

where F0 is the anchoring force holding up the catenary at s = 0, λ(s) is the mass per unit length at position s along the catenary, and \vec{\tau}(s) is the tension at s. Taking the derivative with respect to s and assuming a constant λ(s) = λ0 yields

\vec{0}=\lambda_0\vec{g}+\frac{d\vec{\tau}(s)}{ds}. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change

Breaking this into its constituent x and y components and assuming \vec{g} points in the − y direction yields

\frac{d(\tau \cos{\theta})}{ds}=0,\qquad\quad\qquad(1)
\frac{d(\tau \sin{\theta})}{ds}=\lambda_0 g,\qquad\qquad(2)

where θ is the angle from the horizontal x axis.

From equation (1) we see that τcosθ = c, where c is a constant, implies that

\tau=\frac{c}{\cos{\theta}}.

Substituting into equation (2),

\frac{d(\tan{\theta})}{ds}=\frac{\lambda_0 g}{c}.\qquad\qquad(3)

Now substituting the arc length

ds=\sqrt{1+\left(\frac{dy}{dx}\right)^2}dx

into equation (3) yields the differential equation

\frac{dy'}{dx}=\frac{\lambda_0 g}{c}\sqrt{1+(y')^2},\qquad\qquad(4)

where y' = dy / dx. Determining the length of an irregular arc segment — also called Rectification of a Curve — was historically difficult A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the The solution to equation (4) is

y=\frac{c}{\lambda_0 g}\cosh{\left(\frac{\lambda_0 g}{c} x+\alpha\right)}+\beta,

where α and β are constants to be determined, along with c, by the boundary conditions of the problem. In Mathematics, in the field of Differential equations a boundary value problem is a Differential equation together with a set of additional restraints

General Equation

The intrinsic equation of the shape of the catenary with both ends anchored at equal height is given by the hyperbolic cosine function or its exponential equivalent

y = a \cdot \cosh \left ({x \over a} \right ) = {a \over 2} \cdot \left (e^{x/a} + e^{-x/a} \right ),

in which

a =\frac{T_o}{\lambda}.

where To is the horizontal component of the tension (a constant) and λ is the weight per length unit. In Mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular functions

If you roll a parabola along a straight line, its focus traces out a catenary (see roulette). In Geometry, the foci (singular focus) are a pair of special points used in describing Conic sections The four types of conic sections are the Circle In the Differential geometry of curves, a roulette is a kind of Curve, generalizing Cycloids Epicycloids Hypocycloids and (The curve traced by one point of a wheel (circle) as it makes one rotation rolling along a horizontal line is not an inverted catenary but a cycloid. ) Finally, as proved by Euler in 1744, the catenary is also the curve which, when rotated about the x axis, gives the surface of minimum surface area (the catenoid) for the given bounding circle. Surface area is the measure of how much exposed Area an object has A catenoid is a three- Dimensional Shape made by rotating a Catenary Curve around the x axis Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the

Square wheels can roll perfectly smoothly if the road has evenly spaced bumps in the shape of a series of inverted catenary curves. A literal square wheel is a Wheel that instead of being circular, has the shape of a square. The wheels can be any regular polygon save for a triangle, but one must use the correct catenary, corresponding correctly to the shape and dimensions of the wheels [7].

A charge in a uniform electric field moves along a catenary (which tends to a parabola if the charge velocity is much less than the speed of light c). In Physics, the space surrounding an Electric charge or in the presence of a time-varying Magnetic field has a property called an electric field (that can In Mathematics, the parabola (pəˈræbələ from the Greek παραβολή) is a Conic section, the intersection of a right circular

Suspension bridges

Ponte Hercilio Luz, Florianópolis, Brazil. Suspension bridges follow a parabolic, not catenary, curve.
Ponte Hercilio Luz, Florianópolis, Brazil. Florianópolis ( Nicknamed Floripa) is the capital city of Santa Catarina State in southern Brazil. |utc_offset = -2 to -4 |time_zone_DST = BRST |utc_offset_DST = -2 to -5 |cctld Suspension bridges follow a parabolic, not catenary, curve.

Free-hanging chains follow the curve of the hyperbolic function above, but suspension bridge chains or cables, which are tied to the bridge deck at uniform intervals, follow a parabolic curve, much as Galileo originally claimed (derivation). This article is concerned with a particular type of suspension bridge the suspended-deck type

When suspension bridges are constructed, the suspension cables initially sag as the catenaric function, before being tied to the deck below, and then gradually assume a parabolic curve as additional connecting cables are tied to connect the main suspension cables with the bridge deck below.

The inverted catenary arch

The catenary is the ideal curve for an arch which supports only its own weight. An arch is a structure that spans a space while supporting weight (e When the centerline of an arch is made to follow the curve of an up-side-down (ie. inverted) catenary, the arch endures almost pure compression, in which no significant bending moment occurs inside the material. Physical compression is the result of the subjection of a material to Compressive stress, resulting in reduction of Volume. This article is about structural behavior For other meanings see Bending (disambiguation. If the arch is made of individual elements (eg. , stones) whose contacting surfaces are perpendicular to the curve of the arch, no significant shear forces are present at these contacting surfaces. In Geometry, two lines or planes (or a line and a plane are considered perpendicular (or orthogonal) to each other if they form congruent A shear stress, denoted \tau\ ( Tau) is defined as a stress which is applied Parallel or tangential to a face of a material (Shear stress is still present inside each stone, as it resists the compressive force along the shear sliding plane. ) The thrust (including the weight) of the arch at its two ends is tangent to its centerline. For the tangent function see Trigonometric functions. For other uses see Tangent (disambiguation.

The throne room of the Taq-i Kisra in 1824.
The throne room of the Taq-i Kisra in 1824.

In Antiquity, the curvature of the inverted catenary was intuitively discovered and found to lead to stable arches and vaults. A spectacular example remains in the Taq-i Kisra in Ctesiphon, which was once a great city of Mesopotamia. The Taq-i Kisra ( Persian طاق كسرى, meaning Iwan of Khosrau) is a Persian monument in Al-Mada'in which is the only visible For the Spanish saint see Ctesiphon of Vergium. Ctesiphon (قطسيفون تیسفون was one of the great cities of the Persian Empire Mesopotamia (from the Greek meaning "land between the rivers" is an area geographically located between the Tigris and Euphrates rivers largely corresponding In ancient Greek and Roman cultures, the less efficient curvature of the circle was more commonly used in arches and vaults. The efficient curvature of inverted catenary was perhaps forgotten in Europe from the fall of Rome to the Middle-Ages and the Renaissance, where it was almost never used, although the pointed arch was perhaps a fortuitous approximation of it. An arch is a structure that spans a space while supporting weight (e

Catenary arches under the roof of Gaudí's Casa Milà, Barcelona, Spain
Catenary arches under the roof of Gaudí's Casa Milà, Barcelona, Spain

The Catalan architect Antoni Gaudí made extensive use of catenary shapes in most of his work. Casa Milà, better known as La Pedrera ( Catalan for 'The Quarry' is a building designed by the Catalan Architect, Antoni Gaudí, Catalonia (Cataluña Catalunya Aranese: Catalonha) is an Autonomous Community in the northeast part of Spain. An architect is a licensed individual who leads a design team in the Planning and Design of buildings and participates in oversight of Building Construction In order to find the best curvature for the arches and ribs that he desired to use in the crypt of the Church of Colònia Güell, Gaudí constructed inverted scale models made of numerous threads under tension to represent stones under compression. The Church of Colònia Güell, Antoni Gaudí 's great unfinished work was built as a place of worship for the people in a manufactured suburb in Santa Coloma This technique worked well to solve angled columns, arches, and single-curvature vaults, but could not be used to solve the more complex, double-curvature vaults that he intended to use in the nave of the church of the Sagrada Familia. The Temple Expiatori de la Sagrada Família (official Catalan nameTemplo Expiatorio de la Sagrada Familia "Expiatory Temple of the Holy Family" often simply The idea that Gaudi used thread models to solve the nave of the Sagrada Familia is a common misconception, although it could have been used in the solution of the bell towers.

The Gateway Arch (looking East).
The Gateway Arch (looking East).

The Gateway Arch in Saint Louis, Missouri, United States follows the form of an inverted catenary. The Jefferson National Expansion Memorial is located in St Louis Missouri near Missouri ( or) is a state in the Midwestern region of the United States bordered by Iowa, Illinois, Kentucky, Tennessee The United States of America —commonly referred to as the It is 630 feet wide at the base and 630 feet tall. The exact formula

y = -127.7 \; \textrm{ft} \cdot \cosh({x / 127.7 \; \textrm{ft}}) + 757.7 \; \textrm{ft}

is displayed inside the arch.

In structural engineering a catenary shell is a structural form, usually made of concrete, that follows a catenary curve. Structural engineering is a field of Engineering dealing with the analysis and design of Structures that support or resist loads Structural engineering is Concrete is a construction material composed of Cement (commonly Portland cement) as well as other cementitious materials such as Fly ash and Slag The profile for the shell is obtained by using flexible material subjected to gravity, converting it into a rigid formwork for pouring the concrete and then using it as required, usually in an inverted manner. Gravitation is a natural Phenomenon by which objects with Mass attract one another Formwork is the term given to either temporary or permanent moulds into which Concrete or similar materials are poured

Catenary arch kiln under construction over temporary form
Catenary arch kiln under construction over temporary form

A kiln, a kind of oven for firing pottery, may be made from firebricks with a body in the shape of a catenary arch, usually nearly as wide as it is high, with the ends closed off with a permanent wall in the back and a temporary wall in the front. Kilns are thermally insulated chambers or Ovens in which controlled temperature regimes are produced Pottery is the Ceramic ware made by potters It also refers to a group of materials that includes Earthenware, Stoneware A fire brick, firebrick, or refractory brick is a block of refractory Ceramic material used in lining Furnaces Kilns The bricks (mortared with fireclay) are stacked upon a temporary form in the shape of an inverted catenary, which is removed upon completion. Fire clay is a specific kind of Clay used in the manufacture of Ceramics especially Fire brick. The form is designed with a simple length of light chain, whose shape is traced onto an end panel of the form, which is inverted for assembly. A particular advantage of this shape is that it does not tend to dismantle itself over repeated heating and cooling cycles — most other forms such as the vertical cylinder must be held together with steel bands. A cylinder is one of the most basic curvilinear geometric shapes the Surface formed by the points at a fixed distance from a given Straight line, the axis

Anchoring of marine vessels

The catenary form given by gravity is made advantage of in its presence in heavy anchor rodes which usually consist mostly of chain or cable as used by ships, oilrigs, docks, and other marine assets which must be anchored to the seabed. An anchor is an object often made out of metal that is used to attach a ship to the bottom of a body of water at a specific point

Particularly with larger vessels, the catenary curve given by the weight of the rode presents a lower angle of pull on the anchor or mooring device. This assists the performance of the anchor and raises the level of force it will resist before dragging. With smaller vessels it is less effective[8].

The catenary curve in this context is only fully present in the anchoring system when the rode has been lifted clear of the seabed by the vessel's pull, as the seabed obviously affects its shape while it supports the chain or cable. There is also typically a section of rode above the water and thus unaffected by buoyancy, creating a slightly more complicated curve.

Towed cables

A truss arch bridge designed by Gustav Eiffel employing an inverted catenary arch
A truss arch bridge designed by Gustav Eiffel employing an inverted catenary arch

When a cable is subject to wind or water flows, the drag forces lead to more general shapes, since the forces are not distributed in the same way as the weight. A truss arch Bridge combines the elements of the Truss bridge and the Arch bridge. Alexandre Gustave Eiffel ( December 15, 1832 &ndash December 27, 1923; in French efɛl in English usually ˈaɪfəl was a French A cable having radius a and specific gravity σ, and towed at speed v in a medium (e. g. , air or water) with density ρ0, will have an (x,y) position described by the following equations (Dowling 1988):

\frac{{dT}}{{ds}}=-\rho _{0}\left( {\sigma -1}\right) \pi a^{2}g\sin \phi-\rho _{0}v^{2}\pi aC_{T}\cos \phi ;
T\frac{{d\phi }}{{ds}}=-\rho _{0}\left( {\sigma -1}\right) \pi a^{2}g\cos\phi +\rho _{0}av^{2}\left[ {C_{D}\sin \phi +\pi C_{N}}\right] \sin \phi ;
\frac{{dx}}{{ds}}=\cos \phi ;
\frac{{dy}}{{ds}}=-\sin \phi .

Here T is the tension, φ is the incident angle, g = 9. 81m / s2, and s is the cable scope. There are three drag coefficients: the normal drag coefficient CD (\approx 1.5 for a smooth cylindrical cable); the tangential drag coefficient CT (\approx 0.0025), and CN ( = 0. 75CT).

The system of equations has four equations and four unknowns: T, φ, x and y, and is typically solved numerically.

Critical angle tow

Critical angle tow occurs when the incident angle does not change. In practice, critical angle tow is common, and occurs far from significant point forces.

Setting \frac{{d\phi }}{{ds}}=0 leads to an equation for the critical angle:

\rho _{0}\left( {\sigma -1}\right) \pi a^{2}g\cos \phi =\rho _{0}av^{2}\left[{C_{D}\sin \phi +\pi C_{N}}\right] \sin \phi .

If πCN < < CDsinφ, the formula for the critical angle becomes

\rho _{0}\left( {\sigma -1}\right) \pi a^{2}g\cos \phi =\rho _{0}av^{2}{C_{D}\sin }^{2}{\phi ;}

or

\left( {\sigma -1}\right) \pi ag\cos \phi =v^{2}{C_{D}\sin }^{2}{\phi =}v^{2}{C_{D}}\left( 1-\cos ^{2}\phi \right) ;

or

\cos ^{2}\phi +\frac{\left( {\sigma -1}\right) \pi ag}{v^{2}{C_{D}}}\cos\phi -1=0;

leading to the rule-of-thumb formula

\cos \phi =-\frac{\left( {\sigma -1}\right) \pi ag}{2v^{2}{C_{D}}}+\sqrt{1+\frac{\left( {\sigma -1}\right) ^{2}\pi ^{2}a^{2}g^{2}}{4v^{4}{C_{D}^{2}}}}.

The drag coefficients of a faired cable are more complicated, involving loading functions that account for drag variation as a function of incidence angle.

Other uses of the term

References

A. P. Dowling, The dynamics of towed flexible cylinders. Part 2. Negatively buoyant elements (1988). Journal of Fluid Mechanics, 187, 533-571.

  1. ^ Swetz, Faauvel, Bekken, "Learn from the Masters", 1997, MAA ISBN 0883857030, pp. 128-9
  2. ^ more word origins 8
  3. ^ http://links.jstor.org/sici?sici=0035-9149(200105)55%3A2%3C289%3AMAMSTO%3E2.0.CO%3B2-X
  4. ^ cf. the anagram for Hooke's law, which appeared in the next paragraph. cf is an abbreviation for the Latin -derived (but also modern English) word confer, meaning "compare" or "consult" In Mechanics, and Physics, Hooke's law of elasticity is an approximation that states that the amount by which a material body is deformed (the
  5. ^ Arch Design
  6. ^ The original anagram was "abcccddeeeeeefggiiiiiiiiillmmmmnnnnnooprrsssttttttuuuuuuuux": the letters of the Latin phrase, alphabetized.
  7. ^ "Roulette: A Comfortable Ride on an n-gon Bicycle" by Borut Levart, The Wolfram Demonstrations Project, 2007.
  8. ^ Chain, Rope, and Catenary - Anchor Systems For Small Boats

External links

Dictionary

catenary

-noun

  1. (geometry) The curve described by a flexible chain or a rope if it is supported at each end and is acted upon only by no other forces than a uniform gravitational force due to its own weight.
  2. (nautical) The curve of an anchor cable from the seabed to the vessel; it should be horizontal at the anchor so as to bury the flukes.
  3. A system of power lines hanging above rail tracks that provide trains, trolleys, etc with electricity.
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