Английская Википедия:Catenary
Шаблон:About Шаблон:Redirect Шаблон:Short description
In physics and geometry, a catenary (Шаблон:IPAc-en Шаблон:Respell, Шаблон:IPAc-en Шаблон:Respell) is the curve that an idealized hanging chain or cable assumes under its own weight 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, which it is not.
The curve appears in the design of certain types of arches 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,[1] or, particularly in the materials sciences, an example of a funicular.[2] Rope statics describes catenaries in a classic statics problem involving a hanging rope.[3]
Mathematically, the catenary curve is the graph of the hyperbolic cosine function. The surface of revolution of the catenary curve, the catenoid, is a minimal surface, specifically a minimal surface of revolution. A hanging chain will assume a shape of least potential energy which is a catenary.[4] Galileo Galilei in 1638 discussed the catenary in the book Two New Sciences recognizing that it was different from a parabola. The mathematical properties of the catenary curve were studied by Robert Hooke in the 1670s, and its equation was derived by Leibniz, Huygens and Johann Bernoulli in 1691.
Catenaries and related curves are used in architecture and engineering (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 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 two evanescent waves would form a catenary shape.[6][7][8]
History
The word "catenary" is derived from the Latin word catēna, which means "chain". The English word "catenary" is usually attributed to Thomas Jefferson,[9][10] who wrote in a letter to Thomas Paine on the construction of an arch for a bridge:
It is often said[11] that Galileo 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°.[12] The fact that the curve followed by a chain is not a parabola was proven by Joachim Jungius (1587–1657); this result was published posthumously in 1669.[11]
The application of the catenary to the construction of arches is attributed to Robert Hooke, whose "true mathematical and mechanical form" in the context of the rebuilding of St Paul's Cathedral alluded to a catenary.[13] Some much older arches approximate catenaries, an example of which is the Arch of Taq-i Kisra in Ctesiphon.[14]
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[15] in an appendix to his Description of Helioscopes,[16] 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[17] in his lifetime, but in 1705 his executor provided it as ut pendet continuum flexile, sic stabit contiguum rigidum inversum, meaning "As hangs a flexible cable so, inverted, stand the touching pieces of an arch."
In 1691, Gottfried Leibniz, Christiaan Huygens, and Johann Bernoulli derived the equation in response to a challenge by Jakob Bernoulli;[11] their solutions were published in the Acta Eruditorum for June 1691.[18][19] David Gregory wrote a treatise on the catenary in 1697[11][20] in which he provided an incorrect derivation of the correct differential equation.[19]
Euler proved in 1744 that the catenary is the curve which, when rotated about the Шаблон:Mvar-axis, gives the surface of minimum surface area (the catenoid) for the given bounding circles.[1] Nicolas Fuss gave equations describing the equilibrium of a chain under any force in 1796.[21]
Inverted catenary arch
Catenary arches are often used in the construction of kilns. To create the desired curve, the shape of a hanging chain of the desired dimensions is transferred to a form which is then used as a guide for the placement of bricks or other building material.[22][23]
The Gateway Arch in St. Louis, Missouri, United States is sometimes said to be an (inverted) catenary, but this is incorrect.[24] It is close to a more general curve called a flattened catenary, with equation Шаблон:Math, which is a catenary if Шаблон:Math. 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.[25][26]
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The Sheffield Winter Garden is enclosed by a series of catenary arches.[28]
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The Gateway Arch (St. Louis, Missouri) is a flattened catenary.
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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.[29] The same is true of a simple suspension bridge or "catenary bridge," where the roadway follows the cable.[30][31]
A stressed ribbon bridge is a more sophisticated structure with the same catenary shape.[32][33]
However, in a suspension bridge with a suspended roadway, the chains or cables support the weight of the bridge, and so do not hang freely. In most cases the roadway is flat, so when the weight of the cable is negligible compared with the weight being supported, the force exerted is uniform with respect to horizontal distance, and the result is a parabola, as discussed below (although the term "catenary" is often still used, in an informal sense). If the cable is heavy then the resulting curve is between a catenary and a parabola.[34][35]
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 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.[36]
Cable ferries and chain 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.[37]
Mathematical description
Equation
The equation of a catenary in Cartesian coordinates has the form[34]
<math display=block>y = a \cosh \left(\frac{x}{a}\right) = \frac{a}{2}\left(e^\frac{x}{a} + e^{-\frac{x}{a}}\right),</math> where Шаблон:Math is the hyperbolic cosine function, and where Шаблон:Mvar is the distance of the lowest point above the x axis.[38] All catenary curves are similar to each other, since changing the parameter Шаблон:Mvar is equivalent to a uniform scaling of the curve.
The Whewell equation for the catenary is[34] <math display=block>\tan \varphi = \frac{s}{a},</math> where <math>\varphi</math> is the tangential angle and Шаблон:Mvar the arc length.
Differentiating gives <math display=block>\frac{d\varphi}{ds} = \frac{\cos^2\varphi}{a},</math> and eliminating <math>\varphi</math> gives the Cesàro equation[39] <math display=block>\kappa=\frac{a}{s^2+a^2},</math> where <math>\kappa</math> is the curvature.
The radius of curvature is then <math display=block>\rho = a \sec^2 \varphi,</math> which is the length of the normal between the curve and the Шаблон:Mvar-axis.[40]
Relation to other curves
When a parabola is rolled along a straight line, the roulette curve traced by its focus is a catenary.[41] The envelope of the directrix of the parabola is also a catenary.[42] The involute 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.[41]
Another roulette, formed by rolling a line on a catenary, is another line. This implies that square wheels can roll perfectly smoothly on a road made of a series of bumps in the shape of an inverted catenary curve. The wheels can be any regular polygon except a triangle, but the catenary must have parameters corresponding to the shape and dimensions of the wheels.[43]
Geometrical properties
Over any horizontal interval, the ratio of the area under the catenary to its length equals Шаблон:Mvar, 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 Шаблон:Mvar-axis.[44]
Science
A moving charge in a uniform electric field travels along a catenary (which tends to a parabola if the charge velocity is much less than the speed of light Шаблон:Mvar).[45]
The surface of revolution with fixed radii at either end that has minimum surface area is a catenary revolved about the Шаблон:Mvar-axis.[41]
Analysis
Model of chains and arches
In the mathematical model the chain (or cord, cable, rope, string, etc.) is idealized by assuming that it is so thin that it can be regarded as a curve and that it is so flexible any force of tension exerted by the chain is parallel to the chain.[46] 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.[47] An underlying principle is that the chain may be considered a rigid body once it has attained equilibrium.[48] Equations which define the shape of the curve and the tension of the chain at each point may be derived by a careful inspection of the various forces acting on a segment using the fact that these forces must be in balance if the chain is in static equilibrium.
Let the path followed by the chain be given parametrically by Шаблон:Math where Шаблон:Mvar represents arc length and Шаблон:Math is the position vector. This is the natural parameterization and has the property that
<math display=block>\frac{d\mathbf{r}}{ds}=\mathbf{u}</math>
where Шаблон:Math is a unit tangent vector.
A differential equation for the curve may be derived as follows.[49] Let Шаблон:Math be the lowest point on the chain, called the vertex of the catenary.[50] The slope Шаблон:Math of the curve is zero at Шаблон:Math since it is a minimum point. Assume Шаблон:Math is to the right of Шаблон:Math since the other case is implied by symmetry. The forces acting on the section of the chain from Шаблон:Math to Шаблон:Math are the tension of the chain at Шаблон:Math, the tension of the chain at Шаблон:Math, and the weight of the chain. The tension at Шаблон:Math is tangent to the curve at Шаблон:Math and is therefore horizontal without any vertical component and it pulls the section to the left so it may be written Шаблон:Math where Шаблон:Math is the magnitude of the force. The tension at Шаблон:Math is parallel to the curve at Шаблон:Math and pulls the section to the right. The tension at Шаблон:Math can be split into two components so it may be written Шаблон:Math, where Шаблон:Mvar is the magnitude of the force and Шаблон:Mvar is the angle between the curve at Шаблон:Math and the Шаблон:Mvar-axis (see tangential angle). Finally, the weight of the chain is represented by Шаблон:Math where Шаблон:Mvar is the mass per unit length, Шаблон:Mvar is the gravitational field strength and Шаблон:Mvar is the length of the segment of chain between Шаблон:Math and Шаблон:Math.
The chain is in equilibrium so the sum of three forces is Шаблон:Math, therefore
<math display=block>T \cos \varphi = T_0</math> and <math display=block>T \sin \varphi = \lambda gs\,,</math>
and dividing these gives
<math display=block>\frac{dy}{dx}=\tan \varphi = \frac{\lambda gs}{T_0}\,.</math>
It is convenient to write
<math display=block>a = \frac{T_0}{\lambda g}</math>
which is the length of chain whose weight is equal in magnitude to the tension at Шаблон:Math.[51] Then
<math display=block>\frac{dy}{dx}=\frac{s}{a}</math>
is an equation defining the curve.
The horizontal component of the tension, Шаблон:Math is constant and the vertical component of the tension, Шаблон:Math is proportional to the length of chain between Шаблон:Math and the vertex.[52]
After deriving the equations of the curve (in the next section) <math display="inline">y = a \cosh\left(\frac{x}{a}\right)</math>, one can plug the equation back to obtain the simple equation <math>T = \lambda g s/\sin \varphi = \lambda g y</math>.
Derivation of equations for the curve
The differential equation given above can be solved to produce equations for the curve.[53]
From
<math display=block>\frac{dy}{dx} = \frac{s}{a}\,,</math>
the formula for arc length gives <math display=block>\frac{ds}{dx} = \sqrt{1+\left(\dfrac{dy}{dx}\right)^2} = \frac{\sqrt{a^2+s^2}}{a}\,.</math>
Then
<math display=block>\frac{dx}{ds} = \frac{1}{\frac{ds}{dx}} = \frac{a}{\sqrt{a^2+s^2}}</math>
and
<math display=block>\frac{dy}{ds} = \frac{\frac{dy}{dx}}{\frac{ds}{dx}} = \frac{s}{\sqrt{a^2+s^2}}\,.</math>
The second of these equations can be integrated to give
<math display=block>y = \sqrt{a^2+s^2} + \beta</math>
and by shifting the position of the Шаблон:Mvar-axis, Шаблон:Mvar can be taken to be 0. Then
<math display=block>y = \sqrt{a^2+s^2}\,,\quad y^2=a^2+s^2\,.</math>
The Шаблон:Mvar-axis thus chosen is called the directrix of the catenary.
It follows that the magnitude of the tension at a point Шаблон:Math is Шаблон:Math, which is proportional to the distance between the point and the directrix.[52]
This tension may also be expressed as Шаблон:Math .
The integral of the expression for Шаблон:Mvar can be found using standard techniques, giving[54]
<math display=block>x = a\operatorname{arsinh}\left(\frac{s}{a}\right) + \alpha\,.</math>
and, again, by shifting the position of the Шаблон:Mvar-axis, Шаблон:Mvar can be taken to be 0. Then
<math display=block>x = a\operatorname{arsinh}\left(\frac{s}{a}\right)\,,\quad s=a \sinh\left(\frac{x}{a}\right)\,.</math>
The Шаблон:Mvar-axis thus chosen passes through the vertex and is called the axis of the catenary.
These results can be used to eliminate Шаблон:Mvar giving
<math display=block>y = a \cosh\left(\frac{x}{a}\right)\,.</math>
Alternative derivation
The differential equation can be solved using a different approach.[55] From
<math display=block>s = a \tan \varphi</math>
it follows that
<math display=block>\frac{dx}{d\varphi} = \frac{dx}{ds}\frac{ds}{d\varphi}=\cos \varphi \cdot a \sec^2 \varphi= a \sec \varphi</math> and <math display=block>\frac{dy}{d\varphi} = \frac{dy}{ds}\frac{ds}{d\varphi}=\sin \varphi \cdot a \sec^2 \varphi= a \tan \varphi \sec \varphi\,.</math>
Integrating gives,
<math display=block>x = a \ln(\sec \varphi + \tan \varphi) + \alpha</math> and <math display=block>y = a \sec \varphi + \beta\,.</math>
As before, the Шаблон:Mvar and Шаблон:Mvar-axes can be shifted so Шаблон:Mvar and Шаблон:Mvar can be taken to be 0. Then
<math display=block>\sec \varphi + \tan \varphi = e^\frac{x}{a}\,,</math> and taking the reciprocal of both sides <math display=block>\sec \varphi - \tan \varphi = e^{-\frac{x}{a}}\,.</math>
Adding and subtracting the last two equations then gives the solution <math display=block>y = a \sec \varphi = a \cosh\left(\frac{x}{a}\right)\,,</math> and <math display=block>s = a \tan \varphi = a \sinh\left(\frac{x}{a}\right)\,.</math>
Determining parameters
In general the parameter Шаблон:Mvar is the position of the axis. The equation can be determined in this case as follows:[56]
Relabel if necessary so that Шаблон:Math is to the left of Шаблон:Math and let Шаблон:Mvar be the horizontal and Шаблон:Mvar be the vertical distance from Шаблон:Math to Шаблон:Math. Translate the axes so that the vertex of the catenary lies on the Шаблон:Mvar-axis and its height Шаблон:Mvar is adjusted so the catenary satisfies the standard equation of the curve
<math display=block>y = a \cosh\left(\frac{x}{a}\right)</math>
and let the coordinates of Шаблон:Math and Шаблон:Math be Шаблон:Math and Шаблон:Math respectively. The curve passes through these points, so the difference of height is
<math display=block>v = a \cosh\left(\frac{x_2}{a}\right) - a \cosh\left(\frac{x_1}{a}\right)\,.</math>
and the length of the curve from Шаблон:Math to Шаблон:Math is
<math display="block">L = a \sinh\left(\frac{x_2}{a}\right) - a \sinh\left(\frac{x_1}{a}\right)\,.</math>
When Шаблон:Math is expanded using these expressions the result is
<math display="block">L^2-v^2=2a^2\left(\cosh\left(\frac{x_2-x_1}{a}\right)-1\right)=4a^2\sinh^2\left(\frac{H}{2a}\right)\,,</math> so <math display="block">\frac 1H \sqrt{L^2-v^2}=\frac{2a}H \sinh\left(\frac{H}{2a}\right)\,.</math>
This is a transcendental equation in Шаблон:Mvar and must be solved numerically. Since <math>\sinh(x)/x</math> is strictly monotonic on <math>x > 0</math>,[57] there is at most one solution with Шаблон:Math and so there is at most one position of equilibrium.
However, if both ends of the curve (Шаблон:Math and Шаблон:Math) are at the same level (Шаблон:Math), it can be shown that[58] <math display=block>a = \frac {\frac14 L^2-h^2} {2h}\,</math> where L is the total length of the curve between Шаблон:Math and Шаблон:Math and Шаблон:Mvar is the sag (vertical distance between Шаблон:Math, Шаблон:Math and the vertex of the curve).
It can also be shown that <math display=block>L = 2a \sinh \frac {H} {2a}\,</math> and <math display=block>H = 2a \operatorname {arcosh} \frac {h+a} {a}\,</math> where H is the horizontal distance between Шаблон:Math and Шаблон:Math which are located at the same level (Шаблон:Math).
The horizontal traction force at Шаблон:Math and Шаблон:Math is Шаблон:Math, where Шаблон:Mvar is the mass per unit length of the chain or cable.
Variational formulation
Consider a chain of length <math>L</math> suspended from two points of equal height and at distance <math>D</math>. The curve has to minimize its potential energy <math display=block> U = \int_0^D g\rho y\sqrt{1+y'^2} dx </math> and is subject to the constraint <math display=block> \int_0^D \sqrt{1+y'^2} dx = L\,.</math>
The modified Lagrangian is therefore <math display=block> \mathcal{L} = (g\rho y - \lambda )\sqrt{1+y'^2}</math> where <math>\lambda </math> is the Lagrange multiplier to be determined. As the independent variable <math>x</math> does not appear in the Lagrangian, we can use the Beltrami identity <math display=block> \mathcal{L}-y' \frac{\partial \mathcal{L} }{\partial y'} = C </math> where <math>C</math> is an integration constant, in order to obtain a first integral <math display=block>\frac{(g\rho y - \lambda )}{\sqrt{1+y'^2}} = -C</math>
This is an ordinary first order differential equation that can be solved by the method of separation of variables. 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.[59]
Let Шаблон:Mvar denote the weight per unit length of the chain, then the weight of the chain has magnitude
<math display=block>\int_\mathbf{c}^\mathbf{r} w\, ds\,,</math>
where the limits of integration are Шаблон:Math and Шаблон:Math. Balancing forces as in the uniform chain produces
<math display=block>T \cos \varphi = T_0</math> and <math display=block>T \sin \varphi = \int_\mathbf{c}^\mathbf{r} w\, ds\,,</math> and therefore <math display=block>\frac{dy}{dx}=\tan \varphi = \frac{1}{T_0} \int_\mathbf{c}^\mathbf{r} w\, ds\,.</math>
Differentiation then gives
<math display=block>w=T_0 \frac{d}{ds}\frac{dy}{dx} = \frac{T_0 \dfrac{d^2y}{dx^2}}{\sqrt{1+\left(\dfrac{dy}{dx}\right)^2}}\,.</math>
In terms of Шаблон:Mvar and the radius of curvature Шаблон:Mvar this becomes
<math display=block>w= \frac{T_0}{\rho \cos^2 \varphi}\,.</math>
Suspension bridge curve
A similar analysis can be done to find the curve followed by the cable supporting a suspension bridge with a horizontal roadway.[60] If the weight of the roadway per unit length is Шаблон:Mvar 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 Шаблон:Math to Шаблон:Math is Шаблон:Mvar where Шаблон:Mvar is the horizontal distance between Шаблон:Math and Шаблон:Math. Proceeding as before gives the differential equation
<math display=block>\frac{dy}{dx}=\tan \varphi = \frac{w}{T_0}x\,. </math>
This is solved by simple integration to get
<math display=block>y=\frac{w}{2T_0}x^2 + \beta</math>
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.[61]
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, Шаблон:Mvar, per unit length of the chain can be written Шаблон:Mvar, where Шаблон:Mvar is constant, and the analysis for nonuniform chains can be applied.[62]
In this case the equations for tension are
<math display=block>\begin{align} T \cos \varphi &= T_0\,,\\ T \sin \varphi &= \frac{1}{c}\int T\, ds\,. \end{align}</math>
Combining gives
<math display=block>c \tan \varphi = \int \sec \varphi\, ds</math>
and by differentiation
<math display=block>c = \rho \cos \varphi</math>
where Шаблон:Mvar is the radius of curvature.
The solution to this is
<math display=block>y = c \ln\left(\sec\left(\frac{x}{c}\right)\right)\,.</math>
In this case, the curve has vertical asymptotes and this limits the span to Шаблон:Math. Other relations are
<math display=block>x = c\varphi\,,\quad s = \ln\left(\tan\left(\frac{\pi+2\varphi}{4}\right)\right)\,.</math>
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".[63]
Elastic catenary
In an elastic catenary, the chain is replaced by a spring which can stretch in response to tension. The spring is assumed to stretch in accordance with Hooke's Law. Specifically, if Шаблон:Math is the natural length of a section of spring, then the length of the spring with tension Шаблон:Mvar applied has length
<math display=block>s=\left(1+\frac{T}{E}\right)p\,,</math>
where Шаблон:Mvar is a constant equal to Шаблон:Mvar, where Шаблон:Mvar is the stiffness of the spring.[64] In the catenary the value of Шаблон:Mvar is variable, but ratio remains valid at a local level, so[65] <math display=block>\frac{ds}{dp}=1+\frac{T}{E}\,.</math> The curve followed by an elastic spring can now be derived following a similar method as for the inelastic spring.[66]
The equations for tension of the spring are
<math display=block>T \cos \varphi = T_0\,,</math> and <math display=block>T \sin \varphi = \lambda_0 gp\,,</math>
from which
<math display=block>\frac{dy}{dx}=\tan \varphi = \frac{\lambda_0 gp}{T_0}\,,\quad T=\sqrt{T_0^2+\lambda_0^2 g^2p^2}\,,</math>
where Шаблон:Mvar is the natural length of the segment from Шаблон:Math to Шаблон:Math and Шаблон:Math is the mass per unit length of the spring with no tension and Шаблон:Mvar is the gravitational field strength. Write
<math display=block>a = \frac{T_0}{\lambda_0 g}</math>
so
<math display=block>\frac{dy}{dx}=\tan \varphi = \frac{p}{a} \quad\text{and}\quad T=\frac{T_0}{a}\sqrt{a^2+p^2}\,.</math>
Then <math display=block>\begin{align} \frac{dx}{ds} &= \cos \varphi = \frac{T_0}{T} \\[6pt] \frac{dy}{ds} &= \sin \varphi = \frac{\lambda_0 gp}{T}\,, \end{align}</math> from which <math display=block>\begin{alignat}{3} \frac{dx}{dp} &= \frac{T_0}{T}\frac{ds}{dp} &&= T_0\left(\frac{1}{T}+\frac{1}{E}\right) &&= \frac{a}{\sqrt{a^2+p^2}}+\frac{T_0}{E} \\[6pt] \frac{dy}{dp} &= \frac{\lambda_0 gp}{T}\frac{ds}{dp} &&= \frac{T_0p}{a}\left(\frac{1}{T}+\frac{1}{E}\right) &&= \frac{p}{\sqrt{a^2+p^2}}+\frac{T_0p}{Ea}\,. \end{alignat}</math>
Integrating gives the parametric equations
<math display=block>\begin{align} x&=a\operatorname{arsinh}\left(\frac{p}{a}\right)+\frac{T_0}{E}p + \alpha\,, \\[6pt] y&=\sqrt{a^2+p^2}+\frac{T_0}{2Ea}p^2+\beta\,. \end{align}</math>
Again, the Шаблон:Mvar and Шаблон:Mvar-axes can be shifted so Шаблон:Mvar and Шаблон:Mvar can be taken to be 0. So
<math display=block>\begin{align} x&=a\operatorname{arsinh}\left(\frac{p}{a}\right)+\frac{T_0}{E}p\,, \\[6pt] y&=\sqrt{a^2+p^2}+\frac{T_0}{2Ea}p^2 \end{align}</math>
are parametric equations for the curve. At the rigid limit where Шаблон:Mvar 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 Шаблон:Math acting on the chain, the following analysis can be made.[67]
First, let Шаблон:Math be the force of tension as a function of Шаблон:Mvar. 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, Шаблон:Math must be parallel to the chain. In other words,
<math display=block>\mathbf{T} = T \mathbf{u}\,,</math>
where Шаблон:Mvar is the magnitude of Шаблон:Math and Шаблон:Math is the unit tangent vector.
Second, let Шаблон:Math be the external force per unit length acting on a small segment of a chain as a function of Шаблон:Mvar. The forces acting on the segment of the chain between Шаблон:Mvar and Шаблон:Math are the force of tension Шаблон:Math at one end of the segment, the nearly opposite force Шаблон:Math at the other end, and the external force acting on the segment which is approximately Шаблон:Math. These forces must balance so
<math display=block>\mathbf{T}(s+\Delta s)-\mathbf{T}(s)+\mathbf{G}\Delta s \approx \mathbf{0}\,.</math>
Divide by Шаблон:Math and take the limit as Шаблон:Math to obtain
<math display=block>\frac{d\mathbf{T}}{ds} + \mathbf{G} = \mathbf{0}\,.</math>
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, Шаблон:Math where the chain has mass Шаблон:Mvar per unit length and Шаблон:Mvar is the gravitational field strength.
See also
- Catenary arch
- Chain fountain or self-siphoning beads
- Overhead catenary – power lines suspended over rail or tram vehicles
- Roulette (curve) – an elliptic/hyperbolic catenary
- Troposkein – the shape of a spun rope
- Weighted catenary
Notes
Bibliography
- Шаблон:AnchorШаблон:Cite book
- Шаблон:Cite book
- Шаблон:AnchorШаблон:Cite book
- Шаблон:Cite book
- Шаблон:Cite book
- Шаблон:Cite book
- Шаблон:AnchorШаблон:Cite book
- Шаблон:AnchorШаблон:Mathworld
Further reading
External links
Шаблон:Commons category Шаблон:Wikiquote Шаблон:Wikisource1911Enc
- Шаблон:MacTutor
- Шаблон:PlanetMath
- Catenary curve calculator
- Catenary at The Geometry Center
- "Catenary" at Visual Dictionary of Special Plane Curves
- The Catenary - Chains, Arches, and Soap Films.
- Cable Sag Error Calculator – 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
- ↑ 1,0 1,1 MathWorld
- ↑ e.g.: Шаблон:Cite book
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite book
- ↑ 11,0 11,1 11,2 11,3 Lockwood p. 124
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite book
- ↑ cf. the anagram for Hooke's law, which appeared in the next paragraph.
- ↑ Шаблон:Cite web
- ↑ The original anagram was abcccddeeeeefggiiiiiiiillmmmmnnnnnooprrsssttttttuuuuuuuux: the letters of the Latin phrase, alphabetized.
- ↑ Шаблон:Citation
- ↑ 19,0 19,1 Шаблон:Citation
- ↑ Шаблон:Citation
- ↑ Routh Art. 455, footnote
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite book
- ↑ Шаблон:Citation
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Citation and Шаблон:NHLS url Шаблон:Small
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite journal
- ↑ 34,0 34,1 34,2 Lockwood p. 122
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite web
- ↑ MathWorld, eq. 7
- ↑ Routh Art. 444
- ↑ 41,0 41,1 41,2 Шаблон:Cite book
- ↑ Yates p. 80
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite book
- ↑ Routh Art. 442, p. 316
- ↑ Шаблон:Cite book
- ↑ Whewell p. 65
- ↑ Following Routh Art. 443 p. 316
- ↑ Routh Art. 443 p. 317
- ↑ Whewell p. 67
- ↑ 52,0 52,1 Routh Art. 443 p. 318
- ↑ Following Routh Art. 443 p/ 317
- ↑ Use of hyperbolic functions follows Maurer p. 107
- ↑ Following Lamb p. 342
- ↑ Following Todhunter Art. 186
- ↑ See Routh art. 447
- ↑ Archived at GhostarchiveШаблон:Cbignore and the Wayback MachineШаблон:Cbignore: Шаблон:Cite webШаблон:Cbignore
- ↑ Following Routh Art. 450
- ↑ Following Routh Art. 452
- ↑ Ira Freeman investigated the case where only the cable and roadway are significant, see the External links section. Routh gives the case where only the supporting wires have significant weight as an exercise.
- ↑ Following Routh Art. 453
- ↑ Шаблон:Cite journal
- ↑ Routh Art. 489
- ↑ Routh Art. 494
- ↑ Following Routh Art. 500
- ↑ Follows Routh Art. 455
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