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单词 Catenary
释义

catenary


A wire takes a form resembling an arc of a parabolaMathworldPlanetmathPlanetmath when suspended at its ends.  The arc is not from a parabola but from the graph of the hyperbolic cosineMathworldPlanetmath (http://planetmath.org/HyperbolicFunctions) functionMathworldPlanetmath in a suitable coordinate systemMathworldPlanetmath.

Let’s derive the equation  y=y(x)  of this curve, called the catenaryMathworldPlanetmath, in its plane with x-axis horizontal and y-axis vertical.  We denote the of the wire by σ.

In any point  (x,y)  of the wire, the tangent lineMathworldPlanetmath of the curve forms an angle φ with the positive direction of x-axis.  Then,

tanφ=dydx=y.

In the point, a certain tension T of the wire acts in the direction of the value a.  Hence we may write

T=acosφ,

whence the vertical of T is

Tsinφ=atanφ

and its differentialMathworldPlanetmath (http://planetmath.org/Differential)

d(Tsinφ)=adtanφ=ady.

But this differential is the amount of the supporting   σ1+(y(x))2dx (see the arc lengthMathworldPlanetmath).  Thus we obtain the differential equationMathworldPlanetmath

σ1+y2dx=ady,(1)

which allows the separation of variablesMathworldPlanetmath:

𝑑x=aσdy1+y2

This may be solved by using the substitution (http://planetmath.org/SubstitutionForIntegration)

y:=sinht,dy=coshtdt,1+y2=cosht

giving

x=aσt+x0,

i.e.

y=dydx=sinhσ(x-x0)a.

This leads to the final solution

y=aσcoshσ(x-x0)a+y0

of the equation (1).  We have denoted the constants of integration by x0 and y0.  They determine the position of the catenary in regard to the coordinate axes.  By a suitable choice of the axes and the equation

y=acoshxa(2)

of the catenary.

Some of catenary

  • tanφ=sinhxa,sinφ=tanhxa  (cf. the GudermannianMathworldPlanetmath)

  • The arc length of the catenary (2) from the apex  (0,a)  to the point  (x,y)  is   asinhxa=y2-a2.

  • The radius of curvatureMathworldPlanetmath of the catenary (2) is  acosh2xa, which is the same as length of the normal lineMathworldPlanetmath of the catenary between the curve and the x-axis.

  • The catenary is the catacausticMathworldPlanetmath of the exponentialMathworldPlanetmathPlanetmath curve (http://planetmath.org/ExponentialFunction) reflecting the vertical rays.

  • If a parabola rolls on a straight line, the focus draws a catenary.

  • The involute (a.k.a. the evolvent) of the catenary is the tractrixMathworldPlanetmath.

Titlecatenary
Canonical nameCatenary
Date of creation2014-10-26 21:25:30
Last modified on2014-10-26 21:25:30
Ownerpahio (2872)
Last modified bypahio (2872)
Numerical id29
Authorpahio (2872)
Entry typeDerivation
Classificationmsc 53B25
Classificationmsc 51N05
Synonymchain curve
Related topicEquationOfCatenaryViaCalculusOfVariations
Related topicLeastSurfaceOfRevolution
Related topicHyperbolicFunctions
Related topicTractrix
Related topicEqualArcLengthAndArea
Definescatenary
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更新时间:2025/5/4 20:51:45